Bi lkent Calculus II Exams - Bilkent Universityotekman/all/allspr.pdf · Bilkent Calculus II Exams 1988-2019 3 ¸ Express V in terms of iterated integrals in Cartesian coordinates

Bi.lkent Calculus II Exams

1988-2019

Spring 2019 Final version

Spring 2019 Midterm I . . . . . . . . 1Spring 2019 Midterm II . . . . . . . 2Spring 2019 Final . . . . . . . . . . . 5Spring 2018 Midterm I . . . . . . . . 7Spring 2018 Midterm II . . . . . . . 9Spring 2018 Final . . . . . . . . . . . 10Spring 2017 Midterm I . . . . . . . . 12Spring 2017 Midterm II . . . . . . . 13Spring 2017 Final . . . . . . . . . . . 15Spring 2016 Midterm I . . . . . . . . 18Spring 2016 Midterm II . . . . . . . 20Spring 2016 Final . . . . . . . . . . . 21Spring 2015 Midterm I . . . . . . . . 23Spring 2015 Midterm II . . . . . . . 25Spring 2015 Final . . . . . . . . . . . 27Spring 2014 Midterm I . . . . . . . . 30Spring 2014 Midterm II . . . . . . . 33Spring 2014 Final . . . . . . . . . . . 36Spring 2013 Midterm I . . . . . . . . 38Spring 2013 Midterm II . . . . . . . 41Spring 2013 Final . . . . . . . . . . . 43Spring 2012 Midterm I . . . . . . . . 44Spring 2012 Midterm II . . . . . . . 47Spring 2012 Final . . . . . . . . . . . 48Spring 2011 Midterm I . . . . . . . . 51Spring 2011 Midterm II . . . . . . . 52Spring 2011 Final . . . . . . . . . . . 54Spring 2010 Midterm I . . . . . . . . 56Spring 2010 Midterm II . . . . . . . 57Spring 2010 Final . . . . . . . . . . . 58Spring 2009 Midterm I . . . . . . . . 60Spring 2009 Midterm II . . . . . . . 61Spring 2009 Final . . . . . . . . . . . 62Spring 2008 Midterm I . . . . . . . . 63Spring 2008 Midterm II . . . . . . . 64Spring 2008 Final . . . . . . . . . . . 65Spring 2007 Midterm I . . . . . . . . 65Spring 2007 Midterm II . . . . . . . 66Spring 2007 Final . . . . . . . . . . . 67Spring 2006 Midterm I . . . . . . . . 68Spring 2006 Midterm II . . . . . . . 69Spring 2006 Final . . . . . . . . . . . 70Spring 2005 Midterm I . . . . . . . . 71Spring 2005 Midterm II . . . . . . . 72

Spring 2005 Final . . . . . . . . . . . 73Spring 2004 Midterm I . . . . . . . . 75Spring 2004 Midterm II . . . . . . . 75Spring 2004 Final . . . . . . . . . . . 76Spring 2003 Midterm I . . . . . . . . 77Spring 2003 Midterm II . . . . . . . 77Spring 2003 Final . . . . . . . . . . . 78Spring 2002 Midterm I . . . . . . . . 78Spring 2002 Midterm II . . . . . . . 79Spring 2002 Final . . . . . . . . . . . 80Spring 2001 Midterm I . . . . . . . . 80Spring 2001 Midterm II . . . . . . . 81Spring 2001 Final . . . . . . . . . . . 82Spring 2000 Midterm I . . . . . . . . 83Spring 2000 Midterm II . . . . . . . 84Spring 2000 Final . . . . . . . . . . . 84Spring 1999 Midterm I . . . . . . . . 85Spring 1999 Midterm II . . . . . . . 86Spring 1999 Final . . . . . . . . . . . 86Spring 1998 Midterm I . . . . . . . . 87Spring 1998 Midterm II . . . . . . . 88Spring 1998 Final . . . . . . . . . . . 89Spring 1997 Midterm I . . . . . . . . 90Spring 1997 Midterm II . . . . . . . 90Spring 1997 Final . . . . . . . . . . . 91Spring 1996 Midterm I . . . . . . . . 92Spring 1996 Midterm II . . . . . . . 93Spring 1996 Final . . . . . . . . . . . 93Spring 1995 Midterm I . . . . . . . . 94Spring 1995 Final . . . . . . . . . . . 95Spring 1994 Midterm I . . . . . . . . 95Spring 1994 Midterm II . . . . . . . 96Spring 1994 Final . . . . . . . . . . . 97Spring 1993 Midterm I . . . . . . . . 98Spring 1993 Midterm II . . . . . . . 99Spring 1993 Final . . . . . . . . . . . 100Spring 1992 Midterm II . . . . . . . 101Spring 1992 Final . . . . . . . . . . . 102Spring 1991 Midterm I . . . . . . . . 103Spring 1991 Midterm II . . . . . . . 103Spring 1991 Final . . . . . . . . . . . 104Spring 1990 Midterm II . . . . . . . 104Spring 1989 Midterm II . . . . . . . 105Spring 1988 Midterm II . . . . . . . 106

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Bilkent Calculus II Exams 1988-2019 1

Spring 2019 Midterm I

1. 1a. In this part, just �ll in the boxes. No explanation is required. No partial points will begiven.

Ê Give an example of a plane that contains the x-axis, but does not contain the y- andz-axes by writing its equation in the box below. [ The box should contain nothing except the equation! ]

Ë Give an example of a line that does not intersect the xy-plane, but intersects each of theyz- and xz-planes at exactly one point by writing its parametric equations in the box below.

[ The box should contain nothing except the parametric equations! ]

1b. The positions of two points P1 and P2 in the space as a function of time t are given by:

r1 =−−→OP1 = (4t− 1) i+ t2 j+ tk and r2 =

−−→OP2 = 3t i+ t j+ t3 k

Find all times t when there is a plane P such that

• The plane P passes through the points P1 and P2 at time t, and

• The velocity vectors v1 and v2 of the points P1 and P2 at time t are parallel to the planeP .

2.The Bateman-Burgers equation

∂u

∂t=∂2u

∂x2+ u

∂u

∂x

arises in the study of nonlinear acoustics and gas dynamics in �uid mechanics, and in the studyof tra�c �ow in civil engineering.

Find all possible values of the pair of constants (a, b) for which the function

u(x, t) =x

ax2 + bt+ 1

satis�es the Bateman-Burgers equation for all (x, t) with ax2 + bt+ 1 = 0 .

2 Bilkent Calculus II Exams 1988-2019

3a.You are given the following information about a di�erentiable function f(x, y) :

À The tangent line to the level curve f(x, y) = f(1,−1) at the point (1,−1)

has the equation 5x− 8y = 13 .

Ád

dtf(t3 − 2t2 + 1, t2 − 4/t− 3)

∣∣∣∣t=2

= 1 .

Choose one of the following:

� The given data is not consistent.

� The given data is consistent, but not su�cient to determine ∇f(1,−1).

� The given data is consistent, and su�cient to determine ∇f(1,−1).

Now prove your claim.

3b.Evaluate the limit lim(x,y)→(0,0)

xy5

(x2 + y4)(x4 + y2).

4.Consider the function f(x, y) = x2y(x2 + y2 − 1) .

a.Find all critical points of f . [ Do not classify them! ]

b.Choose one of the critical points of f that lies on the y-axis by �lling in the box:

(x, y) = (0, )

Determine whether this point is a local maximum, local minimum, or a saddle point withoutusing the 2nd Derivative Test.

Spring 2019 Midterm II

1a.Evaluate the iterated integral

ˆ 1/2

0

ˆ (1−2x)/3

0

sin(πx/(1− 3y)

)dy dx .

1b.Evaluate the double integral

¨D

1

(x2 + y2)2dA where D = {(x, y) : xy ≥ 1 and x > 0} .

2.Let V be the volume of the solid bounded by the cylinder x2+ y2 = 1 on the sides, the planez = y at the top, and the xy-plane at the bottom.

a.Only three of Ê-Í will be graded. Mark the ones you want to be graded by putting a 7

in the corresponding � s.

Ê � Express V in terms of iterated integrals in Cartesian coordinates by �lling in therectangles.

V =

ˆ ˆ ˆdz dy dx


Ë � Express V in terms of iterated integrals in Cartesian coordinates by �lling in therectangles.

V =

ˆ ˆ ˆdx dy dz

Ì � Express V in terms of iterated integrals in cylindrical coordinates by �lling in therectangles.

V =

ˆ ˆ ˆr dz dr dθ

Í � Express V in terms of iterated integrals in spherical coordinates by �lling in therectangles.

V =

ˆ ˆ ˆρ2 sinϕ dρ dϕ dθ

b. Compute V .

3. In each of the following, indicate all possible completions of the sentence that will make itinto a true statement by putting a 7 in the corresponding � s.


a.

{1

n

}∞

n=1

=

{1,

1

2,1

3,1

4, · · ·, 1

n, · · ·

}is

� a convergent sequence � a divergent sequence

� a convergent series � a divergent series � none of these

b.∞∑n=1

1

2n−1= 1 +

1

2+

1

4+

1

8+ · · ·+ 1

2n−1+ · · · is



c.∞∑n=1

1 = 1 + 1 + 1 + 1 + · · ·+ 1 + · · · is



d.{(−1)n−1

}∞n=1

={1,−1, 1,−1, · · ·, (−1)n−1, · · ·

}is



e.∞∑n=1

(−1)n−1 = 1− 1 + 1− 1 + · · ·+ (−1)n−1 + · · · is



f. {1}∞n=1= {1, 1, 1, 1, · · ·, 1, · · · } is



g.∞∑n=1

1

n= 1 +

1

2+

1

3+

1

4+ · · ·+ 1

n+ · · · is



h.

{1

2n−1

}∞

n=1

=

{1,

1

2,1

4,1

8, · · ·, 1

2n−1, · · ·

}is



i. Mark only �none of these" in this part.




4.A sequence {an}∞n=1 satis�es

a1 = 1 , a2 = A , and an =an−1 + an−2

an−1 − an−2

· an−1 for n ≥ 3 ,

where A is a real number such that A = 1, A = 0, A = −1.

a. In the following, �ll in the s with real numbers that will make the sentence into

a true statement.

If A = , then a3 = and a4 = .

b. In each of the following, �ll in the with a real number that will make the

corresponding sentence into a true statement.

� Ê If A = , then limn→∞

an = ∞ .

� Ë If A = , then limn→∞

an = 0 .

� Ì If A = , then limn→∞

an = 0 and limn→∞

|an| = ∞ .

c. Now choose exactly one of the statements you made in Part b by putting a 7 in thecorresponding � , and prove it fully and carefully by using correct mathematical reasoning andnotation.

Spring 2019 Final

1.Consider the function f(x, y, z) =x

y− y

zand the point P0(3, 1, 1) .

a.Compute ∇f(P0) .

b. Is there a unit vector u such that Duf(P0) = 5? If Yes, �nd one. If No, prove that itdoes not exist.

c. Is there a unit vector u such that Duf(P0) = 3? If Yes, �nd one. If No, prove that itdoes not exist.

d. Let S be the set of all points P (x, y, z) where f increases fastest in the direction of thevector A = 2i + j + 2k. Show that S is a subset of the union L1 ∪ L2 of two lines L1 and L2,and �nd parametric equations of these lines.

2.Evaluate the following integrals.

a.

¨D

x dA where D = {(x, y) : 1 ≤ x+ y ≤ 2 , x ≥ 0 and y ≥ 0}

b.

˚E

1√x2 + y2 + z2

dV where E = {(x, y, z) : x2 + y2 + z2 ≤ 4 and z ≥ 1}


3. In each of the following, if the given statement is true for all sequences {an}∞n=1 , then markthe � to the left of True with a 8 ; otherwise, mark the � to the left of False with a 8

and give a counterexample. No explanation is required.

a. If {an}∞n=1 is increasing, then limn→∞

an = ∞ .

� True

� False , because it does not hold for an = for n ≥ 1

b. If limn→∞

an = 0 , then∞∑n=1

an converges.

� True


c. If∞∑n=1

an converges conditionally, then∞∑n=1

(−1)nan diverges.

� True


d. If∞∑n=1

a2n converges, then∞∑n=1

a3n converges.

� True


e. If∞∑n=1

a2n diverges, then∞∑n=1

a3n diverges.

� True


4.Determine whether each of the following series converges or diverges.


a.∞∑n=1

n2(arctan(n+ 1)− arctan(n)

)b.

∞∑n=1

n(arctan(n+ 1)− arctan(n)

)c.

∞∑n=1

(arctan(n+ 1)− arctan(n)

)5. Suppose that the function f(x) de�ned on the interval (−R,R) by a power series

f(x) = c0 + c1x+ c2x2 + · · ·+ cnx

n + · · ·

which has radius of convergence R > 0 satis�es f(0) = 1 and f ′(x) = f(x/2) for all x in(−R,R).

a. Express the coe�cients of the following series in terms of c0, c1, c2, . . . , cn, . . . by �lling inthe boxes. No explanation is required.

f ′(x) = + x+ x2 + · · ·+ xn + · · ·

f(x/2) = + x+ x2 + · · ·+ xn + · · ·

b. Give a recurrence relation for {cn}∞n=0 by �lling in the boxes. No explanation is required.

c0 = and cn+1 = · cn for n ≥ 0

c. Find R.

d. Determine whether f(−2) is positive or negative.


1.Write your Bilkent student ID number here:A B C D E F G H

Now �ll in the boxes below with the corresponding digits from above.

P : x+ y + z = 1A B C

L : x = t+ , y = t+ , z = t ; −∞ < t <∞E F G H

Now �nd an equation for the plane perpendicular to the plane P and containing the line L.


2a. Make the sentences Ê and Ë into true statements by choosing one of the possiblecompletions for each of them. Indicate your choice by marking the � in front of it with a 3 .No explanation is required.

Ê The limit lim(x,y)→(0,0)

x sin y − y sinx

x2 + y2� exists � does not exist

Ë The limit lim(x,y)→(0,0)

x sin y − y sinx

x4 + y4� exists � does not exist

2b. Choose one of the statements � Ê and � Ë you made in part (2a), and prove it.Indicate your choice by marking the � in front of it with a 3 .

3. The delayed heat equation

ut(x, t+ 1) = uxx(x, t)

where u(x, t) is the temperature as a function of the position x and the time t, arises in theproblems of heat conduction in media which react to spatial variations in temperature witha time delay. For example, the temperature of meat as it is cooked can be modeled with thedelayed heat equation.

Find all possible values of the pair of positive constants (a, b) for which the function

u(x, t) = sin(ax− bt)

satis�es the delayed heat equation for all (x, t).

4. Suppose that a di�erentiable function f(x, y, z) satis�es the following conditions:

Ê∂f

∂y(3, 1, 3) = 1.

Ë The parametric curve

C1 : r1 = (3 + 2t) i+ (1− t2) j+ (3− 5t+ t2)k (−∞ < t <∞)

is contained in the level surface S of f(x, y, z) passing through the point (3, 1, 3).

Ì The parametric curve

C2 : r2 = (2 + t2) i+ (2t3 − 1) j+ (2t+ 1)k (−∞ < t <∞)

is also contained in the level surface S of f(x, y, z) passing through the point (3, 1, 3).


Find∂f

∂z(3, 1, 3).

5. Find and classify the critical points of the function f(x, y) = x2y + y2 − cxy where c is aconstant.


1. 1.Evaluate the following integrals where D = {(x, y) : x2 + y2 ≤ 1, x ≥ 0 and y ≥ 0}.

a.

¨D

cos(π2(x2 + y2)

)dA

b.

¨D

y3(x− x3) cos(πy4) dA

2.Consider the iterated integral:

I =

ˆ 2

0

ˆ 1− 12z

0

ˆ 3−2x− 32z

0

f(x, y, z) dy dx dz

a. The iterated integral I corresponds to a triple integral

˚E

f(x, y, z) dV

where E is a region in space. Draw a picture of the region E, and clearly label the surfacesbounding it with their equations and clearly label the important points with their coordinates.

b. Express the iterated integral I in terms of iterated integrals with the order of integrationdz dy dx.

3.Consider the triple integral

I =

˚E

1

(x2 + y2 + z2)2dV

where E = {(x, y, z) : x2 + y2 ≥ 1}.

a. Express I in terms of iterated integrals in cylindrical coordinates.

b. Express I in terms of iterated integrals in spherical coordinates.

c. Evaluate I.


4.Evaluate the integral ¨D

(x2 + y2)3 dA

where D is the region bounded by the hyperbolas x2−y2 = 1, x2−y2 = 4, xy = 1, and xy = −1in the right half plane.

Spring 2018 Final

1.Three students are working on a problem involving a di�erentiable function f(x, y, z) at apoint P0.

Student A says: An equation for the tangent plane to the level surface of f passingthrough P0 is 2x+ 3y − 6z = 11.

Student B says: The largest possible rate of change of f at P0 in any direction is 25.

Student C says: The directional derivative of f at P0 in the direction of the vectorA = 2i+ 2j+ k is 5.

Student D, who has been listening in, but does not know anything about the function f , decidesthat at least one of the Students A, B, C must have made a mistake in their calculations.

Explain Student D 's reasoning.

2. The surface area SA of the graph of a function f(x, y) on a region D is de�ned by theformula:

SA =

¨D

√1 + (fx)2 + (fy)2 dA

a.Show that this formula gives the surface area of the upper half of the unit sphere correctlyby considering the function f(x, y) =

√1− x2 − y2 on the unit disk D = {(x, y) : x2+y2 ≤ 1} .


b.Assume that g is a di�erentiable function on the interval [a, b] with 0 < a < b and considerthe function f(x, y) = g(r) on the ring D = {(x, y) : a2 ≤ x2+y2 ≤ b2} where r =

√x2 + y2 .

Express the surface area SA of the graph of f on D as a de�nite integral with respect to rwhose integrand involves only r and g′(r).

3.A sequence {an}∞n=1 satis�es the recursion relation

an =10

9max{an−1, an−2} −min{an−1, an−2}

for n ≥ 3.

a. In this part suppose that a1 = 1 and a2 = 1/9.

Ê Fill in the following boxes with the correct values. No explanation is required.

a3 = a4 = a2018 =

Ë Does the sequence {an}∞n=1 converge or diverge? Carefully prove your claim.

b. In this part suppose that a1 = 1 and a2 = 2/3.

Ê Fill in the following boxes with the correct values. No explanation is required.

a3 = a4 = a2018 =

Ë Does the sequence {an}∞n=1 converge or diverge? Carefully prove your claim.

4.Determine whether each of the following series is convergent or divergent.

a.∞∑n=2

(−1)nlnn

2n

b.∞∑n=1

1

2n−lnn

c.∞∑n=2

1

2n/ lnn

5.Consider the power series f(x) =∞∑n=1

(1

2+

1

n

)n

xn .

a. Find the radius of convergence R of the power series.

b. Determine whether f(R) converges absolutely, converges conditionally, or diverges.

c. Determine whether f(−R) converges absolutely, converges conditionally, or diverges.


d. Choose a suitable positive integer M and show that f(1) < M .

You will earn max{22− 3M, 0} points from this part for a completely correct solution.


1. Evaluate the following limits.

a. lim(x,y)→(0,0)

xy5

x4 + y6

b. lim(x,y)→(0,0)

xy5

x4 + x5y + y6

c. lim(x,y)→(0,0)

xy5

x4 + x3y + y6

2. The Tricomi equation

yuxx + uyy = 0

arises in the study of transonic �ow in �uid mechanics and in the study of isometric embeddingsof 2-dimensional Riemannian manifolds into 3-dimensional Euclidian space in di�erentialgeometry.

Find all possible values of the pair of constants (a, b) for which the function u(x, y) = (ax2+y3)b

satis�es the Tricomi equation for all (x, y) with ax2 + y3 > 0 .

3. Consider the surfaces S1 : xyz = 10 and S2 : z = x2 + y2 , and the point P0(1, 2, 5) .

a. Find an equation of the tangent plane to S1 at P0.

b. Find parametric equations of the tangent line to the curve of intersection of S1 and S2

at P0.

4. Find the absolute maximum and minimum values of the function f(x, y) = x3 − y2 + x2yon the closed triangular region T shown in the �gure below.

x

y

1−2

1

−2

T



1. In each of the following, a double integral

¨D

f(x, y) dA is expressed as an iterated integral

in polar coordinates. In each part, draw a picture of the region D, and clearly label the curvesbounding it with their equations both in Cartesian and polar coordinates.

a.

ˆ π/4

0

ˆ sec θ

0

f(r cos θ, r sin θ) r dr dθ

b.

ˆ π/4

0

ˆ csc θ

0


c.

ˆ π/4

0

ˆ 2 sin θ

0


d.

ˆ π/3

0

ˆ 2

0


e.

ˆ 1

0

ˆ π

arccos r

f(r cos θ, r sin θ) r dθ dr

2a. Evaluate the iterated integral

ˆ ∞

0

ˆ 1/√y

0

e−1/x dx dy .

2b. Evaluate the double integral

¨R

(x2 + y2) dA where R is the region between the unit circle

and the regular hexagon with center at the origin shown in the �gure .

x

y

2

1

R


3. LetD be the region in space bounded by the parabolic cylinder x = y2, the plane x+y+z = 2,the yz-plane, and the xy-plane.

• Choose two of the following rectangular boxes by putting a 7 in the � in front of them,and then

• choose one of the orders of integration in each of the selected boxes by putting a 7 in the� in front of them.

� dx dy dz � dy dx dz � dz dx dy� � �

� dx dz dy � dy dz dx � dz dy dx

Express the volume V of the region D in terms of iterated integrals in each of your selectedorders of integration (a) and (b).

c. Find the volume V .

4a. In Ê-Ë, if there exists a sequence {an}∞n=1 satisfying the given conditions, write its nth termin the box; and if no such sequence exists, write Does Not Exist in the box. No explanation isrequired.

Ê limn→∞

an+1

an= 1 and lim

n→∞an does not exist.

an =

Ë limn→∞

an+1

an= −1 and lim

n→∞an exists.

an =


4b. Let c be a real number, and consider the sequence {an}∞n=1 with a1 = c and satisfying therecursion relation an+1 = an + a2n for all n ≥ 1.

À Show that if the sequence converges, then limn→∞

an = 0.

Á Fill in the boxes so that the sentence below becomes a true statement.

If c = , then the sequence .� �

Write here a real number Write here either

which is not an integer converges or diverges

Â Prove the statement in Á.

Spring 2017 Final

1. Consider the functionf(x, y, z) = x3y2z + ax2y + bxz2 ,

where a and b are constants, the point P0(1,−1, 2) , and the vector A = 2i+ 3j+ 6k .

a. Compute ∇f(P0) .

b. Find all possible values of (a, b) for which f increases the fastest in the direction of A atP0 .

c. Let a = 3 and b = 1. Find the directional derivative of f in the direction of A at P0 .

2a. Evaluate the double integral ¨R

cos(πx2/y) dA

where R is the region bounded by the parabolas y = 3x2, y = 3x2/2, x = y2, x = 2y2 in theplane.


2b. A solid D in space satis�es the following conditions:

• The intersection of D with the xy-plane is the region bounded by the cardioid with theequation r = 1 + cos θ in polar coordinates.

• The intersection of D with each half-plane θ = c in spherical coordinates, where c is aconstant, is a disk with a diameter lying in the xy-plane.

Express the volume V of the solid D as an iterated integral in spherical coordinates by �llingin the rectangles below. No explanation is required.

V =

ˆ ˆ ˆdρ dϕ dθ

3. In each of the following, if the given statement is true for all sequences {an}∞n=1 , then markthe � to the left of True with a 8 ; otherwise, mark the � to the left of False 8 and givea counterexample. No explanation is required.

a. If an < an+1 for all n ≥ 1, then limn→∞

an = ∞ .

� True


b. If limn→∞

an = 0 , then∞∑n=1

an converges.

� True



c. If∞∑n=1

an converges, then {an}∞n=1 converges.

� True


d. If 0 <1

2n< an for all n ≥ 1, then

∞∑n=1

an diverges.

� True


e. If 0 < an <1

nfor all n ≥ 1, then

∞∑n=1

an converges.

� True


4. Determine whether each of the following series is convergent or divergent.

a.∞∑n=1

cos(πn

)

b.∞∑n=1

(1− cos

(πn

))

c.∞∑n=1

1

n2−cos(π/n)

5. Consider the power series f(x) =∞∑n=2

xn

n3 − n.

a. Find the interval of convergence I of the power series and determine whether it convergesabsolutely or conditionally at each point of I.

b. Find the exact value of f(−1).



1. Consider the sequence {an}∞n=1 satisfying the conditions a1 = A and an+1 = 3an −1

anfor

n ≥ 1 where A is a real number such that an = 0 for all n ≥ 1.

a. Assume that the sequence converges and let L = limn→∞

an . Show that, depending on A,

there are at most two possible values for L.

b. Give an example of A for which the sequence converges. Explain your reasoning.

c. Give an example of A for which the sequence diverges and |an| < 1 for all n ≥ 1. Explainyour reasoning.

d. Show that the sequence is increasing if A = 1.

e. Determine whether the sequence converges or diverges if A = 1.

2a. In each of Ê-Í, indicate all possible completions of the sentence that will make it into atrue statement by 3 ing the corresponding � s. No explanation is required.

Ê

∞∑n=1

1

n= 1 +

1

2+

1

3+

1

4+ · · ·+ 1

n+ · · · is



Ë

{1

n

}∞

n=1

=

{1 ,

1

2,1

3,1

4, · · · , 1

n, · · ·

}is



Ì

{1

2n−1

}∞

n=1

=

{1 ,

1

2,1

4,1

8, · · · , 1

2n−1, · · ·

}is



Í

∞∑n=1

1

2n−1= 1 +

1

2+

1

4+

1

8+ · · ·+ 1

2n−1+ · · · is




2b. In each of Î-Ï, if there exists a sequence {an}∞n=1 satisfying the given conditions, writeits general term inside the box; and if no such sequence exists, write Does Not Exist inside thebox. No explanation is required.

Î The sequence {an}∞n=1 diverges and the sequence {(−1)nan}∞n=1 converges.

an =

Ï The series∞∑n=1

an converges and the series∞∑n=1

(−1)nan diverges.

an =

3. Determine whether each of the following series converges or diverges.

a.∞∑n=2

1

(lnn)2016

b.∞∑n=1

(4031

2016− n1/n

)n

c.∞∑n=1

(20161/n − 1

)


xn

(n2 − 1) · n!.


b. Show that f(1) <1

2e− 7

6.

c. Show that f(−1) <3

20.



1. Consider the plane P : 3x− y + 2z = 7 .

a. Give an example of a nonzero vector n normal to the plane P . No explanation is required.

n = i+ j+ k

b. Give an example of a point P0 in the plane P . No explanation is required.

P0

(, ,

)

c. Give an example of a nonzero vector v parallel to the plane P . No explanation is required.

v = i+ j+ k

d. Write inside the box parametric equations of one of the lines lying in the plane P . Noexplanation is required.

L :

e. Find an equation of the plane that passes through the point (1, 1, 1) and contains the lineL in Part d. Show all your work.

2. Suppose that f(x, y, z) is a di�erentiable function with the gradient

∇f = (3x2 − y2z)i− 2xyzj+ (2z − xy2)k

and consider the point P0(1,−1, 2).

b. Find a unit vector u for which the directional derivative Duf(P0) has its largest possiblevalue.

c. Find a unit vector u for which Duf(P0) = 0 .

d. Find a unit vector u for which Duf(P0) = 5 .

e. Give an example of a function f whose gradient is the one given in this question. Noexplanation is required.

f(x, y, z) =


3. Each of the following functions has a critical point at (0, 0) . [You do not need to verify this.]

Determine whether this critical point is a local maximum, a local minimum, a saddle point orsomething else.

a. f(x, y) = (1− x2)(1− y2)

b. f(x, y) = x2 − y4

c. f(x, y) = x2 − xy + y4

4. Find the absolute maximum and minimum values of the restriction of the function

f(x, y, z) = xy + xz

to the unit sphere x2 + y2 + z2 = 1 .

Spring 2016 Final

1. Determine the smallest of the real numbers A, B, C, D, E where :

A =∞∑n=0

(−1)n

2nB =

∞∑n=1

1

n2nC =

∞∑n=1

(−1)n+1

n!

D =∞∑n=1

n

3nE =

∞∑n=0

(−1)n

3n(2n+ 1)

2. The Dym equation

ut = u3 uxxx

is a nonlinear evolution equation which arises in the study of the motion of the interface betweena viscous and a nonviscous �uid with surface tension.

Find all nonzero constants a, b, c such that the function

u(x, t) = (ax+ bt)c

satis�es the Dym equation for all (x, t) with ax+ bt > 0.

3. Suppose that g(x, y) = f(x2 − y2, 2xy) where f(x, y) is a di�erentiable function. Find anequation of the tangent plane to the graph of z = f(x, y) at the point (3,−4, 7) if an equationof the tangent plane to the graph of z = g(x, y) at the point (−2, 1, 7) is z = 5x− 6y + 23 .


4. Consider the double integral

I =

¨R

1

(x2 + y2)2dA

where R is the region in the �rst quadrant lying outside thecircle x2 + y2 = 2 , and bounded by the line x = 2 on theright and the line y = 2 at the top.

a. Express I in terms of iterated double integrals inCartesian coordinates.

x

y

2√2

2

√2

R

b. Express I in terms of iterated double integrals in polar coordinates.

c. Evaluate I.

5. Let V be the volume of the solid cone whose base is the unit disk in the xy-plane and whosetip is at the point (0, 0, 2) in the xyz-space.

y

z

x

2

1−11

−1

a. Only two of Ê-Ì will be graded. Mark the ones you want to be graded by putting a 3



V =

ˆ ˆ ˆdz dy dx

Ë � Express V in terms of iterated integrals in cylindrical coordinates by �lling in therectangles.


V =

ˆ ˆ ˆr dz dr dθ

Ì � Express V in terms of iterated integrals in spherical coordinates by �lling in therectangles.

V =


b. Compute V using its expression in terms of iterated integrals in one of the coordinatesystems in Part a.


1.

1

1

1/2

1/2

1/3

1/3

1/4

1/4 . . .

We have a 1/n×1/n square for each positive integer n. For each of (a-c), indicate by 4ing the� to the left of Yes or No whether it is possible or not to place these squares in the xy-planein such a way that they completely cover the given set. If Yes, describe how this can be done(you might also want to draw a picture) and then fully justify your claim. If No, explain whythis cannot be done.

a. The entire xy-plane : � Yes � No

b. The line de�ned by the equation y = x : � Yes � No

c. The region between the graph of y = e−x and the x-axis for x ≥ 0 : � Yes � No

2. In each of the following, if the reasoning in the given sentence is correct, then 4 thecorresponding � ; otherwise, leave it blank. No explanation is required. For each of the parts

(a-e), to get full points you must check exactly the squares corresponding to the correct reasonings. Note that

in each sentence all the statements to the right of �because" are true. You must decide whether they lead to

the statement to the left of �because", possibly using a test you have seen in this course.

a. �∞∑n=1

1

nconverges because lim

n→∞

1

n= 0 .

�∞∑n=1

1

nconverges because 0 <

1

n+ 1<

1

nfor all n ≥ 1 .


�∞∑n=1

1

ndiverges because 1 +

1

2+

1

3+ · · ·+ 1

n> ln(n+ 1) for all n ≥ 1 .

�∞∑n=1

1

ndiverges because 1 +

1

2+

1

3+ · · ·+ 1

2n> 1 +

n

2for all n ≥ 2 .

b. �∞∑n=0

(−1)n converges because −1 ≤ (−1)n ≤ 1 for all n ≥ 0 .

�∞∑n=0

(−1)n diverges because the sequence 1,−1, 1,−1, 1, . . . diverges.

�∞∑n=0

(−1)n diverges because the sequence 1, 0, 1, 0, 1, . . . diverges.

�∞∑n=0

(−1)n diverges because −1 ≤ (−1)n ≤ 1 for all n ≥ 0 .

c. �∞∑n=1

1

n1+1/nconverges because lim

n→∞

1

n1+1/n= 0 .

�∞∑n=1

1

n1+1/nconverges because 0 < (1/(n+1)1+1/(n+1))/(1/n1+1/n) < 1 for all n ≥ 1 .

�∞∑n=1

1

n1+1/nconverges because 0 <

1

n1+1/n<

1

nfor all n ≥ 2 and

∞∑n=1

1

ndiverges.

�∞∑n=1

1

n1+1/ndiverges because lim

n→∞

((1/n1+1/n)/(1/n)

)= 1 and

∞∑n=1

1

ndiverges.

d. �∞∑n=2

(1− 1

n

)n2

converges because limn→∞

((1− 1

n

)n2)1/n= e−1 < 1 .

�∞∑n=2

(1− 1

n

)n2

converges because limn→∞

(1− 1

n

)n2

= 0 .

�∞∑n=2

(1− 1

n

)n2

converges because 0 <

(1− 1

n

)n2

<1

2nfor all n ≥ 2 and

∞∑n=0

1

2n

converges.

�∞∑n=2

(1− 1

n

)n2

diverges because1

3n<

(1− 1

n

)n2

for all n ≥ 6 and∞∑n=0

1

3nconverges.

e. �∞∑n=0

1

2nconverges because lim

n→∞

(1

2n

)1/n=

1

2< 1 .

�∞∑n=0

1

2nconverges because lim

n→∞

((1/2n+1)/(1/2n)

)=

1

2< 1 .

�∞∑n=0

1

2nconverges because the sequence 1,

3

2,7

4,15

8,31

16, . . . converges.

�∞∑n=0

1

2ndiverges because

1

2n= 0 for in�nitely many n ≥ 0 .


3. Determine whether each of the following series converges or diverges. Explain your reasoning in

full by clearly stating the name and the conditions of the test you are using, and explicitly checking the validity

of these conditions for the given series.

a.∞∑n=1

(5n − 3n

)−1

b.∞∑n=2

sin5(π/ 3

√n)

c.∞∑n=1

cos5(π/ 3

√n)

4. Consider f(x) =∞∑n=0

xn

5n(n2 + 1).


b. Show that4

3< f(3) <

3

2.

c. Show that3

4< f(−3) <

4

5.


1. In each of the following if the series converges, then write the exact value of its sum in aform as simpli�ed as possible in the box; otherwise, write Div. No explanation is required.

a.∞∑n=0

3n

n!=

b.∞∑n=0

(−1)n3n

2n+ 1=

c.∞∑n=0

(−1)nπ2n

(2n+ 1)!=


d.∞∑n=1

(−1)nn

3n=

e.∞∑n=1

1

3nn=

2. In the xyz-space where a �yscreen lies along the plane with the equation

2x+ y − 2z = 1 ,

the trajectory of a bee as a function of time t is given by

r = t i+ t2 j+ t3 k

for −∞ < t <∞.

a. Find the velocity v of the bee as a function of time.

b. Give an example of a nonzero vector n perpendicular to the screen.

c. Find all times t when the bee is �ying parallel to the screen.

d. Find all times t when the bee is �ying perpendicular to the screen.

e. There are holes in the screen through which the bee passes. Find the coordinates of allof these holes.

3a. Evaluate the following limits. [ Show all your work. ]

Ê lim(x,y)→(0,0)

x3y2

x6 + y6

Ë lim(x,y)→(0,0)

x3y2

x6 x2y2 + y6 � Your choice: Fill in the with �+" or �−", then evaluate!

3b. Evaluate the following statements. [ Just check and fill, no further explanation is required. ]

Ì If f(x, y) approaches 0 as (x, y) approaches (0, 0) along any line through (0, 0), then lim(x,y)→(0,0)

f(x, y) = 0 .

� True and can be proven using the

� False because does not hold for f(x, y) =


Í If, for (x, y) = (0, 0), the line y = x and every circle tangent to it at (0, 0) are level curves of f(x, y)

belonging to di�erent values, then lim(x,y)→(0,0)

f(x, y) does not exist.

� True and can be proven using the

� False because does not hold for f(x, y) =

4. In Genetics, Fisher's Equation,

∂p

∂t= p (1− p) +

∂2p

∂x2

describes the spread of an advantageous allele in a population with uniform density along a1-dimensional habitat, like a shoreline, as a result of both reproduction and dispersion of theo�spring. Here p(x, t) is the frequency of the allele as a function of the position x and the timet.

Find all possible values of the pair of constants (a, b) for which the function

p(x, t) =1

(1 + eax+bt)2

satis�es the Fisher's Equation.

Spring 2015 Final

1. Consider the following conditions for a di�erentiable function f(x, y) :

Ê f(2, 1) = 8

Ë An equation for the tangent line to the level curve f(x, y) = 8 in the xy-plane at thepoint (2, 1) is 3x− 5y = 1

Let P be the tangent plane to the graph of z = f(x, y) at the point (2, 1, 8) .

In each of the parts (a-e) below a Ìrd condition is given.

• If there is no function satisfying the conditions Ê-Ì , then 3 the � next to None.

• If there are functions satisfying the conditions Ê-Ì , but they do not all have the sametangent plane P, then 3 the � next to Many.


• If there are functions satisfying the conditions Ê-Ì and all of these functions have thesame tangent plane P, then 3 the � next to Unique and write an equation of P insidethe box.

a. Ì f(3, 2) = 11

� None � Many � Unique P :

b. Ì fx(2, 1) = −1


c. Ìd

dtf(t2 + 1, t3)

∣∣∣∣t=1

= 6


d. Ì The line with parametric equations x = 4t+2 , y = 2t+1 , z = t+8 , (−∞ < t <∞) ,lies in P


e. Ì The line with parametric equations x = −t+2 , y = 2t+1 , z = t+8 , (−∞ < t <∞) ,is perpendicular to P


2. In the �gure below some of the level curves and the corresponding values of a function f areshown.

(The �gure is on the next page.)

a. Indicate the signs of the following derivatives at (0, 0) by 3 ing the corresponding � .No explanation is required.

Ê fx(0, 0) is � positive � negative

Ë fy(0, 0) is � positive � negative

Ì fxx(0, 0) is � positive � negative

Í fyy(0, 0) is � positive � negative

Î fxy(0, 0) is � positive � negative

b. Draw the gradient vector ∇f(0, 0) at the origin as best you can on the �gure. Noexplanation is required.


3. Find the absolute maximum and minimum values of the function f(x, y) = 2(x2+y2−1)2+x2 − y2 on the unit disk D = {(x, y) : x2 + y2 ≤ 1} .

4. Evaluate the following integrals.

a.

¨R

(1 + x− y) dA where R = {(x, y) : |x− y| ≤ 2/3 and 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1}

b.

ˆ 1

−1

ˆ √1−x2

0

sin(π(x2 + y2)

)dy dx

5. Let V be the volume of the ball B = {(x, y, z) : x2 + y2 + z2 ≤ 2z} .





V =

ˆ ˆ ˆdz dy dx


V =

ˆ ˆ ˆr dr dz dθ


V =


b. Compute V using its expression in terms of iterated integrals in one of the coordinatesystems in part (a).


1a. In each of Ê-Ñ , indicate the kind of geometric object de�ned in the xyz-space by the givenset of equations by marking the corresponding � with a 3 . No explanation is required.

Ê x+ 3z = 7y + 5

� A point � A line � A plane � Something else

Ë x = 3t− 2 , y = 4 , z = −5t+ 1 ; −∞ < t <∞� A point � A line � A plane � Something else

Ì x− 4 = (y − 3)/5 = (z − 11)/8


Í 2x+ y − 3z = 0 , −x+ 6y + z = 8



Î y = 13


Ï x = t3 + 2 , y = 4t3 + 5 , z = t3 − 1 ; −∞ < t <∞


Ð x = 1 , z = 3


Ñ x2 + y2 + z2 = 0


1b. Write inside the box an equation for the plane that passes through the point (3, 2, 1) andcontains the line of intersection of the planes with equations x+ y+ z = 3 and x+2y+3z = 6.No explanation is required.

1c. The equation of the tangent plane to the graph of the function z = f(x, y) at the pointwith (x, y) = (3, 5) is 4z − 7x + y = 2. Write inside the box an equation of the tangent lineto the level curve of the function f in the xy-plane that passes through the point (3, 5). Noexplanation is required.

2. Mark two � s in each of the sentences in parts (a) and (b) with 3 s to make them truestatements. Then prove your claim by using the method you chose.

a. The limit lim(x,y)→(0,0)

x3y2 − x2y3

x4 + y6� is 0 � does not exist , and this can be shown by using

� the Sertöz Theorem � the 2-Path Test � the Squeeze Theorem .

b. The limit lim(x,y)→(0,0)

x4y2

x4 + y6 − x2y2� is 0 � does not exist , and this can be shown by

using � the Sertöz Theorem � the 2-Path Test � the Squeeze Theorem .


Sertöz Theorem:

Let a and b be nonnegative integers, let c and d be positive even integers,and let

f(x, y) =xayb

xc + yd.

Then:

• Ifa

c+b

d> 1 , then lim

(x,y)→(0,0)f(x, y) = 0 .

• Ifa

c+b

d≤ 1 , then lim

(x,y)→(0,0)f(x, y) does not exist.

3a. Find the largest possible value of the directional derivative Duf(2,−1, 1) for the function:

f(x, y, z) = yzexy+2z2

3b. A �y walks with a speed of 1 cm/s in any direction on this page which is identi�ed withthe xy-plane. If the �y walks from the origin in the direction of the vector A = i − 7j , thetemperature it measures increases at a rate of 3◦C/s. If the �y walks from the origin in thedirection of the vector B = i+j , the temperature it measures decreases at a rate of 2◦C/s. Findhow fast the temperature it measures changes if the �y walks from the origin in the positivex-direction.

4a. Find and classify the critical points of f(x, y) = 2x2 + y2 + 2x2y .

4b. Next week in Section 14 of Math 102 the following quiz will be given:

Q. Find the absolute minimum value of f(x, y) = 2x2+ y2+2x2y

on the square S = {(x, y) : |x|+ |y| ≤ 4} .



A friend of yours in Section 14 tells you their solution to this quiz:

�f has three critical points in the interior of S where its values are 0, 1 and 1,

respectively. Moreover f(4, 0) = f(−4, 0) = 32 and f(0, 4) = f(0,−4) = 16. So the

absolute minimum value of f on S must be 0."

You respond by saying that:

�Your answer cannot be correct, because ."

Fill in the box with less than 13 characters to make your response mathematically valid.



ˆ ∞

0

ˆ ∞

√x

1

(y2 + 1)2dy dx .

1b. Evaluate the double integral

¨R

1

(x2 + y2)2dA where R is the region shown in the �gure.



2. The region R bounded by the curve z2 = y2 − y4 in the right half of the yz-plane is rotatedabout the z-axis to obtain a solid D in the xyz-space. Let V be the volume of the solid D.




V =

ˆ ˆ ˆdz dy dx



V =

ˆ ˆ ˆr dz dr dθ


V =


b. Compute V .

3a. Fill in the rectangles to make each of the following sentences a true statement.

is a convergent sequence.

is a convergent series.

is a divergent sequence.

is a divergent series.

3b. Consider the question:

Q. Find the sum of the series

∞∑n=2

ln

(n

n+ 1

).

Your friend from Section 14 shows you their solution:∞∑

n=2

ln

(n

n+ 1

)= ln

(2

3

)+ ln

(3

4

)+ln

(4

5

)+ . . .

= (ln 2−��ln 3) + (��ln 3−HHln 4) + (HHln 4−��ln 5) + . . .

= ln 2


Express your opinion of your friend's solution by 3 ing the corresponding � .

� The solution is correct.

� The solution is incorrect, but the answer is correct.

� Both the solution and the answer are incorrect.

If you 3 ed the �rst � , you are done with Question 3b. If you 3 ed the second or third � ,write the correct solution below.

4. In each of the parts (a-d), mark the appropriate � s with 3 s and then give a briefexplanation consisting of complete and meaningful mathematical sentences.

a. The series∞∑n=1

1

2nis � convergent � divergent .

A positive integer n such that 2014 ≤n∑

i=1

1

2i≤ 2015 � exists � does not exists ,

because: � Explain!

b. The series∞∑n=1

5n is � convergent � divergent .


i=1

5i ≤ 2015 � exists � does not exist ,


c. The series∞∑n=1

(2999

3000

)n

is � convergent � divergent .


i=1

(2999

3000

)i

≤ 2015 � exists � does not exist ,


d. The series∞∑n=1

1

nis � convergent � divergent .


i=1

1

i≤ 2015 � exists � does not exist ,


Spring 2014 Final

1. Suppose that f(x, y) is a di�erentiable function that satis�es f(1, 2) = 7, fx(1, 2) = −3,fy(1, 2) = 11, and

f

(x

x2 + y2,

y

x2 + y2

)= f(x, y)

for all (x, y) = (0, 0).

a. Find an equation of the tangent plane to the graph of z = f(x, y) at the point with(x, y) = (1, 2).

b. Find parametric equations of the normal line to the surface z = f(x, y) at the point(x, y, z) = (1, 2, 7).


c. Find an equation of the tangent line to the level curve f(x, y) = 7 at the point (x, y) =(1, 2).

d. Compute fx(1/5, 2/5).

2. Consider the transformation T : x =u

u+ v + 1, y =

v

u+ v + 1.

a. Compute the Jacobian∂(x, y)

∂(u, v)of T .

b. Show that there is a constant C such that the inequality

¨G

∣∣∣∣∂(x, y)∂(u, v)

∣∣∣∣ du dv ≤ C holds

for all regions G contained in the �rst quadrant of the uv-plane.

3. In each of the following, if there exists a sequence {an} with nonzero terms satisfying thegiven conditions, write its general term inside the box; and if no such sequence exists, writeDoes Not Exist inside the box. No explanation is required. No partial credit will be given.

a.∞∑n=1

an converges and∞∑n=1

1

anconverges.

an =

b.∞∑n=1

an converges and∞∑n=1

1

andiverges.

an =

c.∞∑n=1

an diverges and∞∑n=1

1

andiverges.

an =

d. {an} converges and

{1

an

}converges.

an =


e. {an} converges and

{1

an

}diverges.

an =

f . {an} diverges and

{1

an

}diverges.

an =

4. Determine whether each of the following series converges or diverges.

a.∞∑n=0

(ln 2)n

b.∞∑n=1

1

nln 2

c.∞∑n=1

(−1)n−1 n√ln 2

d.∞∑n=1

(1− n√ln 2)

e.∞∑n=1

(1− n√ln 2)n

5. Consider the power series∞∑n=2

xn

n2 − 1.

a. Determine the interval of convergence of the power series.

b. Find the exact sum of the power series at x = −1.


1. Consider the sequence de�ned by:

a1 = � Write the last digit of your Bilkent ID number in the box!

an+1 =√

90 + an for n ≥ 1


In Parts a-c, mark the � in the appropriate box or �ll in the to make these into true

statements and then prove them.

a. The sequence {an}∞n=1 is bounded below by .

b. The sequence {an}∞n=1 is bounded above by .

c. The sequence {an}∞n=1 is � increasing � decreasing.

d. Show that the sequence {an}∞n=1 converges.

e. Let L = limn→∞

an. Find L.

2. Determine whether each of the following series is convergent or divergent. Explain yourreasoning in full.

a.∞∑n=1

(n

n+ 1

)n2

b.∞∑n=1

(21/n − 1)

c.∞∑n=2

1

(lnn)lnn

3. In each of the following,

• If the given statement is true for all sequences {an} , then mark the � to the left ofTrue with a 8 and �ll in the blank with the name of a test;

• Otherwise, mark the � to the left of False and give a counterexample.

a. If∞∑n=1

an converges, then {an}∞n=1 converges.

� True , as can be shown using .

� False , because it does not hold for an = for n ≥ 1.

b. If {an}∞n=1 converges, then∞∑n=1

an converges.




c. If∞∑n=1


nan diverges.



d. If∞∑n=1


nan converges.



4. Consider the function f de�ned by the following power series:

f(x) =∞∑n=1

nn

(n!)2xn

a. Write the �rst �ve nonzero terms of the power series with their coe�cients in a form assimpli�ed as possible.

b. Find the radius of convergence of the power series.

c. Show that f(1) ≤ 15

4. You may use the fact that

nn

(n!)2≤ 64

9 · 2nfor n ≥ 1.

5. For each of the following series, in the

• Write the exact value of the sum in a form as simpli�ed as possible if the series converges;and

• Write Div if the series diverges.

No explanation is required. No partial credit will be given.


a.∞∑n=0

(−1)nπ2n

4n(2n)!=

b.∞∑n=0

(−1)n

3n(2n+ 1)=

c.∞∑n=1

23n

32n=

d.∞∑n=1

1

1 + 2 + · · ·+ n=


1. Use power series to determine the value of the constant a for which the limit

limx→0

cosx− eax2

sin(x4)

exists and evaluate the limit for this value of a. (Do not use L'Hôpital's Rule!)

2. Consider the following lines:

L1 : x = 2t− 1, y = −t+ 2, z = 3t+ 1

L2 : x = s+ 5, y = 2s+ 3, z = −s

a. Find a nonzero vector v perpendicular to both L1 and L2 .

b. Find a parametric equation of the line L that intersects both L1 and L2 perpendicularly.

3. Consider the function

f(x, y) =

xayb

x4 + y6if (x, y) = (0, 0)

0 if (x, y) = (0, 0)


where a and b are nonnegative integers. In each of the following, markthe corresponding � with a 8 and �ll in the s where necessaryto form a true statement. No explanation is required.

a. f(x, y) is continuous at (0, 0)

� for a = and b = .

� for no values of a and b.

b. f(x, y) goes to 1 as (x, y) approaches (0, 0) along the line y = x, and f(x, y) goes to−1 as (x, y) approaches (0, 0) along the line y = −x



c. f(x, y) goes to 0 as (x, y) approaches (0, 0) along any line through the origin, and thelimit lim

(x,y)→(0,0)f(x, y) does not exist



d. f(x, y) goes to 0 as (x, y) approaches (0, 0) along any line through the origin exceptthe y-axis, and f(x, y) goes to 1 as (x, y) approaches (0, 0) along the y-axis



e. fx(0, 0) and fy(0, 0) exist, and f(x, y) is not di�erentiable at (0, 0)



4. Let z = f(x, y) be a di�erentiable function such that

f(3, 3) = 1, fx(3, 3) = −2, fy(3, 3) = 11,

f(2, 5) = 1, fx(2, 5) = 7, fy(2, 5) = −3.

a. Find an equation of the tangent plane to the graph of z = f(x, y) at the point (2, 5, 1).


b. Suppose w is a di�erentiable function of u and v satisfying the equation

f(w,w) = f(uv, u2 + v2)

for all (u, v). Find∂w

∂uat (u, v, w) = (1, 2, 3).

5. Find the values of the constants c and k for which the function

u(x, y) =1

1 + ex−cy

satis�es the equation

u∂u

∂x+∂u

∂y= k

∂2u

∂x2

for all (x, y).

Spring 2013 Final

1. Consider the functions f(x, y) = x2y and g(x, y) = x3 + 5y2.

a. Compute ∇f and ∇g.

b. Find all unit vectors u along which the directional derivatives of f and g at the pointP0(2, 1) are equal.

c. Find all points P (x, y) in the plane at which the directional derivatives of f and g in thedirection of v = i+ j are zero.

2. Find and classify the critical points of f(x, y) = xy2 − x2 + 2x− 3y2.

3. Three hemispheres with radiuses 1, x and y, where 1 ≥ x ≥ y ≥ 0, are stacked on top ofeach other as shown in the �gure. Find the largest possible value of the total height h.

1

y

x

h



a.

ˆ ∞

0

ˆ 2x

x

e−y2 dy dx

b.

¨D

1

(x2 + y2)2dA where D = {(x, y) : x2 + y2 ≥ 1 and 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1}

5a. Express the given triple integral as an iterated integral by �lling in the boxes, where D isthe region shown in the �gure.

˚D

f(x, y, z) dV =

ˆ ˆ ˆf(x, y, z) dy dx dz

5b. Evaluate

˚D

1

1 + (x2 + y2 + z2)3/2dV where D = {(x, y, z) : x2 + y2 + z2 ≤ 1} .


1a. Write in the box an equation for one of the planes that contain the line x = −2t + 3, y =5t+ 1, z = 4t− 1,−∞ < t <∞. No explanation is required.

1b.Write in the box parametric equations for one of the lines that are contained in the plane7x− y − 2z = 11. No explanation is required.


1c. Find an equation of the plane that passes through the point P (−3, 2, 1) and is perpendicularto both of the planes with equations x+ 3y − 8z = 2 and 2x− y + 6z = 1.

2a. Find the length of the parametric curve x = (1+cos t) cos t, y = (1+cos t) sin t, 0 ≤ t ≤ 2π.

2b. Find parametric equations of the tangent line to the parametric curve r = t i+ t2 j+ t3 k,−∞ < t <∞, at the point with t = 2.

3. The level curves of the following �ve functions are shown in the �gures below. Match thesewith their functions by �lling in the boxes with the corresponding letters.

A. f(x, y) = sin x+ 2 sin y B. f(x, y) = (4x2 + y2)e−x2−y2 C. f(x, y) = x2y2

D. f(x, y) = xye−y2 E. f(x, y) = 3x− 2y

(The �gures are on the next page.)



4. Let f(x, y) =x3y4

x4 + x3y2 + y10.

a. Show that the limit of f(x, y) as (x, y) approaches (0, 0) along any line through the originis the same.

b. Show that lim(x,y)→(0,0)

f(x, y) does not exist.

5. Let u = x+ y+ z, v = xy+ yz+ zx, w = xyz, and suppose that f(u, v, w) is a di�erentiablefunction satisfying f(u, v, w) = x4 + y4 + z4 for all (x, y, z). Find fu(2,−1,−2).


1a. In (i-iii), if there is a di�erentiable function f(x, y) whose derivatives at (0, 0) in thedirections of the vectors A, B, C are all positive, give an example of such a function; if thereis no such function, write Does not exist in the box. No explanation is required.

i . A = i+ 2j , B = i− j , C = i

f(x, y) =

ii . A = i+ 2j , B = i− j , C = −i− j

f(x, y) =

iii . A = 3i+ j , B = i− j , C = −i− j

f(x, y) =

1b. A bug is standing on the ground at a point P . If it moves towards north from P ,the temperature decreases at a rate of 4 C◦/m. If it moves towards southeast from P, thetemperature increases at a rate of 3

√2 C◦/m. In which direction should the bug move to go to

cooler points as fast as possible? (Choose a coordinate system on the ground with the positivex-axis pointing east and the positive y-axis pointing north, and express your answer as a unitvector.)

2. Find the absolute maximum value of f(x, y) = x3−xy− y2+2y on the square R = {(x, y) :0 ≤ x ≤ 1 and 0 ≤ y ≤ 1}.

3. Find and classify the critical points of f(x, y) = x3 − 2x2 + xy2.



a.

¨R

| cos(x+ y)| dA where R = {(x, y) : 0 ≤ x ≤ π/2 and 0 ≤ y ≤ π/2}.

b.

ˆ 1

0

ˆ √y−y2

0

dx dy

(x2 + y2 − (x2 + y2)2)1/2

5. Let D be the region in the �rst octant bounded by the coordinate planes, the plane x+y = 4,and the cylinder y2 + 4z2 = 16.

• Choose two of the following rectangular boxes by putting a 7 in the small square infront of them, and then

• choose one of the orders of integration in each of the selected boxes by putting a 7 inthe small square in front of them.



Express the volume V of the region D as iterated integrals in both of your selected orders ofintegration (a) and (b). (Do not evaluate the integrals! )

Spring 2012 Final

1a. Find all points on the surface z = x2 + y2 where the tangent plane is parallel to the planex+ 2y + 3z = 6.

1b. A bug is standing on the ground at a point P . If it moves towards north from P ,the temperature decreases at a rate of 4 C◦/m. If it moves towards southeast from P, thetemperature increases at a rate of 3

√2 C◦/m. In which direction should the bug move to go to

cooler points as fast as possible? (Choose a coordinate system on the ground with the positivex-axis pointing east and the positive y-axis pointing north, and express your answer as a unitvector.)

2a. Suppose that T is a one-to-one transformation from the xy-plane to the uv-plane whoseJacobian satis�es the condition

0 <∂(u, v)

∂(x, y)≤ 1

(x2 + y2 + 1)2

for all (x, y). Show that if G is a region in the xy-plane then the area of its image T (G) in theuv-plane is not more than π.

2b. Let f(x) = 1− ex + e2x − e3x + · · · . Find the domain and the range of f .

3. Let D be the region in space bounded on the top by the sphere x2 + y2 + z2 = 2 and on thebottom by the paraboloid z = x2+ y2. Fill in the boxes in parts (a-c) so that the right sides of


the equalities become iterated integrals expressing the volume V of D in the given coordinatesand orders of integration. No explanation is required.

a. V =

ˆ ˆ ˆdz dy dx

b. V =

ˆ ˆ ˆdz dr dθ

c. V =

ˆ ˆ ˆdρ dϕ dθ

+

ˆ ˆ ˆdρ dϕ dθ

4. In each of the following indicate whether the given series converges or diverges, and alsoindicate the best way of determining this by choosing exactly one of the tests and �lling in thecorresponding blank if any. (You must choose a test to get any points.)

a.∞∑n=1

1

n2 +√n

� converges � diverges

� nth Term Test � Direct Comparison Test with∑

� Integral Test � Limit Comparison Test with∑

� Ratio Test � nth Root Test � Alternating Series Test

b.∞∑n=0

3n sin(π−n) � converges � diverges





c.∞∑n=0

2n2

n!� converges � diverges




d.∞∑n=2

(−1)n lnn � converges � diverges




e.∞∑n=2

1

n(lnn)√2





5. Let f(x) =∞∑n=0

cnxn for all x in the interval of convergence of the power series where c0 = 1

and cn+1 = cn

(1 +

1

n+ 1

)−(n+1)

for n ≥ 0.

a. Find c1, c2 and c3.

b. Find the radius of convergence of the power series∞∑n=0

cnxn.

c. Find f ′′(0).


d. Express the sum of the series∞∑n=1

n2cn in terms of A = f(1), B = f ′(1) and C = f ′′(1).



xn2

2nn.

a.Write the �rst three nonzero terms of the power series.

b. Find the radius of convergence of the power series.

c. Find the exact value of the sum of the power series at x = −1 .

2a. Find the coe�cient of x2011 in the Maclaurin series of f(x) = xe−x2.

2b. Exactly one of the following statements is true. Choose the true statement and mark thebox in front of it with a 3 .

� If∞∑n=1


(−1)n+1an converges.

� If∞∑n=1


(−1)n+1a3n converges.

Now either

� Prove the true statement, or

� Give an example that shows the other statement is not true.

Indicate the task you choose with a 7 .

3a. Find the equation of the plane containing the line L1 : x = 2t+ 3, y = 4t− 1, z = −t+ 2 ,−∞ < t <∞ , and parallel to the line L2 : x = 2s+ 3, y = s+ 2, z = 2s− 2 , −∞ < s <∞ .

3b. When a wheel of unit radius rolls along the x-axis the path traced by a point P on itscircumference is given by r = (t − sin t) i + (1 − cos t) j , −∞ < t < ∞ . Find the distancetraveled by P during one full turn of the wheel.

4a. Show that lim(x,y)→(0,0)

x3y2

x6 + y2= 0 .

4b. Show that lim(x,y)→(0,0)

x6y4

(x6 + y2)3does not exist.

5. Find all possible values of the constants a and b such that the function

f(x, y) = yaebx2/y


satis�es the equation∂2f

∂x2+

2

x

∂f

∂x=∂f

∂y

for all (x, y) with x > 0 and y > 0 .


1. Let P0(2, 2, 1) and suppose that f(x, y, z) and g(x, y, z) are di�erentiable functions satisfyingthe following conditions:

i. f(P0) = 1 and g(P0) = 6.

ii.∂g

∂x

∣∣∣∣P0

= −1.

iii. At P0, f increases fastest in the direction of the vector A = 4i− j− 8k and its derivativein this direction is 7.

iv. The tangent plane of the surface de�ned by the equation

f(x, y, z) + 2g(x, y, z) = 13

at the point P0 has the equation 5x+ y − z = 11.

Find∂g

∂z

∣∣∣∣P0

.

2. Each of the following functions has a critical point at (0, 0). Indicate the type of this criticalpoint by marking the corresponding box with a 7 . No explanation is required. No partialcredit will be given.

a. f(x, y) = x3y3 has a

� local maximum

� local minimum

� saddle point

� none of the above

at (0, 0).

b. f(x, y) = 1− x2y2 has a

� local maximum

� local minimum

� saddle point


at (0, 0).


c. f(x, y) = y2 − yx2 has a

� local maximum

� local minimum

� saddle point


at (0, 0).

d. f(x, y) = x2 − x2y + y2 has a

� local maximum

� local minimum

� saddle point


at (0, 0).

3. A �at circular plate has the shape of the region x2 + y2 ≤ 1. The plate, including theboundary where x2 + y2 = 1, is heated so that the temperature at a point (x, y) is

T (x, y) = x2 + 2y2 − x.

Find the temperatures at the hottest and coldest points on the plate.


a.

¨R

√x2 + y2 dA where R is the region shown in the �gure.

x

y

x2 + y2 = 4

y =√3x

R

b.

ˆ 1

0

ˆ 1/x2

1

xy2e−y2 dy dx

5. Let D be the region in space lying inside the sphere x2 + y2 + z2 = 4, outside the conez2 = 3(x2 + y2), and above the xy-plane. Fill in the boxes in parts (a-c) so that the rightsides of the equalities become iterated integrals expressing the volume V of D in the givencoordinates and orders of integration. No explanation is required.


a. V =

ˆ ˆ ˆdρ dϕ dθ

b. V =

ˆ ˆ ˆdz dr dθ

+

ˆ ˆ ˆdz dr dθ

c. V =

ˆ ˆ ˆdr dz dθ

y

z

x

Spring 2011 Final

1a.Write the �rst three nonzero terms of the Maclaurin series ofx

1 + ax2where a is a constant.


1b.Write the �rst three nonzero terms of the Maclaurin series of sin(bx) where b is a constant.

1c. Find the constants a, b, c, d ifx

1 + ax2− sin(bx) = x3 + cx4 + dx5 + · · · on some open

interval containing x = 0.

2. Let u = x2y3, v = sin(πx), and z = f(u, v) where f is a function with continuous secondorder partial derivatives satisfying:

f(4, 0) = 10 fu(4, 0) = 5 fv(4, 0) = 7

fuu(4, 0) = −2 fuv(4, 0) = −1 fvv(4, 0) = 3

a. Find∂z

∂x

∣∣∣∣(x,y)=(−2,1)

.

b. Find∂2z

∂y∂x

∣∣∣∣(x,y)=(−2,1)

.

3. Evaluate

¨R

e(y−x)/(y+x) dA where R is the region shown in the �gure.

x

y

1

1

R

4a. Evaluate

˚D

dV√x2 + y2 + (z − 2)2

where D is the unit ball x2 + y2 + z2 ≤ 1.

4b. Evaluate the line integral

‰C

(6xy + sin(x2)) dx+ (5x2 + sin(y2)) dy where C is the

boundary of the region R shown in the �gure.



x

y

x2 + y2 = 4

x2 + y2 = 1

R

C

5. Consider the parametrized surface S : r = u2i +√2uvj + v2k,−∞ < u < ∞ , 0 ≤ v < ∞ .

Find the area of the portion of the surface S that lies inside the unit ball x2 + y2 + z2 ≤ 1.



xy2

x6 + y2= 0 .


xy

x6 + y2does not exist.

1c. Consider lim(x,y)→(0,0)

x |y|a

x6 + y2where a is a constant.

There is a real number A such that this limit is 0 if a > A, and this limit does not exist ifa < A. What is A?

Write your answer here � A =

No explanation is required and no partial points will be given in this part.

2. Assume that f(x, y, z) is a di�erentiable function and at the point P0(1,−1, 2), f increasesfastest in the direction of the vector A = 2i+ j− 2k.

Exactly one of the following statements can be true about this function.

• Mark this statement with an 3 and �nd (∇f)P0 assuming the statement to be true.

• Mark the other statement with an 7 and explain why it cannot be true.

� The directional derivative of f at P0 in the direction of the vector B = 2i+ 6j+ 3k is 5.

� The directional derivative of f at P0 in the direction of the vector B = 3i+ 2j+ 6k is 5.

3. Find the points on the surface xy+ yz+ zx− x− z2 = 0 where the tangent plane is parallelto the xy-plane.


4. Find all possible values of the constants C and k such that the function f(x, y) = C(x2+y2)k

satis�es the equation fxx + fyy = f 3 for all (x, y) = (0, 0).

5. Find the absolute maximum and minimum values of the function f(x, y) = 2x3+2xy2−x−y2on the unit disk D = {(x, y) : x2 + y2 ≤ 1}.


1. Evaluate the integral

ˆ π

0

ˆ 1

x/π

y4 sin(xy2) dy dx .

2. Let D be the region in space bounded by the plane y + z = 1 on the top, the paraboliccylinder y = x2 on the sides, and the xy-plane at the bottom.

• Choose two of the following rectangular boxes by putting a 7 in the small square infront of them, and then

• choose one of the orders of integration in each of the selected boxes by putting a 7 inthe small square in front of them.



Express the volume V of the region D as iterated integrals in both of your selected orders ofintegration (a) and (b). (Do not evaluate the integrals! )

3.V =

ˆ √3

−√3

ˆ √3−x2

−√3−x2

ˆ √12−x2−y2

x2+y2dz dy dx

expresses the volume V of a region D in space as an iterated integral in Cartesian coordinates.

Fill in the boxes in (a) and (b) so that the right sides of the equalities become iteratedintegrals expressing the volume of D in cylindrical and spherical coordinates, respectively. Noexplanation is necessary in this question. (Do not evaluate the integrals! )

a. V =

ˆ ˆ ˆdz dr dθ

b. V =

ˆ ˆ ˆdρ dϕ dθ


+

ˆ ˆ ˆdρ dϕ dθ

4. Find the value of the line integral

‰C

(3x2y2 + y) dx+ 2x3y dy

where C is the cardioid r = 1 + cos θ parameterized counterclockwise.

5. Let D be the closed ball x2 + y2 + z2 ≤ 16.

a. Show that for any vector �eld F that has continuous partial derivatives and satis�es thecondition |F| ≤ 5 on D, ˚

D

∇ · F dV ≤ 320π.

b. Give an example of a vector �eld F that has continuous partial derivatives and satis�esthe condition |F| ≤ 5 on D such that

˚D

∇ · F dV = 320π,

and verify that the equality holds.

Spring 2010 Final

1. Let T be a transformation from the uv-plane to the xy-plane given by x = f(u, v) andy = g(u, v) where f and g are functions with continuous partial derivatives. Assume that Tsatis�es the condition that

(Area of T (G)) =

¨G

(u2 + v2) du dv

for every closed subset of the uv-plane on which T is one-to-one.

a. Let G = {(u, v) : 1 ≤ u2 + v2 ≤ 4 , u ≥ 0 and v ≥ 0}. Find the area of the image of Gunder T assuming that T is one-to-one on G.

b. If f(u, v) = uv, �nd a g(u, v) that satis�es the conditions of the question. (You do not

have to explain how you found g(u, v), but you must verify that the conditions are satis�ed.)


a.∞∑n=0

1

(ln 2)n


b.∞∑n=2

(lnn)3

n2

c.∞∑n=0

(5n)!

30n(2n)!(3n)!


a.∞∑n=1

(−1)n+1 cos

(π

n

)

b.∞∑n=1

1

n+√n sinn

c.∞∑n=1

5n − 2n

7n − 6n


xn

9n2 − 1.

a. Find the radius of convergence of the power series.

b. Determine whether the power series converges or diverges at the right endpoint of itsinterval of convergence. If it converges, determine the type of convergence.

c. Determine whether the power series converges or diverges at the left endpoint of itsinterval of convergence. If it converges, determine the type of convergence.

5. In parts (a-b) of this question, if the series converges, write the exact sum of the seriesinside the box; and if the series diverges, write Diverges inside the box. No explanation isrequired. No partial points will be given.

a.∞∑n=0

1

4n(2n+ 1)=

b.∞∑n=2

1

n2 − 1=

In parts (c-d) of this question, if there exists a sequence {an} satisfying the given conditions,write its general term inside the box; and if no such sequence exists, write Does Not Exist

inside the box. No explanation is required. No partial points will be given.

c. 1 ≤ an < an+1 for all n ≥ 1 and limn→∞

an = ∞ .


an =

d. limn→∞

nan = 0 and∞∑n=2

an diverges.

an =



a.∞∑n=0

(−1)n+12n

3n

b.∞∑n=1

1

1 + (lnn)2

c.∞∑n=1

cos(1/n)


a.∞∑n=0

n!(n+ 1)!

(2n+ 1)!

b.∞∑n=0

10−n2/(n+1)

c.∞∑n=1

(e1/n2 − 1)


xn

(n2 + 1)(2n + 1).


b. Determine whether the power series is absolutely convergent, conditionally convergent ordivergent at the right endpoint of its interval of convergence.


c. Determine whether the power series is absolutely convergent, conditionally convergent ordivergent at the left endpoint of its interval of convergence.

4. Find the coe�cient of the �rst nonzero term in the Maclaurin series generated by

f(x) = sin x− x

1 + x2/6.

5a. Determine whether the sum of the series∞∑n=0

(−4)n

n!(n+ 1)!is positive or negative.

5b. Find the sum of the series∞∑n=2

1

2n(n2 − 1).



x4y4

x2 + y4= 0 .


x4y4

(x2 + y4)3does not exist.

2. A di�erentiable function f(x, y, z) increases fastest at the point P0(2, 5,−1) in the directionof i+ 2j− 3k and at a rate of 7.

a. Find∂f

∂z

∣∣∣∣P0

.

b. Find the equation of the tangent plane to the level surface of f passing through the pointP0 .

3a. Show that if f(z) is a di�erentiable function and u(x, y) = f(x2 − y2), then yux + xuy = 0for all (x, y).

3b. Show that the parametric curve r(t) = et cos t i+ et sin t j , −∞ < t <∞, cuts every circlewith center at the origin at the same angle, and �nd this angle.

4. Find and classify the critical points of f(x, y) = x3 − 6x+ x2y2.

5. Evaluate the following integrals

a.

ˆ π1/6

0

ˆ π1/3

x2

x3 sin(y3) dy dx

b.

¨R

x2 sin(x2 + y2) dA where R = {(x, y) : x2 + y2 ≤ π}.


Spring 2009 Final

1. In each of the following indicate whether the given series converges or diverges, and alsoindicate the best way of determining this by marking the corresponding boxes and �lling in thecorresponding blanks.

a.∞∑n=1

1

n1+1/n� converges � diverges


� Ratio Test � Limit Comparison Test with∑

� nth Root Test � Alternating Series Test

b.∞∑n=2

n√n5 − 1





c.∞∑n=1

n√2

2√n





d.∞∑n=2

(−1)nn

lnn� converges � diverges




e.∞∑n=1

en − 2n

πn − 3n� converges � diverges





2. Find the closest and farthest points on the sphere x2+y2+z2 = 4 to the point P (3, 1,−1) .

3a. Evaluate the integral

˚D

z2

(x2 + y2 + z2)3dV where D = {(x, y, z) : x2 + y2 ≥ 1} .

3b. Express the iterated integral in cylindrical coordinates

ˆ π

0

ˆ 1

0

ˆ √4−r2

0

dz dr dθ in terms of

iterated integrals in spherical coordinates. Do not evaluate.

4a. Find a function f(x, y) such that, for any region G in the �rst quadrant of the xy-plane, the

double integral

¨G

f(x, y) dx dy gives the area of T (G) where T is the transformation u = x2/y,

v = x/y2.

4b. Evaluate

ˆC

F · dr where F = 3x2y2z i + 2(x3z + z2)y j + (x3 + 2z)y2 k and

C : r = (t3 − 2t) i+ (t4 − 4t2 − 1) j+ cos(πt)k , 0 ≤ t ≤ 1 .

5. Let C be the unit circle in the xy-plane, and S be the boundary of the squareK = {(x, y) : 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1}. If a is a constant such that

‰C

(y3 + xy2 + xy) dx+ (3xy2 + x2y + ax) dy = 1 ,

evaluate ‰S

(y3 + xy2 + xy) dx+ (3xy2 + x2y + ax) dy .


1. a. Evaluate limn→∞

(n!)2(3n)n

nn(2n)!.

b. Does the series∞∑n=2

(sinn

n2+

1

n(lnn)2

)converge or diverge? Justify!

2. a. Find the Taylor series generated by f(x) = (1 + x2)1/3 at x = 0.

b. Use the series in part (a) to estimate

ˆ 1/2

0

(1 + x2)1/3 with error less than 0.01.


3. Find the radius and the interval of convergence of the series∞∑n=1

(−1)n(x+ 2)n√n(n+ 1)

.

4. a. Find parametrization for the line in which the planes 5x − 2y = 11 and 4y − 5z = −17intersect.

b.Write u = j+ k as the sum of a vector parallel to v = i+ j and a vector orthogonal to v.

5. a. Let r1(t) = cos t i + sin t j + tk and r2(t) = sin s i + cos sk be two curves. Find theintersection points of r1 and r2. Moreover, �nd the angle between the tangent vectors at eachintersection point.

b. Find the point of intersection of the lines x = 2t+1, y = 3t+2, z = 4t+3 and x = s+2,y = 2s+ 4, z = −4s− 1. Then �nd the plane determined by these lines.


1. Let D be the solid bounded by the surfaces x2 + y2 + z2 = 9 and x2 + y2 + z2 = 1, andlying above the surface z2 = 3(x2 + y2) with z ≥ 0. Write down three integrals in rectangular,cylindrical, and spherical coordinates that give the volume of the solid. Do not evaluate theseintegrals.

2. a. Evaluate

ˆ 9

0

ˆ 3

√y

sin(x3) dx dy .

b. Evaluate

ˆ 2

0

ˆ √1−(x−1)2

0

x+ y

x2 + y2dy dx .

3. Find the shortest distance from the origin to the surface xyz2 = 2. (Explain why it is theshortest and not the longest.)

4. a. Let f(x, y) = x2y − 2xy + y2 − 15y . Find and classify the critical points of f .

b. Find the direction of most rapid increase for f at the point (1, 1) and the rate of changeof f in this direction.

5. a. Let z = f(x, y) be a di�erentiable function of two independent variables x and y such thatf(2, 1) = 3, fx(2, 1) = 2, fy(2, 1) = −1. De�ne another function z = g(x, y) of two independentvariables x and y as follows:

g(x, y) = f

(2

x2 + y2,y

xexy)

Find the equation of the tangent plane to the surface z = g(x, y) at the point (x, y) = (1, 0).

b. Using the tangent plane in part (a) approximate g(1.1,−0.2).


Spring 2008 Final

1. Find the radius and the interval of convergence of the series

∞∑n=2

(−2)n(n+ 1)

(3n2 + 1) lnn(x− 5)n .

Explain your reasoning and be sure to check the endpoints.

2. Find the absolute maximum and minimum values of the function f(x, y) = yx2 − y2 + 4 onthe unit disk D = {(x, y) : x2 + y2 ≤ 1}.

3. a. Find the work done by the vector �eld

F(x, y, z) = (2xz + 2)i+ (2y + z)j + (x2 + y)k

over the curve r(t) = (sin t+ 2 cos t− 2)i+ et2−πtj+ (t/π − 5)k for t in [0, π]. (Hint: First try

to answer the question �Is F a conservative �eld?")

b. Integrate f(x, y, z) = x2 + y + 3z over the line segment joining (0, 1, 2) to (−1,−1,−2).

4. Let C be a curve that encloses a region R such that the area of the region R is 10π and theinterior of the region contains the unit disk D = {(x, y) : x2 + y2 ≤ 1}. Compute the integral

‰C

x− 2y

x2 + y2dx+

(2x+ y

x2 + y2+ 3x

)dy .

(Hint: You might want to use Green's Theorem to compute this integral, but note the problemsabout the point (x, y) = (0, 0). So you have to use Green's Theorem carefully.)

5. a. Evaluate

‰C

F·dr where C is the intersection of z = x2 + y2 + 1 and z = 2y + 1 oriented

clockwise as viewed from above and F =< sin(x2), y3, z ln z − x > .

b. Let Q be the solid bounded by the paraboloid z = 4 − x2 − y2 and the xy-plane. Findthe outward �ux of the vector �eld F =< x3, y3, z > over the boundary of Q.


1. Determine if each of the following series is convergent or divergent.

a.∞∑n=3

1

n · lnn · (ln(lnn))3

b.∞∑n=2

(lnn)n

n


c.∞∑n=1

(−1)n+1

ˆ n+1

n

e−x

xdx


a.∞∑n=1

(−1)n+1nn

n!

b.∞∑n=0

(4n)!

10nn!(3n)!

c.∞∑n=0

πn sin2(2−n)

3. Consider the series∞∑n=2

1

(n− 1)√n+ 1 + (n+ 1)

√n− 1

.

a. Show that this series is convergent.

b. Find the sum of this series by interpreting it as a telescoping series.

4a. Find all solutions of the equationx

4=

∞∑n=1

n(n+ 1)

xn.

4b. Show that if 1 > an > 0 for all n ≥ 1 and the series∞∑n=1

an converges, then the series

∞∑n=1

an1− an

converges.

5a. Find the sum of the series∞∑n=0

(n− 1)(n+ 1)

n!exactly.

5b. Determine whether the improper integral

ˆ 1

0

dx

x− sinxconverges or diverges.


1. Find and classify the critical points of f(x, y) = x2y2 − 2x2 + x3 − 4y2.

2. Find the equations of (a) the tangent plane and (b) the normal line to the surface x2 +3y2 − 4z2 = 5 at the point P (3, 2,−2).

3. Find the points on the surface xyz = 1 closest to the origin.


4a. The derivative of a di�erentiable function f(x, y) at a point P in the direction of i + j is2, and in the direction of 3i − 4j is −3/

√2. Find the derivative of f at P in the direction of

7i− j.

4b. Let h(t) be a di�erentiable function of t and let g(x, y) = 3h(x2 − y2)− h(2xy). Find h′(3)if (∇g)(2,1) = 6i+ 5j.


a.

ˆ π2

0

ˆ π

√y

x5 sin

(x2y

π3

)dx dy

b.

¨R

sin(x+ y) dA where R = {(x, y) : x+ y ≤ π/2, x ≥ 0 and y ≥ 0}

Spring 2007 Final

1. Compute the inward �ux of the vector �eld F = xi + yj + k across the closed surface Scomposed of the portion of the paraboloid z = x2 + y2 lying below the plane z = 4, and theportion of the plane z = 4 lying inside the paraboloid z = x2 + y2.

2a. Suppose f(x, y) is a function with continuous second order partial derivatives on the entireplane, and fxx + fyy = ex

2+y2 for all (x, y). Evaluate the line integral

‰C

∇f · n ds

where C is the circle x2 + y2 = 1 parametrized counterclockwise, and n is the outward unitnormal �eld on C.

2b. Evaluate the integral

¨R

(x− y)2 cos(x+ y) dA, where R is the square with vertices

(π/2, 0), (0, π/2), (−π/2, 0) and (0,−π/2), using the change of variables u = x+y and v = x−y.

3. Evaluate the integral ˚D

1

(x2 + y2 + z2)2dV

where D is the region lying outside the cylinder x2 + y2 = 1 and inside the half-cone z =√x2 + y2.

4a. Estimate the value of the integral

ˆ 1

0

x sin(x3) dx with an error of magnitude less than

10−3 using series.

4b. Evaluate the limit

limx→0

ex − e−x − 2x

x− sinx

using series.


5a. Find the exact sum of the series∞∑n=0

32n+1

(2n+ 1)52n+1.

5b. Suppose that g(t) is a di�erentiable function of t, and h(x, y) = y g(y/x). Find hy(3, 2) ifh(3, 2) = 7 and hx(3, 2) = 4.


1. Determine whether each of the following is convergent or divergent.

a.

ˆ 1

0

dx

ex − e−x

b.

ˆ ∞

1

dx

ex − e−x

c.∞∑n=1

1

1 + 2 + · · ·+ n

d.∞∑n=1

ln

(1 +

1

n

)

e.∞∑n=1

n5

3n

f .∞∑n=1

2nn

4n + n2

g.∞∑n=0

(2n)!

5n(n!)2

h.∞∑n=1

(−1)n+1 arctann

2. Let un =1 · 3 · 5 · · · · · (2n− 1)

2 · 4 · 6 · · · · · (2n)for n ≥ 1.

a. Show that1

2n≤ un for all n ≥ 1.

b. Show that un <1√nfor all n ≥ 1.

c. Show that un > un+1 for all n ≥ 1.

d. Show that the series∞∑n=1

un diverges.

e. Show that the series∞∑n=1

(−1)n+1un converges.


f . Choose the correct statement: (No explanation is required for this part.)

� The series∞∑n=1

(−1)n+1un converges absolutely.

� The series∞∑n=1

(−1)n+1un converges conditionally.


1. For each of the following, write the exact sum of the series in the box if it converges andwrite diverges if it diverges. No further explanation is required. No partial credit will begiven.

a.∞∑n=0

(−1)n3n

n!=

b.∞∑n=0

(−1)n4nπ2n

9n(2n)!=

c.∞∑n=1

n

2n(n+ 1)=

d.∞∑n=0

(−1)n

3n(2n+ 1)=


(−1)nxn

n2 + 2n.

a. Find the radius of convergence of the series.

b. Determine whether the series converges absolutely, converges conditionally or diverges atthe left endpoint of its interval of convergence.

c. Determine whether the series converges absolutely, converges conditionally or diverges atthe right endpoint of its interval of convergence.

3. Assume that f(x, y, z) is a di�erentiable function and P (2, 1,−1) is a point such that

• (∇f)P is parallel to the xz-plane,

• The derivative of f at P in the direction of A = 2i− 3j+ 6k is 2,

• The derivative of f at P in the direction of B = 2i+ 2j+ k is −7.


a. Find (∇f)P .

b. In which direction does f increase fastest at P? What is the rate change of in thisdirection?

4. Suppose that f(x, y) is a twice-di�erentiable function satisfying f(1, 2) = 3, fx(1, 2) =5, fy(1, 2) = −4, fxx(1, 2) = 7, fxy(1, 2) = fyx(1, 2) = −1 and fyy(1, 2) = −2.

a. Findd

dtf(t2, 2t3)

∣∣∣∣t=1

.

b. Assume that g(t) is a twice-di�erentiable function such that g(1) = 2 and f(t, g(t)) = 3for all t. Find g′′(1).

5. Find the absolute maximum and the absolute minimum values of f(x, y) = 4x3+9y2−18xyon the square R = {(x, y) : 0 ≤ x ≤ 2 and 0 ≤ y ≤ 2}.

Spring 2006 Final

1. For each of the following, write the exact sum of the series in the box if it converges andwrite diverges if it diverges. No further explanation is required. No partial credit will begiven.

a.∞∑n=0

1

2n=

b.∞∑n=1

1

2nn=

c.∞∑n=0

1

2nn!=

d.∞∑n=0

(−1)n(n+1)/2 πn

4nn!=

2. Find the points P0 on the paraboloid z = x2 + y2 such that the tangent plane to theparaboloid at P0 passes through the points (1, 0, 0) and (0, 2, 0).

3. Evaluate the following iterated integrals.

a.

ˆ 8

0

ˆ 2

3√x

dy dx

y4 + 1


b.

ˆ 2

0

ˆ 0

−√

1−(y−1)2

x2

ydx dy

4. Express the triple integral ˚D

z

(x2 + y2 + z2)5/2dV

where D = {(x, y, z) : x2 + y2 ≤ 1 and z ≥ 1} as an iterated integral in (a) cylindrical and (b)spherical coordinates, and (c) evaluate it using any coordinate system you wish.

5a. Evaluate the line integral ‰C

−y dx+ x dy

1 + x2 + y2

where C is the unit circle x2 + y2 = 1.

5b. Find a function f(x, y) such that

‰C

−y dx+ x dy

1 + x2 + y2=

¨R

f(x, y) dA

for every simple closed curve C in the plane and region R it encloses.


1. Determine whether each of the following series is convergent or divergent. State clearly thename and the conditions of the test you are using.

a.∞∑n=1

n√2n4 + 1

b.∞∑n=2

1

n lnn

c.∞∑n=1

n 2n

3nd.

∞∑n=1

2n

n 3n

e.∞∑n=0

(3n)!

7nn!(2n)!f .

∞∑n=1

(−1)n+1 n√n2 + 1


(−1)n3nxn

n2 + 1.


b. Determine whether the power series is absolutely convergent, conditionally convergent ordivergent at the left endpoint of its interval of convergence.

c. Determine whether the power series is absolutely convergent, conditionally convergent ordivergent at the right endpoint of its interval of convergence.


3. Find the sums of the following series exactly:

a.∞∑n=1

n2

2nb.

∞∑n=1

1

2n n(n+ 1)

4. In each of the following, if there exists a sequence {an} which satis�es the given condition,then give an example of such a sequence; otherwise write Does Not Exist. No furtherexplanation is required.

a. 0 <an+1

an< 1 for all n ≥ 1 and

∞∑n=1

an is divergent.

b. The sequence {an} is convergent and∞∑n=1

an is divergent.

c.∞∑n=1

an is divergent and∞∑n=1

(an)2 is convergent.

d.1

n< an for all n ≥ 1 and

∞∑n=1

an is convergent.

e.∞∑n=1

an is convergent and∞∑n=1

(an)2 is divergent.


1. Let f(x, y, z) = x2y − y3z + xz , P0(1, 2, 3) and A = 2i− j− 2k .

a. Find the directional derivative of f at P0 in the direction of A .

b. Find the direction at P0 in which f increases the fastest.

2. Find and classify the critical points of the function f(x, y) = y2 − x2y − 1

2y4 .

3. Let x = s2t , y = st2 and z = f(x, y) where f is a function with continuous second orderpartial derivatives satisfying

f(4, 2) = 1 , fx(4, 2)= −3 , fy(4, 2) = 5 ,

fxx(4, 2) = −7 , fxy(4, 2)= 11 , fyy(4, 2) = −13 .

Find∂2z

∂s2

∣∣∣∣(s,t)=(2,1)

.


ˆ 1

0

ˆ 1

√x

ey3

dy dx .



¨R

x√x2 − y2

dA where R = {(x, y) : 1 ≤ x2 + y2 ≤ 4 and − x ≤

y ≤ x} .

5a. Find the volume of the region in the �rst octant bounded by the cylinders z = y2 andz = 1− x2 , and the coordinate planes.


˚D

1

(x2 + y2 + z2)2dV where D is the region inside the cylinder

x2 + y2 = 1 and above the hemisphere z =√

2− x2 − y2 .

Spring 2005 Final

1. Determine whether each of the following series is absolutely convergent, conditionallyconvergent or divergent.

a.∞∑n=0

101102n

n!

b.∞∑n=1

(−1)n

n

c.∞∑n=0

(2n)!

(n!)2

d.∞∑n=2

1

n lnn

e.∞∑n=1

(−1)n(n+1)/2

n2

2a. Find (∇f)(1,2) where f(x, y) = xy2 + x3 − y.

2b. Find the equation of the tangent plane to the surface z = xy2 + x3 − y at (1, 2, 3).

2c. Find the equation of the tangent line to the curve xy2 + x3 − y = 3 at (1, 2).

3. Let V be the volume of the region bounded on the sides by the cylinder x2 + y2 = 3, at thetop by the plane z = 1 and at the bottom by the plane z = 0. In each of the following, �llin the boxes so that the right side of the equality becomes the iterated integral for V in thecorresponding coordinate system and the order of integration.

a. V =

ˆ ˆ ˆdz dy dx


b. V =

ˆ ˆ ˆdz dr dθ

c. V =

ˆ ˆ ˆdρ dϕ dθ

+

ˆ ˆ ˆdρ dϕ dθ

4a. Let u = x2 − y2 and v = 2xy . Compute the Jacobian∂(u, v)

∂(x, y).


¨R

(x2 + y2) dA where R = {(x, y) : −1 ≤ x2 − y2 ≤ 1 , xy ≤ 1 , x ≥

0 and y ≥ 0} .

5. S : The surface cut from the cone z2 = x2 + y2, z ≥ 0 , by the cylinderx2 + y2 = 2x

n : The unit normal vector �eld on S pointing away from the positive z-axisC : The curve of intersection of the cone z2 = x2 + y2 , z ≥ 0 , and the cylinder

x2 + y2 = 2x , parametrized counterclockwise as seen from a point onthe positive z-axis

F = yz i− xz j

a. Find the area of S.

b. Choose and evaluate one of the integrals

¨S

∇× F · n dσ and

‰C

F · dr .

c. Use the result of part (b) to �nd the value of the other integral.

6. Show that there is a constant c such that every region D in space enclosed by an orientedsurface S with outward unit normal vector �eld n satis�es the equality¨

S

r · n dσ = c V

where r = xi+ yj+ zk and V is the volume of D.



1. Find all values of the constant p for which the improper integralˆ ∞

0

dx

xp 3√x2 + 1

converges.

2. Determine whether each of the following series is convergent or divergent:

a.∞∑n=2

sin(πn

)

b.∞∑n=0

3n

πn


a.∞∑n=1

(−1)n ln

(n

n+ 1

)

b.∞∑n=1

(−1)n+1

n√n


a.∞∑n=0

3n(n!)2

(2n+ 1)!

b.∞∑n=1

(n

n+ 1

)n2

5a. Determine the interval of convergence of the power series∞∑n=1

xn√n

and determine the type

of convergence at each point.

5b. How many terms of∞∑n=1

(−1)n√n

should be used to estimate its sum with an error less then

0.01?


1a. Find the Taylor series generated by f(x) = sin x centered at x =π

4.

1b. Show that this series converges to f(x) for all x.

2. Assume thatxez + y2z = sin x+ 1


de�nes x as a di�erentiable function of y and z and �nd∂x

∂yand

∂x

∂zat the point (x, y, z) =

(0,−1, 1).

3. Find and classify the critical points of f(x, y) = x3 − 3xy + y3.

4. Consider the function f(x, y, z) = x3z + ln(x2 + y2) + cz2 where c is a constant.

a. Find the value of c if the tangent plane to the level surface f(x, y, z) = f(1,−1, 2) at thepoint P0(1,−1, 2) passes through the origin.

b. Let c = 1. Find the directional derivative of f at P0(1,−1, 2) in the direction of A =−5i+ j+ 7k.

5. Find∂2w

∂t∂s

∣∣∣∣(t,s)=(2,1)

if x = t2, y = ts, z = s2, and w = f(x, y, z) is an in�nitely di�erentiable

function satisfying:

fx(4, 2, 1) = 2 fy(4, 2, 1) = −3 fz(4, 2, 1) = 5

fxx(4, 2, 1) = −7 fxy(4, 2, 1) = 11 fxz(4, 2, 1) = −13

fyy(4, 2, 1) = 17 fyz(4, 2, 1) = −19 fzz(4, 2, 1) = 23

Spring 2004 Final

1. Prove that ˆ 1

0

ln(1 + x)

xdx =

∞∑n=1

(−1)n+1

n2

and determine how many terms should be used to estimate this sum with an error less than10−2 .

2. Find the absolute maximum and the absolute minimum values of the function f(x, y) =(y − x2)(y − 2x2) on the square R = {(x, y) : |x| ≤ 1 and |y| ≤ 1} .

3. Find the absolute minimum value of the function f(x, y, z) = x2 + y2 + z2 on the surface1

x+

1

y+

1

z= 1 for x, y, z > 0 .

4. Evaluate the following integrals:

a.

¨R

xe−y dA where R = {(x, y) : 1 ≤ x2 + y2 ≤ 4 and x ≥ 0} .

b.

ˆ π3/8

0

ˆ 1/ 3√y

2/π

sin(x2y) dx dy


5. Let C be the unit circle. Find the value of the constant a such that‰C

((x+ ay) dx+ (x2 + 3x+ y) dy

)= 0 .

where C is oriented counterclockwise.


1a. Find the equation of the line of intersection of the planes 4x+y+z = 0 and 2x+3y−2z = −5.

1b. Find the distance between the planes x− 2y + z = 3 and x− 2y + z = 4.

2. Find the point of intersection of the plane passing through the points P1(0,−2,−6),P2(−1, 1, 5) and P3(2, 3,−6), and the line passing through the points P4(2,−1, 0) andP5(3,−4, 3).

3. Determine whether each of the following improper integrals is convergent or divergent.

a.

ˆ ∞

1

3√x4 − 1

x3dx b.

ˆ 1

0

dx

lnx


a.

ˆ 1

0

√x lnx dx b.

ˆdt

t+√1− t2

5. Evaluate the improper integralˆ ∞

0

dx

(ax2 + 1)(x+ a)

where a is a positive constant.


1. Find and classify the critical points of the function f(x, y) = 3x2y + y3 − 108y.

2. Find the directional derivative of f(x, y, z) = 3x2yz+2yz2 at P0(1, 1, 1) in a direction normalto the surface x2 − y + z2 = 1.

3. Find the absolute maximum value of f(x, y, z) = xy3z5 on the sphere x2 + y2 + z2 = 1 usingthe Lagrange multipliers method.

4a. Determine if the limit lim(x,y)→(0,0)

sin2 x sin2 y

(x2 + y2)2exists.


4b. Estimate the change in z = ln(x2+y2) corresponding to the change dx = 0.2 and dy = −0.1from (x, y) = (3, 4).

5a. Find the values of the constant c for which w = e−2t sin cx cos y satis�es 5wt = wxx + wyy

for all (x, y, t).

5b. Find fxx at the point (x, y) = (√3, 1) if z = f(x, y), x = r cos θ, y = r sin θ, and zr =

1, zθ = −2, zrr = 0, zrθ = 0, zθθ = 0, zrrr = 3, zrrθ = −5, zrθθ = 7, zθθθ = −11 at this point.

Spring 2003 Final

1a. Find the points P0(x0, y0, z0) on the sphere x2 + y2 + z2 = 9 such that the tangent plane tothe sphere at P0 passes through the points (4, 0, 1) and (0, 0, 9).

1b. Find the cosine of the acute angle between the tangent planes to the paraboloid 2z = x2+y2

at the points of intersection of the paraboloid and the line x = t, y = −t, z = t+2,−∞ < t <∞.

2. Find the values of the constant k for which the function f(r, θ) = rk cos(5θ) satis�es theequation fxx + fyy = 0 for all (x, y) = (0, 0) where x = r cos θ and y = r sin θ.


a.

ˆ π3

0

ˆ 1/ 3√y

1/π

cos(x2y) dx dy

b.

ˆ ∞

0

ˆ x

0

1

(1 + x2 + y2)2dy dx

4. Evaluate

¨R

y−1(ex + e−x)−2 dA where R = {(x, y) : ex ≤ y ≤ 4ex and e−x ≤ y ≤ 4e−x}.

5. Find the values of the constants a and b for which the limit

limx→0

x−3 sin(ax+ x3)− 1

(1 + x2) ln(1 + bx2)− 2x2

is a nonzero real number and compute this number.



a.∞∑n=1

sin(πn

2) sin(

2

πn)


b.∞∑n=1

2nn100

3n


1

n2 − 9

a. Find sn.

b. Find the sum of the series.

3. Find the radius of convergence R and the interval of convergence I of the power series

∞∑n=0

(x− 3)n

4n + n3

and determine the type of convergence at each point of I.

4. Find the values of the following expressions:

a. limx→0

ex2 − 1− x2

(1 + x4)1/2 − 1

b.∞∑n=0

(−1)n

22n+3(n+ 2)

5. a. Show that if∑∞

n=1 an is convergent and an ≥ 0 for all n ≥ 1, then∑∞

n=1 a2n is also

convergent.

b. Give an example of a convergent series∑∞

n=1 an for which∑∞

n=1 a2n is divergent.


1. Let f(x, y) = cyexy + (x+ 1)2 cos(πy) where c is a real constant and A = 3i− 4j . Find c ifthe directional derivative of f at (0, 1) in the direction of A is 2.

2. Find and classify the critical points of f(x, y) = x4 + y4 + 4axy where a is a real constant.

3. Find the absolute maximum and minimum values of the function f(x, y) = xy − x− y + 3on the triangular region with vertices at (0, 0), (2, 0) and (0, 4) .

4. Find∂2z

∂x∂y

∣∣∣∣(2,1)

if z = f(u, v) where u = x2y and v = x/y , and f is a twice differentiable

function with fu(4, 2) = 4 , fv(4, 2) = −5 , fuu(4, 2) = −1 , fuv(4, 2) = fvu(4, 2) = 3 , andfvv(4, 2) = 2 .



a.

ˆ 1

0

ˆ y1/3

y

sin(x2) dx dy

b.

¨R

1

(x2 + y2)2dA where R = {(x, y) : x ≥

√y2 + 1 , x ≥

√3 y , y ≥ 0} .

Spring 2002 Final

1. Consider

f(x) =∞∑n=1

(−1)n+1xn

4nn2

Determine the interval of convergence of the series and �nd the value ofdf

dx

∣∣∣∣x=−1/2

explicitly.

2. Let

f(x, y, z) =

ˆ xyz

0

e−t2 dt

Compute the value of fxx + fyy + fzz at the point (2, 1/2,−1) .

3. Find

¨R

y−3 dA where R = {(x, y) : sin x ≤ y ≤ 2 sin x , cosx ≤ y ≤ 2 cos x , 0 ≤ x ≤ π/2} .

4. LetD be the region lying inside the sphere x2+y2+z2 = 4, outside the sphere x2+y2+z2 = z ,and above the xy-plane . Express the triple integral

¨D

(x2 + y2 + z2) dV

in spherical coordinates and evaluate it.

5. Let C be the boundary of R = {(x, y) : 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1} and F = (ax2 + y)i+ xy2jwhere a is a constant. Find the value of a for which the outward �ux of F over C and thecounterclockwise circulation of F around C are equal.



ln

(n2 + 2n+ 1

n2 + 2n

).

a. Show that the series converges.



2. Determine if each of the following series converges or diverges.

a.∞∑n=1

n! (2n)!

(3n)!

b.∞∑n=1

sinn

en − e−n

3. Find the sum of each of the following series if it exists. If not, explain why.

a. π − π3

3!+π5

5!− π7

7!+ · · ·+ (−1)n

π2n+1

(2n+ 1)!+ · · ·

b. π − π3

3+π5

5− π7

7+ · · ·+ (−1)n

π2n+1

2n+ 1+ · · ·

4. Find the interval of convergence of the power series

∞∑n=2

xn

n lnn

and for each point of this interval determine if the convergence is absolute or conditional.

5. Consider the function de�ned by:

f(x) =∞∑n=0

xn

n! 2n(n−1)/2

a. Find the domain of f .

b. Evaluate the limit limx→0

f(x)− ex

1− cosx.

c. Show that f(2) < e+3

2.

d. Show that f(−2) < 0 .


1. Evaluate each of the following limits if it exists, and explain why if it does not.

a. lim(x,y)→(0,0)

xy√xy + 1− 1

b. lim(x,y)→(0,0)

x2y2

x2y2 + (x− y)2

2. Find and classify the critical points of

f(x, y) = x2y + x2 + y2 − xy − x .


3. Consider the function f(x, yz) = xy2 + exy − yz and the point P0(0, 1, 2).

a. Find the direction in which f increases fastest at the point P0.

b. Find a unit vector u which is parallel to the xy-plane and which satis�es

(df

ds

)u,P0

.

4. Find the extreme values of f(x, y, z) = 8x− 4z subject to the constraint x2 +10y2 + z2 = 5.

5. Find the volume of the region that lies under the cone z = (x2 + y2)1/2 and above the disk(x− 1)2 + y2 ≤ 1.

6. a. Change the order of integration in the following iterated integral:

ˆ 2

1

ˆ √2x−x2

2−x

f(x, y) dy dx .

b. Express the double integral

¨R

f(x, y) dA as an iterated integral in polar coordinates if

R = {(x, y) : 0 ≤ y ≤ x2 , 0 ≤ x ≤ 1}.

Spring 2001 Final


a.∞∑n=1

sin

(1

n

)

b.∞∑n=0

n!

2n2

2. Find nonzero real numbers a and b such that the function f(x, t) = taebx2/t satis�es the

equation∂f

∂t=∂2f

∂x2for all (x, t) with t > 0.

3. Let D = {(x, y, z) : x2 + y2 + z2 ≤ 4 and z ≥ 1}. Express -do not evaluate- the volume of Das an iterated integral in

a. cylindrical coordinates,

b. spherical coordinates.

4. Let D = {(x, y, z) : 1 ≤ x ≤ 2 , 0 ≤ xy ≤ 2 , 0 ≤ z ≤ 1}. Evaluate the triple integral˚D

(x2y + 3xyz) dV by applying the transformation u = x, v = xy, w = 3z.


5. Find the area of the piece of the cylinder x2+z2 = 1 which lies inside the cylinder x2+y2 = 1and in the �rst octant.

6a. Use Stokes's Theorem to evaluate the line integral

‰C

y dx+ z dy + x dz where C is the

intersection curve of the plane x+ y + z = 0 with the sphere x2 + y2 + z2 = 4 parametrized inthe counterclockwise direction as seen from the positive z-axis.

6b. Use the Divergence Theorem to �nd the outward �ux of the vector �eld F = xzi + yzj +(3z − z2)j across the boundary of the ball D = {(x, y, z) : x2 + y2 + z2 ≤ 4}.


1. Find the sum of the series∞∑n=1

2n+ 1

n2(n+ 1)2.

2. Find the radius of convergence and the interval of convergence of the power series

∞∑n=0

xn√n2 + 3

.

Also determine the type of convergence (absolute or conditional) for each x in the interval ofconvergence.

3. Evaluate the following limit:

limy→0

arctan y − sin y

y3 cos y

4. Determine if each of the following limits exist.

a. lim(x,y)→(0,0)

x4 − y2

x4 + y2

b. lim(x,y)→(0,0)

x3 − xy2

x2 + y2

5. Find the values of the constant α such that the function

w = (x2 + y2 + z2)α

satis�es the equation wxx + wyy + wzz = 0 for all (x, y) = (0, 0) .



1. Find the directional derivative of the function f(x, y, z) = cos(xy)+eyz+ln(xz) at the pointP0(1, 0, 1/2) in the direction of A = i+ 2j+ 2k .

2. Find the absolute maximum and minimum values of the function

f(x, y) = 6xy − x3 − 3y2

on the region R = {(x, y) : 0 ≤ x ≤ 2 and 0 ≤ y ≤ 1} .


a.

ˆ 1

0

ˆ 1

y1/3

sin(πx2)

x2dx dy

b.

¨R

1

(1 + x2 + y2)2dA where R = {(x, y) : x2 + y2 ≤ 1}.

4. Let D be the region bounded below by the xy�plane, on the sides by the sphere ρ = 2and above by the cone ϕ = π/3. Express (Do not evaluate!) the volume of D in terms ofiterated integrals in:

a. spherical coordinates

b. cylindrical coordinates

c. Cartesian coordinates

5. Evaluate the integral ¨R

1

1 + x2y2dA

where R = {(x, y) : x/2 ≤ y ≤ 2x and xy ≥ 1}.

Spring 2000 Final


a.∞∑n=1

(−1)nsinn

n2

b.∞∑n=1

3n

n22n

2. Find the absolute maximum value of the function f(x, y, z) = x − y + 2z on the ellipsoidx2 + y2 + 16z2 = 16 .

3. Let S be the part of the cone z2 = 4x2 + 4y2 which lies above the xy−plane and inside the

cylinder x2 + y2 = 2x . Evaluate the surface integral

¨S

(x2 + y2) dσ .


4. LetD be the region lying in the �rst octant between the cones z2 = 3x2+3y2 and z2 = x2+y2 ,and inside the sphere x2 + y2 + z2 = 4 . Let S be the boundary of D . Let n be the outwardpointing unit normal vector �eld on S .

a. Find the volume of D .

b. Evaluate the integral

¨S

F · n dσ where

F = (x2 + sin y)i+ (ez − 3xy)j+ (xz + 7z)k .

5. Let S be the surface 4x2 + 9y2 + 36z2 = 36, z ≥ 0 . Let n be the unit normal vector �eld

on S pointing away from the origin. Find

¨S

(∇× F) · n dσ where

F = yi+ x2j+ (x2 + y4)3/2 sin(exyz)k .


1. Find the constants a and b such that the limit

limx→0

tan−1(x+ ax3 + bx5)− x

x7

exists and is �nite. Find the limit in this case. (Do not use L'Hôpital's rule.)

2. Evaluate the sum∞∑n=1

(−1)(n+1)

2n n(n+ 1).

3. Determine the values of the constant α for which the function

f(x, y) =

x4 + y4

(x2 + y2)αif (x, y) = (0, 0),

0 if (x, y) = (0, 0),

is continuous at (x, y) = (0, 0) .

4. Assume that z = f(x, y) is a di�erentiable function of x and y, and x = g(r, s) and y = h(r, s)are di�erentiable functions of r and s. Use the following table

f(2, 3) = 10 fx(2, 3) = 13 fy(2, 3) = 16f(2, 4) = 11 fx(2, 4) = 14 fy(2, 4) = 17f(3, 5) = 12 fx(3, 5) = 15 fy(3, 5) = 18

g(0, 1) = 2 gr(0, 1) = 6 gs(0, 1) = 0g(1, 0) = 3 gr(1, 0) = 7 gs(1, 0) = 1

h(0, 1) = 4 hr(0, 1) = 8 hs(0, 1) = 0h(1, 0) = 5 hr(1, 0) = 9 hs(1, 0) = 1


to compute∂z

∂r

∣∣∣∣(r,s)=(0,1)

and∂z

∂s

∣∣∣∣(r,s)=(1,0)

.

5. Find ∂x/∂w when (x, y, z, w) = (2,−2, 1,−1) if x and z are de�ned as functions of y and wby the equations: {

x3 + y2z + xz2w − z4 = 9

x2z2 − x5w + w3 + y3 = 27


1. Find and classify the critical points of the function:

f(x, y) = x3y + x3 + 4y2

2. Find the absolute maximum and the absolute minimum values of the function

f(x, y) = x3 + y3 + 6xy + 8

on the rectangular region R bounded by the lines y = x+2, y = x−2, x+y = 1 and x+y = −2 .


ˆ 1

0

ˆ x

0

3√

(2y − y2)2 dy dx .

4. Find the volume of the region D which lies between the spheres x2 + y2 + z2 = 9 andx2 + y2 + z2 = 2z , and inside the cone z =

√(x2 + y2)/3 .


¨R

1

xe

3√

xy2 dA where R is the region in the �rst quadrant bounded

by the curves y = 8x, y = 27x, y =1

8√xand y =

1√x.

Spring 1999 Final

1. Find the radius of convergence of the power series

∞∑n=1

nx3n+1

and �nd f(x) explicitly.


2. Evaluate the integral ˚D

z

(x2 + y2 + z2)5/2dV

where D = {(x, y, z) : x2 + y2 ≤ z2 and z ≥ 1}.

3. Find the area of the portion of the cylinder x2 + y2 = 2x that lies inside the spherex2 + y2 + z2 = 4 and in the �rst octant.

4. Verify the Stokes's Theorem for the vector �eld F = zi + xj + yk and the surface S ={(x, y, z) : z = 1− x2 − y2 and z ≥ 0} with the unit normal vector �eld n satisfying n · k > 0.

5. Let f(x, y, z) be a function with continuous second order partial derivatives and assume thatf(x, y, z) = 0 for all (x, y, z). Assume also that |∇f |2 = 4f and ∇ · (f∇f) = 10f . Evaluate

¨S

∇f · n dσ

where S is the sphere x2 + y2 + z2 = 1 and n is the outward unit normal to S.


1. Determine whether each of the following series converges or diverges:

a.∞∑n=3

1

n lnn ln(lnn)

b.∞∑n=1

√2n− 1 lnn

n(n+ 1)

c.∞∑n=1

(n!)2

(2n)!

d.∞∑n=0

sin((2n+ 1)π/2)√n+ 1

2. Find all values of x for which the following series is convergent and also determine the typeof convergence:

∞∑n=1

1√n

(x+ 3

x

)n

3. Find the �rst three non-zero terms of the Maclaurin series of f(x) = sin(sin x).


4. a. Find

ˆ 1

0

cosx3 dx approximately with an error of magnitude less than 5× 10−4.

b. Find the sum of the following series as a function of x for |x| < 1:

x2

2− x3

3 · 2+

x4

4 · 3− x5

5 · 4+ · · ·

5. a. Find the intersection points of the curves r = 1 + cos θ and r = cos θ.

b. Find the area of the region which lies inside the curve r = 1+cos θ and outside the curver = cos θ.


1. Find the equation of the line L which passes through the point P0(1, 0, 2) and intersects the

line L1 :x− 1

2=y + 1

3, z = 1 orthogonally.

2a. Let r(t) be the position vector of a curve in space and let v(t) =dr

dtbe the corresponding

velocity vector. Prove that v(t) · a(t) = 1

2

d(v(t)2)

dt, where a(t) =

dv

dtis the acceleration vector

of r(t) and v is the speed (= magnitude of v).

2b. Let A and B be two �xed vectors making an angle ofπ

3radians with each other and

|A| = 2, |B| = 3. A particle moves on a space curve C in such a way that its position vector

r(t) and velocity vector v(t) are related by the equation v(t) = A× r(t) for all t ∈ R. Moreover

assume that r(0) = B. Show that the speed of the particle is constant and �nd its value.

2c. Prove that the curvature κ of the curve C in part (b) is constant and calculate its value.

Hint : κ =|v(t)× a(t)|

|v(t)|3.

3a. Find the set of all points on the surface (y + z)2 + (z − x)2 = 16 where the normal line isparallel to the yz-plane. Describe this set.

3b. Find the extremum points of the function f(x, y) = x3 + 3xy + y3.

4. Let s(x) be an even di�erentiable function of x ∈ R and

f(x) =

s(x)− s(y)

x2 − y2, y = ±x,

sinx

x, y = ±x and x = 0,

a , (x, y) = (0, 0) .

be a di�erentiable function of (x, y) ∈ R2.


a. Find s(x) and the constant a.

b. Find∂f(x, y)

∂xat the origin.

5. Find the points on the curve 5x2+6xy+5y2 = 9 which are nearest to and farthest from theorigin.

Spring 1998 Final

1a. Find the set of all real numbers x for which the following series is convergent:

∞∑n=0

2nx

n2 + 1

1b. Find the sum of the series∞∑n=0

xn

5n(n+ 1)as a function of x.

2. Evaluate the double integral ¨R

x2

y4dx dy

where R is the region in the plane bounded by the curves xy = 2, xy = 4, y2 = x and y2 = 3x.

3. Let F = (2xy2z + xy3)i+ (2x2yz +3

2x2y2)j+ (x2y2 + 3z2)k. Evaluate

ˆ (−1,−1,1)

(1,1,−1)

F · dr .

4. Let f(x, y) and g(x, y) have continuous �rst order partial derivatives. Let

F = g(x, y)i+ f(x, y)j and G =

(∂f

∂x− ∂f

∂y

)i+

(∂g

∂x− ∂g

∂y

)j .

It is known that for the points (x, y) on the circle x2+ y2 = 1, we have f(x, y) = 1, g(x, y) = y.Let R be the region in the plane de�ned by x2 + y2 ≤ 1. Find

¨R

F ·G dx dy .

5. Verify the Divergence Theorem for the vector �eld F = (z2 + 2)k and the surface S whichconsists of the upper half of the sphere x2 + y2 + z2 = a2 together with its base, the disk ofradius a centered at the origin in the xy-plane.



1. Find all values of x for which∞∑n=1

2nxn

n(3x+ 1)nconverges and determine whether the

convergence is absolute or conditional.

2. a. Use the Taylor series of tan−1 x at x = 0 to �nd the �rst four nonzero terms of the Taylorseries of tan−1(ax+ bx3) at x = 0 where a and b are nonzero constants.

b. Find a and b for which the limit

limx→0

tan−1(ax+ bx3)− x

x5

is �nite and �nd the value of this limit. (Do not use L'Hôpital's Rule.)

3. Let A and B be vectors in space. Show that if |a+ xB| ≥ |A| for all real numbers x, thenA and B are orthogonal.

4. Consider the limit

lim(x,y)→(0,0)

x|y|k

x2 + y4

where k is a positive constant.

a. Find all values of k for which the limit does not exist.

b. Find all values of k for which the limit exists. Show the existence of the limit by the ε-δmethod.

5. Find the linear approximation to f(x, y) = xy2 + x3y at (1, 2) and �nd an upper bound forthe magnitude of the error over the rectangle R = {(x, y) : |x− 1| ≤ 1/2, |y − 2| ≤ 1}.


1. Find and classify the critical points of the function

f(x, y) = x3 − 3xy + y3 .

2. a. Let h(x, y) = f(x + y) + g(x − y) where f and g are twice di�erentiable one-variablefunctions. Show that

∂2h

∂x2=∂2h

∂y2

for all (x, y).

b. Find u(x, y) if u(x, 0) = 0 and∂u

∂x(x, 0) =

x

1 + x2for all x, and

∂2u

∂x2=∂2u

∂y2for all (x, y).



a.

¨R

ln(x2 + y2) dA where R = {(x, y) : x2 + y2 ≤ 2 and 0 ≤ y ≤√3x}.

b.

ˆ ∞

0

ˆ ∞

y

e−x2

dx dy

4. Evaluate

ˆ 0

−1

ˆ 2y+3

−y

x+ y

(x− 2y)2ex−2y dx dy.

5. Sketch the region which lies above the surface z = 2√x2 + y2 and inside the surface

x2 + y2 + z2 = 4z, and express its volume as a triple integral in (a) Cartesian, (b) cylindrical,(c) spherical coordinates. (Do not evaluate.)

Spring 1997 Final

1. Find the �rst �ve nonzero terms of the Taylor series of f(x) = esinx centered at x = 0 .

2. a. Find the absolute maximum of f(x, y, z) = ln x + ln y + 3 ln z on the portion of thesphere x2 + y2 + z2 = 5r2 where x > 0, y > 0 and z > 0 . (Here r is a positive constant.)

b. Use part (a) and show that

abc3 ≤ 27

(a+ b+ c

5

)5

for all positive real numbers a, b and c.

3. Let a be a positive constant. Find the area of the portion of the sphere x2 + y2 + z2 = a2

lying inside the cylinder x2 + y2 = ay .

4. Evaluate the line integral‰C

(y3

x− ex

)dx+

(yx+ ey

2)dy

where C is the boundary of the region lying between the curvesy = x2 , y = 2x2 , y = 1/x , y = 3/x , traced counterclockwise.

5. a. If S is an oriented closed surface with unit normal �eld n and F is a vector �eld with

continuous derivatives, then show that

¨S

(∇× F) · n dσ = 0 .

b. Find a vector �eld whose curl is 2xi + 3yj + 5zk or show that there is no such vector�eld.




1

n(n+ 1)(n+ 2).

a. Find the nth partial sum sn explicitly.

b. Use part (a) to evaluate the sum of the series.

2. Letn∑

k=1

1

k− lnn for n ≥ 1.

a. Show that an > 0 for n ≥ 1.

b. Show that {an} is a decreasing sequence.

c. Show that the sequence {an} converges.


(2n)!

(n!)2xn .


b. Show that (1− 4x)f ′(x) = 2f(x) for all x in the interval of convergence.

c. Use part (b) to express f(x) explicitly.

4. Consider the identity

1

1− t2= 1 + t2 + t4 + · · ·+ t2n + r(t, n)

where n is a positive integer.

a. Find r(t, n).

b. Integrate the identity above to obtain tanh−1 x =2n+1∑k=0

akxk +R(x, n) for |x| < 1 and �nd

ak for 0 ≤ k ≤ 2n+ 1.

c. Show that limn→∞

R(x, n) = 0 when |x| < 1.

5. Determine whether each of the following series or convergent or divergent:

a.∞∑n=1

n!

nn

b.∞∑n=1

1

2lnn


c.∞∑n=1

(1− 1

n2

)n

d.∞∑n=1

n2e−n


1. Find the area of the region that lies inside the circle r = 1 and outside the cardioidr = 1− cos θ.

2. Find the parametric equations for the line that is tangent to the curve of intersection of thesurfaces x3 + 3x2y2 + y3 + 4xy − z2 = 0 and x2 + y2 + z2 = 11 at the point (1, 1, 3).

3. Find all maxima, minima and saddle points of the function f(x, y) = x3+ y3+3x2− 3y2− 8in the entire plane.

4. Find the absolute minimum and maximum values of the function f(x, y) = 4x−8xy+2y+1on the triangular region bounded by the lines x = 0, y = 0 and x+ y = 1 in the �rst quadrant.

5. Find the points on the sphere x2 + y2 + z2 = 25 where f(x, y, z) = x + 2y + 3z has itsmaximum and minimum values.

Spring 1996 Final


xn

n(n+ 1).


b. Evaluate the sum of the series at both end points of the interval of convergence.


a.

ˆ 1

0

ˆ 1

y

x2exy dx dy

b.

¨D

1

x2 + y2 + z2 + 1dV where D is the intersection of the unit ball with the �rst octant.

3. Let R = {(x, y) : x ≤ y ≤ 2x, 1/x ≤ y ≤√3/x}. Evaluate the integral

ˆR

1

1 + x2y2dA by

using a change of variables which transforms R into a rectangle.


4. Let D be the region bounded by the surfaces z = x2 + y2 + 1 and z2 = 4x2 + 4y2. Find thevolume of D.

5. Let C be the curve which traces the graph of y2 = (1 − x2)(1 + x2)(1 + x4)(1 + x6) once

counterclockwise. Let F(x, y) =x− y

x2 + y2i+

x+ y

x2 + y2j . Evaluate the integral

‰C

F · dr.

6. Suppose f(x, y) is a function on R = {(x, y) : (x, y) = (0, 0)} which has continuous secondorder partial derivatives and satis�es the equations

∂2f

∂x2+∂2f

∂y2= 0 and

∂f

∂r= 0

throughout R, where x = r cos θ and y = r sin θ.

a. Find∂2f

∂θ2.

b. Find f(0, 1) if f(1, 0) = 5.


1a. Explain whether∞∑n=1

(sin1

2n− sin

1

2n+ 1) converges or diverges.

1b. Find the interval of convergence of∞∑n=1

n tan(1/n)xn .

2a. Given A = i+ j− 2k, B = −i− k, C = 2i+4j− 2k, �nd the volume of the parallelepipeddetermined by A, B and C.

2b. Find the area of the projection of the parallelogram determined by B and C into thexz-plane.

2c. Find the planes determined by B and C.

3a. Find the area of the region between the circles r = 1 and r = 2 cos θ.

3b. Plot the graph of r = eθ for 0 ≤ θ ≤ 2π, and calculate its length.

4. Evaluate

ˆ 0.1

0

sinx

xdx within an error of magnitude less than 10−11.

5. Evaluate

limx→0

1− x+ x4 + ln(1 + x− x2)− cos(√3x)

x5/2 sin√x

.


Spring 1995 Final

1. Let F = y2/xi+ 2y lnxj− 2/z2k .

a. Show that F is a conservative vector �eld.

b. Find the potential function.

c. Evaluate

ˆ (2,2,2)

(1,1,1)

F · dr .

d. Under what circumstances is your answer to part (c) valid?

2a. Evaluate the surface integral

¨S

(x+ z) dσ where S is the �rst octant portion of the cylinder

z2 + y2 = 9 between the planes x = 0 and x = 1.

2b. Suppose∞∑n=1

an is a convergent series of positive numbers. Does the series∞∑n=1

ln(1 + an)

converge? Explain your answer.

3. Evaluate

¨R

e(x−y)/(x+y) dx dy , where R is the region bounded by the lines x = 0, y = 0 and

x+ y = 1.

Hint : Use the transformation u = x− y, v = x+ y.

4a. Find the extremal values of f(x, y, z) = x − 2y + 2z among the points (x, y, z) withx2 + y2 + z2 = 9.

4b. Find the area of the triangle with vertices located at the points (a, 0, 0), (0, b, 0) and(0, 0, c).

5. Let F = (xi+ yj)/r where r2 = x2 + y2. Calculate

‰C

F · dr when

a. C is enclosing the origin.

b. C is not enclosing the origin.


1. Given the series∞∑n=1

(n

n+ 1− n+ 2

n+ 3

)

a. Find a formula for the partial sums sn .



2. Determine whether absolutely convergent, conditionally convergent or divergent. Explain.

a.∞∑n=1

(−1)nn√n2 + 1

b.∞∑n=1

(−1)n

n lnn ln(lnn)

3. �nd the domain of convergence of the power series

∞∑n=1

(−1)n(x− 2)n

4nn.

4. a. [missing ]

b. Approximate the value of the following integral with error less than 0.0001:

ˆ 0.3

0

cosx2 dx

Hint : Use alternating series estimate.

5. Consider the curves r = −1 + cos θ and r = 3 cos θ.

a. Sketch the curves.

b. Find the intersection points of these curves in polar and Cartesian coordinates.

c. Find the length of the curve r = 3 cos θ between two intersection points. (Choose anytwo you want.)

6. a. Find the length of the curve x = 2et, y = e3t/3 + e−t, 0 ≤ t ≤ 1.

b. Find the area of the region inside r = 1 + cos θ and outside r = 1.


1. Find the shortest distance between the lines L1 and L2 where

L1 :

x = 1 + 2t

y = −1 + t

z = t

and L2 :

x = 2 + s

y = 1− s

z = 2s

.

2a. Find lim(x,y)→(0,0)

arctan

(x− y

x2 + y2

)or show that it does not exist.


2b. Suppose w = x3 − x2y5 + 3z + 2t and x+ 5z + 3t = 10 �nd all possible values of∂w

∂x.

3a. Find the angle between ∇u and ∇v at all points with x = 0 and y = 0 if

x = eu cos v and y = eu sin v .

3b. Show that the curve r =√ti +

√tj− (t + 3)/4k is normal to the surface x2 + y2 − z = 3

at their intersection point.

4. A particle moves with position vector

r = tA+ t2B+4

3t3.2a×B , t ≥ 0

where A and B are two �xed unit vectors making an angle ofπ

2radians with each other.

a. �nd the speed of the particle at time t.

b. How long does it take for the particle to move 12 units of arc length from the initialposition r(0)?

c. Find the curvature at time t > 0.

5. Let f(x, y) = cos(y − ex).

a. Find the linear approximation of f(x, y) at the point(0, 1) and estimate the error madein this approximation if |x| ≤ 0.1 and |y − 1| ≤ 0.1.

b. Find the quadratic approximation of the same function at (0, 1).

6. Find the absolute extreme values of f(x, y) = x4 + y4 − 4xy in the region bounded by thelines x = 2, y = −2 and y = x.

Spring 1994 Final

1. The plane x+ y + 2z = 0 intersects the sphere x2 + y2 + z2 = 1 along a curve C. Find‰C

(x+ 2z) dx+ (x+ y + z) dy + (x+ y + z) dz

if C is traversed in a direction that is counterclockwise when viewed from high above thexy-plane.

2a. Find the radius of convergence of∞∑n=0

(n!)3

(3n)!xn.

2b. Find the sum of the in�nite series

x+x3

3+x5

5+ · · ·+ x2n+1

2n+ 1+ · · · for |x| < 1.


2c. Is the series∞∑n=2

1

(lnn)lnnconvergent or divergent?

3. Consider ‰C

(− y3

(x2 + y2)2dx+

xy2

(x2 + y2)2dy

).

a. Evaluate the above line integral when C is the circle x2 + y2 = a2 traversed in thecounterclockwise direction.

b. Let C be an arbitrary smooth simple closed curve in the plane that does not pass throughthe origin. Show that above line integral has two possible values depending on whether theorigin lies inside or outside C.

4. Find the volume of the parallelepiped bounded by the six planes x+y+2z = ±3, x−2y+z =±2 and 4x+ y + z = ±6.

5. The plane x+y+z = 12 intersects the paraboloid z = x2+y2 in an ellipse. Find the highestand lowest points on this ellipse (the points with greatest and least z-coordinates).

6a. Let f be a function of x and y. Express (fx)2 + (fy)

2 in polar coordinates (in terms of thepartial derivatives with respect to the polar variables).

6b. Show that if w = f(u, v) satis�es the Laplace equation fuu+fvv = 0, and if u = (x2−y2)/2and v = xy, then w satis�es the Laplace equation wuu + wvv = 0.


1. Each of the following series is the value of the Maclaurin series of a function at a point.What function and what point? What is the sum of the series?

a. 1− π2

9 · 2!+

π4

81 · 4!− · · ·+ (−1)n

π2n

32n(2n)!+ · · ·

b.2

3− 4

18+

8

81− · · ·+ (−1)n−1 2n

3nn+ · · ·

2. Use series to �nd the values of a and b for which the limit

limx→0

x− sin(ax+ bx3)

x5

exists and compute that limit.

3. a. Find the interval of convergence of the series

y = 1 +1

6x3 + · · ·+ 1 · 4 · 7 · · · · · (3n− 2)

(3n)!x3n + · · · .


b. Show that the function de�ned by the series satis�es a di�erential equation of the form

d2y

dx2= xay + b

and �nd the values of the constants a and b.

4. Determine for each of the following series whether it converges or diverges. Give reasons foryour answers.

a.∞∑n=0

1√n3 + 2

b.∞∑n=0

n2e−n sinn

5. a. Using vectors in the plane, �nd the angle between the tangent to the curve y = f1(x) atx = x1 and the tangent to the curve y = f2(x) at x = x2 .

b. Write B = −i+ 3j+ 4k as the sum of a vector parallel to A = 2i− 3j+ k and a vectororthogonal to it.

6. a. Find the area of the region which lies inside the curve r = 2b sin θ and outside the curver = a where 2b > a.

b. Find the length of the piece of the curve r = 2b sin θ lying outside the curve r = a.


1. a. Find the equation of the line normal to the surface z = x2 + 3y2 at the point (1,−1, 4).Find the coordinates of all intersection points of this line with the surface.

b. Find the parametric equation for the tangent line to the curve of intersection of thesurfaces z = x2 + 3y2 and z = 6− x2 − y2 at the point (1,−1, 4).

2. a. Show that the curvature is given by κ =−1

r ·Nfor a curve on the sphere x2+y2+z2 = R2.

b. Show that a curve in space with zero curvature at all points is a straight line.

3. Let

f(x, y) =

x2y

x4 + y2if (x, y) = (0, 0) ,

0 if (x, y) = (0, 0) .

a. Is f continuous at (0, 0)? Prove your statement.

b. Show that −1

2≤ f(x, y) ≤ 1

2for all (x, y).


4. Use Lagrange multipliers to �nd the absolute maximum of f(x, y, z) = x3 + 12yz on thesphere x2 + y2 + z2 = 25.

5. Let f(x, y) = x3 + y2 − 2x.

a. Find and classify the critical points of f .

b. Find the absolute maximum and minimum of f on the line segment {(x, y) : x = t, y =t+ 1,−1 ≤ t ≤ 1}.

6. Let u(x, y) and v(x, y) be two functions with continuous second partial derivatives which

satisfy the di�erential equations∂u

∂x=∂v

∂yand

∂u

∂y= −∂v

∂x.

a. Show that∂2u

∂x2+∂2u

∂y2= 0 and

∂2v

∂x2+∂2v

∂y2= 0.

b. Show that all critical points of u are saddle points. You may assume that at a criticalpoint at least one of the second partial derivatives is not zero.

Spring 1993 Final

1. a. Find the absolute minimum of the function de�ned by the series∞∑n=0

n2xn on the interval

(−1, 0].

b. Find the radius of convergence of the series∞∑n=1

n!xn

nn.

2. Let r and θ be the polar coordinates in the plane. Suppose f(x, y) has continuous secondpartial derivatives. Express fxx + fyy in terms of r and θ, and partial derivatives of f withrespect to r and θ.

3. Let R be the region bounded by the curves xy = 1, xy = 4, y =√3x2 and y = x2 in the

�rst quadrant. Evaluate ¨R

x2y

x4 + y2dA

by using a coordinate transformation which maps R onto a rectangular region in the newcoordinate plane.

4. a. Let S be the boundary of the region bounded by the sphere x2 + y2 + z2 = 1 on the topand by the cone z2 = x2 + y2 on the sides. Let n be the outward normal �eld of S. Evaluate¨

S

F · n dσ for F = x3i+ y3j+ z3k.


b. Let C be the circle in the plane 2x+ 2y + z = 2 with center (0, 0, 2) and radius 3 in thecounterclockwise direction as viewed from the origin. Evaluate

‰C

2y dx+ 3x dy − x dz .

5. a. Find the area of the surface cut from the bottom of the paraboloid z = x2 + y2 by theplane z = 3/4.

b. Evaluate

ˆ a

0

ˆ b

0

ef(x,y) dy dx where a and b are positive numbers and

f(x, y) =

{b2x2 if b2x2 ≥ a2y2 ,

a2y2 if b2x2 < a2y2

6. a. Assume that ∇ · F > 0 for all (x, y) where F = M(x, y)i + N(x, y)j is a vector �eld inthe plane whose components have continuous partial derivatives. Show that there is no smoothsimple closed curve in the plane whose tangent vector is parallel to F at all its points.

b. Evaluate

‰C

F · dr where

curl F =−y

4x2 + 9y2i+

x

4x2 + 9y2j

and C is the unit circle parametrized in the counterclockwise direction.


1. Let F (x, y, z) = 2x3 + 3y4 + 5z6 + 7 where x = cos t + sin t, y = tan t + t2 + 1, and

z = 1− t+ 2 ln(2 + t). FinddF

dt

∣∣∣∣t=0

.

2. Find the distance between the following two lines:

L1 :

x = 3 + t

y = 2− 4t

z = t

and L2 :

x = 4− s

y = 3 + s

z = −2 + 3s

3. Describe the points P (ρ, ϕ, θ) whose coordinates satisfy θ = 3π/2, ρ = 3 cosϕ, and sketch.

4. Find the distance from the point(2, 0, 3) to the plane which is tangent to the surface 4x2 −y2 + 4z2 = 4 at the point (1, 2, 1).


5. A function is de�ned as follows:

f(x, y) =

x3y

x5 + 2y3if (x, y) = (0, 0),

0 if (x, y) = (0, 0) .

Show that f is continuous at the origin.

6. A space curve is given by the parametrization r(t) = (e2t, t, t2). Find the equation of theosculating plane at the point corresponding to t = 1.

Spring 1992 Final

1. Find the points on the paraboloid z = 4x2 + 9y2 at which the normal line is parallel to theline through P (−2, 4, 3) and Q(5,−1, 2).

2. Do the following series converge or diverge? Give reasons.

a.∞∑n=0

n!

1000n

b.∞∑n=5

n1/2

(lnn)3

c.∞∑n=3

1

n lnn3

d.∞∑n=1

ln

(n2 + 1

n2

)

3. Find the dimensions of the rectangular box of maximum volume that has three of its facesin the coordinate planes, one vertex at the origin and another vertex in the �rst octant on theplane 2x+ 3y + 5z = 90.

4. Find the minimum, maximum and the saddle points of the function f(x, y) = 2x4+xy+ y2.

5. Find the volume of the region bounded by the plane z = 0, the cylinder x2 + y2 = 4 and thecylinder 2z = 4− y2.

6. Evaluate the integral ¨1

(x2 + y2)5/2dx dy

over the region which is bounded by the lines y =√3x, y = x, y = 1 and y = 2.



1. Does the integral

ˆ ∞

0

lnx

x2dx converge or diverge?

2. a. Find the �rst three non-vanishing terms in the Taylor series for the function f(x) =sin−1 x.

b. Find the radius of convergence of the above series.

3. Let sn = 1 +1

2+ · · ·+ 1

n. Show that lim

n→∞(sn − lnn) exists.

4. Find the volume of the tetrahedron with vertices at (2, 1, 1), (1,−1, 2), (0, 1,−1) and(1,−2, 1).

5. [missing ]

6. Find the distance between the point P (0, 1, 1) and the line x = 1 + 2t, y = −1− t, z = 3t.


1. Prove that the radius of curvature of a curve parameterized by its arc length is given byρ = (x2 + y2 + z2)−1/2 where dot denotes derivative with respect to the arc length.

2.We have 100 vectors in R100 de�ned as follows:

v1 = (1, 0, 0, . . . , 0, 0, 1)

v2 = (0, 1, 0, . . . , 0, 1, 0)

...

v50 = (0, 0, . . . , 1, 1, . . . , 0, 0)

v51 = (1, 0, 0, . . . , 0, 0,−1)

v52 = (0, 1, 0, . . . , 0,−1, 0)

...

v100 = (0, 0, . . . , 1,−1, . . . , 0, 0)

Let A be the vector (1, 2, 3, . . . , 100). If A = a1v1 + · · ·+ a100v100 where ais are real constants,�nd ai, 1 ≤ i ≤ 100.

3. Find the equation of the locus of the center of the circle of curvature of the curve y = x2.

4. Let u and v be two distinct nonzero vectors. Show that the vector w = |v|u + |u|v bisectsthe angle between u and v.


5. Find the curvature of the cycloid x = a(t− sin t), y = (1− cos t), t ≥ 0, at the highest pointof the arc.

Spring 1991 Final

1. a . Find the Taylor expansion of f(x) = cx, c > 0, around x = 0.

b. Test for convergence:∞∑n=1

n!

nn.

2. Find the curvature, the torsion and the normal vector for the space curve

R(t) = a cosωti+ a sinωtj+ btk

where a, b, ω are positive constants.

3. Let f(x, y) = (y − x2)(y − 2x2).

a. Show that (0, 0) is a critical point of f at which f has a saddle point.

b. Show that on any line through the origin f has a local minimum at (0, 0).

5. Find the volume bounded by the surfaces z = x2 + y2 and z = (x2 + y2 + 1)/2.

6. Evaluate ‰C

(y3

x− ex

)dx+

(yx+ ey

2)dy

where C is the boundary of the region bounded by the curves y = x2, y = 2x2, y = 1/x,y = 3/x, traced counterclockwise.


1. Let Π be a plane in 5-dimensional space. Suppose that the vector v = (1, 2, 3, 4, 5) isperpendicular to the plane Π and that the point P (5, 4, 3, 2, 1) is on Π. Find the distance of Πto the origin.

2. Find the unit tangent, a unit principal normal and a unit binormal vector along the curve

r(t) = (3t− t3)i+ 3t2j+ (3t+ t3)k .

3. Find the intersection of the xy-plane with the tangent line to the curve

r(t) = (1 + t)i− t2j+ (1 + t3)k


at t = 1.

4. Finddw

dtif w = f(x, y, z) and x = t, y = g(t), z = h(t, g(t)).

5. Let R be the distance from a �xed point A(a, b, c) to any point P (x, y, z). Show that

∇R =

−→AP

R.

6. Let f(x, y) = ϕ(x− cy) + ψ(x+ cy) where c is a constant. Show that

c2fxx = fyy .


1. Let c > 0 be a constant. Find the set of all x for which the following power series converges.Check also the endpoints.

∞∑n=0

cnx2n

2. By using the series �nd

limx→0

6 sinh x− 6x− x3

x5.

3. Let z = f(u, v) where f is of classC2, u = x2 + y, v = x− 2y2. Find∂2z

∂y∂xin terms of the

partial derivatives of z with respect to u and v.

4. Let p > 0 be a constant and f be of class C1. Assume that ∀t ∈ R and ∀(x, y) ∈ R2 wehave

f(tx, ty) = tpf(x, y) .

Show that

x∂f

∂x+ y

∂f

∂y= pf(x, y) .

5. Find and classify all the critical points of

f(x, y) = 3x2y − 9y3 − x2 .

6. Find the shortest distance from the origin to the surface xyz2 = 2 by using the method ofLagrange multipliers.


7. Find the volume of the solid bounded by y = 4− x2, y = x2 − 4 and −y + 4z = 8.

8. Let a > 0 be a constant. Show thatˆ a

0

ˆ x

0

f(y) dy dx =

ˆ a

0

(a− y)f(y) dy .


1. Suppose 3 < an < 4 for all n and limn→∞

an = 4. Find the interval of convergence of the power

series∞∑n=1

xn

a1a2 · · · an.

Do not forget to check the end points.

2. Let a > 0, b > 0 be constants. Find the open interval I of convergence of the power series

∞∑n=1

(an+ bn

)xn .

For x ∈ I, let f(x) be the sum of the series. Find a closed form for the function f(x).

3. Let U and V be vector spaces over R with zero elements 0U and 0V , respectively. LetT : U → V be a linear mapping, i.e. ∀u1, u2 ∈ U,∀c ∈ R, T (cu1) = cT (u1) and T (u1 + u2) =T (u1) + T (u2). We de�ne

ker(T ) = {u ∈ U : T (u) = 0V } .

a. Show that 0U ∈ ker(T ).

b. Show that ker(T ) is a subspace of U .

4. Let L be a line and P1 a point in R3. Show that the distance between P1 and L is

d(P1, L) =||u×

−−→P0P1||

||v||where u is any vector parallel to L and P) is any point on L.

5. Given the line L :x− 2

1=

y

−2, z = 2 and the point P0(1, 2 − 1) �nd the equation of the

plane Π which contains the line L and the point P0 .

6. Given the plane curve R(t) = et cos ti+ et sin tj , show that the angle between R(t) and theacceleration vector a(t) is constant. Find the angle.

7. For the space curveR(t) = 3 sin ti+ 3 cos tj+ 4tk

�nd T, κ, N, B.

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Bi lkent Calculus II Exams - Bilkent Universityotekman/all/allspr.pdf · Bilkent Calculus II Exams 1988-2019 3 ¸ Express V in terms of iterated integrals in Cartesian coordinates