CALCULUS I - Sogang

CALCULUS I

Exam I, March 25, 2011

You must show all your work. Each problem is worth 10 points.

(1) Using the Squeeze Theorem, prove that

limx→0

sinx

x= 1.

(2) Using (1), prove that

limx→0

cos x− 1x

= 0.

(3) Using (1) and (2), prove that ddx cos x = − sinx.

(4) Using the ε-δ definition, prove that

limx→−3

x2 = 9.

(5) Prove that if a function f is differentiable at a, then it is continuous at a.

(6) Find y′′ by implicit differentiation if√

x +√

y = 1.

(7) Car A is traveling west at 90 km/h and car B is traveling north at 100 km/h.Both are headed for the intersection of the two roads. At what rate are thecars approaching each other when car A is 60 m and car B is 80 m from theintersection?

(8) Find the linearization L(x) of the function f(x) = 1/(1 + 2x)4 at a = 0 anduse it to approximate the number 1/1.024.

(9) Consider an isosceles triangle 4AOB with A(1/2, a), O(0, 0), B(1, 0). Thebisector of angle O intersects the side AB at the point P (x, y). Supposethat the base OB remains fixed but the altitude |AM | = a of the triangleapproaches 0, so A approaches the midpoint M of OB. What happens toP (x, y) during this process? (Hint: tan 2θ = 2 tan θ/(1− tan2 θ).)

(10) The figure shows a circle with radius 1 inscribed in the parabola y = x2.Find the center of the circle.

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CALCULUS I

Exam II, May 6, 2011

You must show all your work.

(1) [10 pts] Find the linearization L(x) of the function f(x) = 1/(1+2x)4

at a = 0 and use it to approximate the number 1/1.024.(2) [6 pts] Find the absolute maximum and minimum values of the func-

tionf(x) = x3 − 3x2 + 1, −1

2 ≤ x ≤ 4.

(3) [6 pts] Suppose that f(0) = −3 and f ′(x) ≤ 5 for all values of x. Howlarge can f(2) possibly be? (Hint: Use the Mean Value Theorem.)

(4) [2+2+7+4+5+5 pts] Considering the function f given by

f(x) =x3

1 + x2,

answer the following:(a) Find the domain of f .(b) Find the x- and y-intercepts.(c) Find the asymptotes (horizontal, vertical, slant) of y = f(x).(d) Find the interval(s) on which f is increasing.(e) Find the inflection points of of the graph of y = f(x).(f) Sketch the graph of y = f(x) including the inflection points.

(5) [10 pts] Define a definite integral.(6) [15 pts] Let f : [a, b] → R be continuous, and define the function g

by

g(x) =∫ x

af(t)dt a ≤ x ≤ b.

Show that g is differentiable on (a, b) and g′(x) = f(x).(7) [10 pts] Find

d2

dx2

∫ x

0

(∫ π2

√t3

sin2011 u du

)dt.

(8) [10+8 pts] Let R denote the region bounded by y = x−x2 and y = 0.(a) Find the volume of the solid obtained by rotating the region R

about the line x = 2.(b) Find the volume of the solid obtained by rotating the region R

about the x-axis.

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CALCULUS I

Final Exam, June 13, 2011

Show all work.(1) Let f : [a, b] → R be continuous, and define the function g by

g(x) =∫ x

af(t)dt a ≤ x ≤ b.

Show that g is differentiable on (a, b) and g′(x) = f(x).

(2) Using Laws of Logarithms, show that

limx→∞ lnx = ∞.

(3) Given the function y = f(x) = ln(4− x2),(a) Find the domain of the function f .(b) Find the x- and the y-intercepts of the graph of f .(c) Find all the asymptotes of the graph of f .(d) Using the first and the second derivative tests, and using clearly

marked axes, sketch the graph of y = ln(4− x2).

(4) Show thate = lim

x→0(1 + x)

1x .

(5) Show thatd

dxsin−1 x =

1√1− x2

.

(6) Show thatsinh−1 x = ln(x +

√x2 + 1)

and then find the derivative of sinh−1 x.

(7) Find the length of the arc y = x2 − 18

lnx, 1 ≤ x ≤ e.

(8) Evaluate the following:(a) y′ =? when y = x

√x

(b) limx→0+

xx

(c)∫ 1

0tan−1 xdx

(d)∫

x√3− 2x− x2

dx

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CALCULUS I

EXAM I, APRIL 3, 2013


(1) (a) Using the ϵ-δ definition, prove that [10]

limx→2

√x =

√2.

(b) Given ϵ > 0, which δ > 0 does not satisfy the condition [5]

0 < |x− 2| < δ ⇒ |x2 − 4| < ϵ?

(i) δ = min{1, ϵ/5} (ii) δ = min{1/5, ϵ/4}(iii) δ = min{1/2, ϵ/5} (iv) δ = min{2, ϵ/6}

(2) Using the Squeeze Theorem:

(a) Prove that limx→0

sinx

x= 1. [10]

(b) Suppose f is a function with the property that |f(x)| ≤ x2 for all x.Show that f(0) = 0 and then show that f ′(0) = 0. [5]

(3) Using the definition of derivative:(a) Find the exact value of [10]

limx→0

sin(3 + x)2 − sin 9

x.

(b) If f and g are differentiable functions with f(0) = g(0) = 0 and g′(0) ̸=0, show that [5]

limx→0

f(x)

g(x)=

f ′(0)

g′(0).

(4) Find the linearization L(x) of the function f(x) = 1/(1 + 2x)4 at a = 0 anduse it to approximate the number 1/1.024. [10]

(5) State precisely the following theorems:• The Intermediate Value Theorem [5]• The Extreme Value Theorem [5]• Fermat’s Theorem [5]• The Mean Value Theorem [5]

(6) Let f be a non-constant function that satisfies the following three conditions:• f is continuous on [a, b]• f is differentiable on (a, b)• f(a) = f(b)

Prove that there is a number c ∈ (a, b) such that f ′(c) = 0. [15]

(7) Using the first and second derivative tests, sketch the graph of the function

f(x) = x2/3(6− x)1/3. Find inflection points. [10]

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CALCULUS I

EXAM II, MAY 13, 2013


(1) Using the first and second derivative tests, sketch the graph ofthe function f(x) = x2/3(6− x)1/3. Find inflection points. [10]

(2) Consider the regions R1 and R2 given below.

(a) Find the volume VA of the solid obtained by rotating R1

about the y-axis. [10]

(b) Let VB be the volume of the solid obtained by rotatingR2 about the line x = k. Find the value of k for whichVA = VB. [10]

(3) Let R be the region bounded by the curve y = x− x2 and thex-axis. Find the volume of the solid obtained by rotating Rabout the line x = 2. [10]

(4) Show that [10]

limx→∞

∫ x

1

1

tdt = ∞ and then lim

x→0+

∫ x

1

1

tdt = −∞.

(5) Sketch the graph of y = ln(4− x2) in detail. [15]

(6) Let f(x) = expx be the inverse of g(x) = ln x. Let e be a

number for which g(e) = 1. Show e = limn→∞

(1 +

1

n

)n

. [10]

(7) Sketch the graph of y = e1/x in detail. [15]

(8) Find y′ when y =√x√x. [10]

1

CALCULUS I


Show all work.

(1) State precisely the following theorems:• The Intermediate Value Theorem [5]• The Mean Value Theorem (for derivative) [5]

(2) Consider the region R given by

R ={(x, y) | 0 ≤ y ≤ 8, 0 ≤ x ≤ − 1

16(y − 4)2 + 4

}.

Find the volume V of the solid obtained by rotating R aboutthe y-axis. [10]

(3) Prove the following formulas:

(a)d

dxsin−1 x =

1√1− x2

[10]

(b) cosh−1 x = ln(x+√x2 − 1), x ≥ 1 [10]

(c)

∫1√

x2 − 1dx = ln(x+

√x2 − 1) + C [10]

(4) (a) Show that [7]∫sec3 θ dθ =

1

2(sec θ tan θ + ln | sec θ + tan θ|) + C

(b) Find the area of the surface generated by rotating the curvey = ex, 0 ≤ x ≤ 1, about the x-axis. [13]

(5) Using Integral Test, show that the p-series∑ 1

npis convergent

for p > 1. [10]

(6) Using Ratio Test, find the set of convergence for the power series∑ n

3n+1(x+ 2)n. [10]

(7) Find a power series of f(x) = tan−1 x and then find its set ofconvergence. [10]

1

CALCULUS I



(1) (a) Using the ε-δ definition, prove that limx→2

1x

=12. [10]

(b) Give a precise definition of limx→a

f(x) = +∞. [5]

(c) Using (b), prove that limx→0

1x2

= +∞. [5]

(2) Define the following:(a) A function f is continuous at a. [5](b) A function f is differentiable at a. [5](c) The differential dy for a differentiable function y = f(x). [5]

(3) (a) Using the Squeeze Theorem, prove that limx→0

sinx

x= 1. [10]

(b) Using (a), prove that limx→0

cos x− 1x

= 0. [5]

(c) Prove thatd

dx(cos x) = − sinx. [5]

(4) State precisely the following theorems:• The Intermediate Value Theorem [5]• The Extreme Value Theorem [5]• The Mean Value Theorem for derivative [5]

(5) Using the implicit differentiation, find dy/dx as a function of x if [10]

sin y = x, −π

2≤ y ≤ π

2.

(6) (a) Air is being pumped into a spherical balloon so that its volume increasesat a rate of 100 cm3/s. How fast is the radius of the balloon increasingwhen the radius is 25 cm? Hint: V (t) = 4

3πr(t)3 and use the ChainRule. [8]

(b) When the radius of the spherical balloon was measured to be 25 cm, ithas a possible error in measurement of at most 0.05 cm. What is themaximum error in using this value of the radius to compute the volumeof the balloon? Hint: Use the differential. [7]

(7) Using the first and second derivatives, sketch the graph of the functionf(x) = cos x/(2 + sin x) over [0, 2π]. Indicate clearly the concavity of thegraph and find the coordinates of all inflection points and (local) maximaand minima. [20]

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CALCULUS I


Show all work.

(1) (a) If f is continuous on [a, b], then prove that the function g definedby

g(x) =∫ x

af(t) dt

is differentiable on (a, b), and g′(x) = f(x). [10](b) Using (a), prove that lnx has derivative 1/x. [5]

(2) Prove the following formulas:

(a)d

dxsec−1 x =

1x√

x2 − 1[10]

(b) cosh−1 x = ln(x +√

x2 − 1), x ≥ 1 [10]

(3) (a) Evaluate the following integral using a trigonometric substitu-tion [10]

∫ 3√

3/2

0

x3

(4x2 + 9)3/2dx

(b) Evaluate the following improper integral [10]∫ ∞

0

1x2

dx

(4) Find the arc function for the curve y = x2− (lnx)/8 taking (1, 1) asthe starting point. [10]

(5) Find the area of the surface generated by ratating the curve y =3 cosh(x/3),−3 ≤ x ≤ 3, about the x-axis. [10]

(6) Given the polar equation r = 1 + cos θ,(a) sketch the polar curve, [5](b) find the slope of the tangent to the curve when θ = π/6, [10](c) find the length of the curve. [10]

1

CALCULUS I



(1) (a) Using the ε-δ definition, prove that limx→1

√x = 1. [10]

(b) Give a precise definition of limx→a−

f(x) = −∞. [5]

(c) Using (b), prove that limx→0−

1x3

= −∞. [5]

(2) (a) Prove that f is continuous at a if and only if limh→0

f(a + h) = f(a). [5]

(b) Using (a), prove that the sine function f(x) = sinx is continuous ev-erywhere. [5]

(c) Use the Squeeze Theorem to evaluate limx→0+

√xesin(π/x). [5]

(3) Determine whether each f is differentiable at 0. When it is possible, findf ′(0).

(a) f(x) =

{x sin 1

x if x 6= 00 if x = 0

[5]

(b) f(x) =

{x2 sin 1

x if x 6= 00 if x = 0

[5]

(c) f(x) = x|x| [5]

(4) (a) State precisely The Intermediate Value Theorem [5]

(b) Using (a), show that there is a root of the equation 4x3 − 6x2 + 3x = 2between 1 and 2. [5]

(5) If x2 + xy + y3 = 1, find the value of y′′ at the point where x = 1. [10]

(6) When f is a one-to-one differentiable function, it is known that f has theinverse function f−1 and f−1 is differentiable.

Now use implicit differentiation to show that (f−1)′(x) =1

f ′(f−1(x))pro-

vided that the denominator is not 0. [10]

(7) A man walks along a straight path at a speed of 1.5 m/s. A searchlight islocated on the ground 6m from the path and is kept focused on the man.At what rate is the searchlight rotating when the man is 8m from the pointon the path closest to the searchlight? [10]

(8) Use a linear approximation to estimate the number 3√

7.9. [10]

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CALCULUS I

EXAM II, MAY 15, 2015


(1) When f is a one-to-one differentiable function, it is known that fhas the inverse function f−1 and f−1 is differentiable.

Now use implicit differentiation to show that (f−1)′(x) =1

f ′(f−1(x))provided that the denominator is not 0. [10]

(2) (a) State the Mean Value Theorem. [5](b) Prove that if f ′(x) = 0 for all x in (a, b), then f is constant on

(a, b). [10]

(3) (a) If f is continuous on [a, b], then prove that the function g definedby

g(x) =

∫ x

af(t) dt

is differentiable on (a, b), and g′(x) = f(x). [15](b) Using (a), prove that lnx has derivative 1/x. [5]

(4) Let R be the region enclosed by the curves y = x and y =√x.

(a) Find the volume of the solid obtained by rotating the region Rabout the line y = 2. [10]

(b) Find the volume of the solid obtained by rotating the region Rabout the line x = 2. [10]

(5) Let f(x) = ln(1− lnx) be given.(a) Find the domain of f . [4](b) Find the x-intercepts and y-intercepts, if any, of y = f(x). [4](c) Find the vertical asymptotes giving the reason. [4](d) Find f ′(x) and f ′′(x) [4](e) Using all the information obtained above (including the first

and second derivative tests), sketch the graph of y = f(x). [4]

(6) Given y = x√x, x > 0,

(a) find y′. [5](b) find lim

x→0+y and lim

x→∞y. [5]

(c) Taking into accounts of (a) and (b), sketch the graph of thefunction y. [5]

1

CALCULUS I


Show all work.The marks for each question are indicated in square brackets [ ].

(1) (a) State the Intermediate Value Theorem. [5]

(b) State the Mean Value Theorem. [5]

(c) Prove that if f ′(x) = 0 for all x ∈ (a, b), then f is constant on(a, b). [5]

(2) (a) Find the log form of y = cosh−1 x [5]

(b) Using (a), find the derivative of y = cosh−1 x [5]

(c) Using (b) and (a), show that [5]∫1√

x2 − a2dx = ln(x+

√x2 − a2) + C.

(d) Find the exact value of

∫ 3

2

dx√4x2 − 9

. [5]

(3) Find the arc length for the curve y = ln(sinx), π/2 ≤ x ≤ 3π/4. [15]

(4) (a) Show that [10]∫ √1 + u2du =

1

2u√

1 + u2 +1

2ln |u+

√1 + u2|+ C

(b) By using improper integral, find the area of the surface gener-ated by rotating the infinite curve y = e−x (x ≥ 0) about thex-axis. [20]

(5) Find the MacLaurin series

c0 + c1x+ c2x2 + c3x

3 + · · ·for the following functions and the radii R of convergence.

(a) f(x) =1

x+ 2, and its radius of convergence. [5]

(b) f(x) = ln(1 + x), and its radius of convergence. [5]

(c) f(x) = tan−1 x, and its radius of convergence. [5]

(6) Use the definition of the MacLaurin series to find the Maclaurinseries for y = e−x first and then y = xe−x. [5]

1

CALCULUS I

EXAM I, MARCH 24, 2017


The marks for each question are indicated in square brackets [ ].

(1) (a) Give a precise definition of limx→a+

f(x) = −∞. [5]

(b) Using (a), prove that limx→0+

(− 1

x3

)= −∞. [10]

(2) Find y′ if(a) tan(x+ y) = y2 secx. [5]

(b) y =x

tan−1√x. [5]

(3) If x2 + xy + y3 = 1, find the value of y′′ at the point where x = 1.[10]

(4) (a) Find the linearization of f(x) = 3√x+ 7 at a = 1. [5]

(b) Use (a) to estimate the number 3√7.9. [5]

(5) (a) Using the exponential definition of hyperbolic functions, showthat [5]

sinh(2x) = 2 sinhx coshx.

(b) Derive the formula cosh−1(x) = ln(x+√x2 − 1), x ≥ 1. [10]

(c) Find y′ if y = x cosh−1(x3

)−

√9 + x2. [10]

(6) (a) State precisely The Mean Value Theorem [5]

(b) Using (a), show that if f ′(x) = 0 for all values x ∈ (a, b) then fis a constant function on (a, b). [10]

(7) Evaluate the following:

(a) limx→π−

sinx

1− cosx[5]

(b) limx→1+

lnx · tan(πx

2

)[5]

(c) limx→0

tanx

tanhx[5]

1

CALCULUS I

EXAM II, APRIL 21, 2017

You must show all your work.The marks for each question are indicated in square brackets [ ].

(1) Find y′ if

(a) y =x

tan−1√x

[5]

(b) y = x cosh−1(x3

)−

√9 + x2 [5]

(2) Let f(x) = ln(sinx) be given.

(a) Find the domain of f . [4]

(b) Explain about the symmetric and periodic properties of thegraph of y = f(x). [4]

(c) Find the x-intercepts and y-intercepts, if any, of y = f(x). [4]

(d) Find the vertical asymptotes giving the reason. [4]

(e) Find f ′(x) and f ′′(x) [5]

(f) Using all the information obtained above (including the firstand second derivative tests), sketch the graph of y = f(x). [4]

(3) Let R be the region surrounded by the curve y =√x and the straight

lines x = 0, x = 1 and y = 0.

(a) Find the volume of the solid obtained by rotating the region Rabout the line y = 2. [10]

(b) Using the cylindrical shell method, find the volume of thesolid obtained by rotating the region R about the line x = −1.

[10]


(a) limx→1+

lnx · tan(πx

2

)[5]

(b)

∫ π4

0sec3 x dx [10]

(c)

∫x√

3− 2x− x2dx [10]

(d)

∫2x2 − x+ 4

x(x2 + 4)dx [10]

(e)

∫ ∞

1

1

xpdx when p > 1 [10]

1

CALCULUS I

Exam 3, May 19, 2017

Show all your work.The marks for each question are indicated in square brackets [ ].

(1) Find y′ if(a) tan(x+ y) = y2 secx. [5]

(b) y =x

tan−1√x. [5]

(c) y = x cosh−1(x3

)−

√9 + x2. [5]


(a)

∫x√

3− 2x− x2dx [10]

(b)

∫ ∞

1

1

xpdx when p > 1 [10]

(3) Consider the cycloid given by the parametric equations

x = r(θ − sin θ), y = r(1− cos θ)

where r > 0.(a) Find an equation of the tangent line at θ = π

3 in the formy = mx+ b. [10]

(b) Find the area under one arch. [10]

(c) Find the length of one arch. [10]

(d) Find the area of the surface obtained by rotating one arch aboutthe x-axis. [10]

(4) Given the polar equation r = 1 + sin(2θ),(a) Find a symmetry of the polar curve r = 1 + sin(2θ). [5]

(b) State the polar equation of the tangent line at the pole. [5]

(c) Draw the polar curve r = 1 + sin(2θ). [5]

(5) Using the Integral test, show that the series

∞∑n=2017

1

n lnnis diver-

gent. [10]

1

CALCULUS I

Exam 4, June 16, 2017

Show all your work. No calculator is allowed.The marks for each question are indicated in square brackets [ ].

(1) Derive the formula [5]

cosh−1 x = ln(x+√

x2 − 1), x ≥ 1.


(a)

∫x√

3− 2x− x2dx [5]

(b)

∫ 0

−π4

sec3 θ dx [5]

(3) Consider the cycloid given by the parametric equations

x = r(θ − sin θ), y = r(1− cos θ)

where r > 0.(a) Find the length of one arch. [5]

(b) Find the area of the surface obtained by rotating one arch aboutthe x-axis. [5]

(4) Determine the convergence of the following series. (State the nameof the tests and check carefully the conditions of the tests that youare using.) [10+10+10]

(a)

∞∑n=1

(−√2)n

n4 + 1(b)

∞∑n=2

1

n ln n√ln n

(c)

∞∑n=1

cos nπ√n

(5) Find the first three nonzero terms of the MacLaurin series for f(x) =

ex2−4x. [5]

(6) Write down the Maclaurin series of ln(5− x) and find its interval ofconvergence. [10]

(7) For a vector function r(t) = ⟨f(t), g(t), h(t)⟩, prove that if |r(t)| = c(a constant) then r(t) is orthogonal to r(t) for all t. [10]

(8) Show that1

ris the curvature of the space curve

r(t) = ⟨r cos t, r sin t, 0⟩ (0 ≤ t ≤ 2π)

where r is a positive constant. [10]

(9) Consider the curve r(t) = ⟨et sin t, et cos t,√2 et⟩, 0 ≤ t ≤ 10.

(a) Find the arc length function s = s(t) of r(t). [5]

(b) Write down the reparametrization of r(t) in terms of the arclength s. [5]

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CALCULUS I - Sogang