BROWNSVILLE CITY MATH MEETHOMER HANNA HIGH SCHOOL

APRIL 14,2004

.. ,~ .

CALCULUS

N~E _ SCHOOL, _

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UTBITSCMathematicsDepartment (956)57~36Fax (V5e) 574-8637

Spring 2OG4

OISD MATIiEMATlC5 MJo:Jo:T

Solve the initial value problem.

1) ~ = -csc2 x, Y'[~J = 10and {~J = 0dx2 2 2

1\) Il~ (US x I • l f l x III 111/ Sill .x' i l Ox - 5n

dr 22) -= csc 150 cot 150,dO1 2 91A) r = - -tan 159 + -

30 301 89

C) r = -cot2 e + -30 30

dv 13) ~ = r:: + 7, y(9) = -5

dx 2,\/x .

1 205A) y = r:: + 7x - -

'\IX 3-1 r:: 269C) Y =--;-.JX + 7x --4 4

CALcULUS

C) Inj see x I -t- 5x - Ian: 0) Inj sin x I - 5;t

1 2 91B)r = - -cot 159 +-30 30

10) r = -csc3 159 cot2 159 + 3

6

B)y =...rx + 7x - 71

D)y=..jX+7x+61

Evaluate the integral by using trigonometric identities and substitutions to reduce it to standard fonn.4) !(sin x + cos x)2 dx

A) x + sin 2x + C B)x + C

Find the magnitude of the given vector.5) v = (3, i') Round to the nearest tenth, it necessary.

A) 3.~ 8)58

Find the area of the shaded region.6) y = x2 - 4x + 3

10 Y

Jy=x-l

A)~6

B)~6

UTBITSC MATIIEMA TICS DEPARTMENT (9!16)51~

C)x - cos x + C 10) x - r= 2x + C

C) 10 0)7.6

C)3 D)~2

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srsn MATHEMATICS MEET

7)iii

(9.4)

)(

A)~3

B)5

CALCULUS

C)B.3 D)3

Write u as the sum of a vector parallel to v and a vector orthogonal to v.

8) u = 3i + 4;, v = -3; - 4k

A)U=f36j+ 48 k)+(3i+ 64j_ 48kJ25 25) 25 25

C) u = - ~ i- : i) +[~ i-~;-12k)Solve the lntial value problem.

9) (t2 + 2t) : = Sx + 7, x(t) = 1

11111A) 5 In 1Sx + 71 = I In t - I In It + 21.+5" In 12 + I In 3

1 1 1 1B)51n 1Sx + 71 = Iin 1t + 21-51n 12 +Iln 3

1 1 1 1C) 5 In 1Sx + 71 = I In t + Iln I t + 21 + "2In 3

1 1D) In 1Sx + 71 = 2" In t - "2 In 1t + 21 + In 12 + In 3

Evaluate the integral.10) f cosxdx

1 + 4 sin x

A) 4 In 11 + 4 sin x] + C B)!In 11.+ 4 sin x I+ C

2• 11) f 18x24X3 dx

1360A)1n4 B) 393,192

In 4

2U12) f log5(x + 5) dx

o x + 5

A) In 20 -In 52

B)21n 5

tJTBf1'SC' MATHEMA ncs DEPARTMVlT (75') 5744Qf

B)

t36. 48 i) ( 36. 123 )u = - 25"1- 25"Jj + - 25"1 - 25j - 12k

0) 36. 48k) (3' 136. 48)U = -J + - + 1+ -J + -k25 25 25 25

C)ln 11 + 4 sin x I + C 0)4 sin x + C

C) 393,192 60)-1-+Cnx

C) In 5 - 1 D) 31n 52

FAX (75') 57~1 SPRING 2004 PAGE 2

nrsn MATIIF:Mt\T1C'S /'.U:Jo:r CALCULUS

213) f sin2 5~x dx

-2 2

A)22

B)5 C)2 D) 1

Find all possible functions with the given derivative.

14) y' = 6x2 + 1

A) 2x3 + C (3) 12x +- C C)x + C D) 2x3 + x + C

Find the derivative ot the function.

1 [11-115) Y = S(7x +- 7) 3 + 1 - -;33 (3.)-

A) s' = S(7x) 2 - x4 J3 (1 .)-2C)y'=S«7x+7»2- 1- x3J

21 3 ( 1 J-2B)y' =7«7x +- 7»2 - - 1--5 x4 x3

7 3 [ 1)-2O)y' =t«7x + 7}}2+ x4 1--;3")

The position vector of a particle is r(O.Find the requested vector.16) The velocity at t = 2 for r(t) = (2 - 2t2)i + (8t + 10)j - e-3t}(

A) v(2) = -4i +8j + 3e-6kC) v(2) = -8i of· 8j - 3e-6k

B)~2) = 81+ 8j .•.3e-6k

O)v(2) = -8i + 8j + 3e-6k

Evaluate the integral using the given substitution.17) /t2(4x - 6)-6 dx, u = 4x- 6

A) -l(4x - 6)-5 + C B)(4x - 6)-5 + C5

6C)--(4x - 6)-5 + C5

3O) - -(4x - 6)-7 + C7

Solve the problem.18) At time t ~ 0, the velocity of a body moving along the s-axis is v = t2 - 8t +7. When is the body's velocity

increasing? .A)t<7 B)t<4 01>7 O)t>4

19) At lift-off, a rocket weighs 40.2 tons, including the weight of fuel.'lt is fired vertically, and the fuel is consumedat the rate of 2.28 tons per 1,000ft of ascent How much work is done in lifting the rocket to an altitude of 11,000ft?A) 3.04 x 105 ft-ton B)3.73 x 105 ft-ton C)3.50 x 105 ft-ton 0) 5.80 x lOS It-ton

20) A charged particle of mass m and charge q moving in an electric field E has an acceleration a given by a = ..9E.,m

F· d daIII drn

A)~=~dm m2

da mC)-=--dm qE

da0)- =qEmdm

, trrBII'SC MATHEMA rics DEPARTMI.NT <"'> 57~ ,.AX ~ 574-4iQ7 SPRINC 20004 PAGE 3

nrsn MATHEMATICS MElIT CALCULUS

. ~21) Find the tangent to y = cot x at x = '4'

ItA) y = 2x - - + 1

2B) y = 2x + 1

~C)y = -2x +-

2D) y = -2x + .:: + 1

. 2

22) If you invest P dollars in an account which earns an annual interest rate of 1"%, after 3 years the amount wit I be

A = 1'[1 + 1~O J 3 Find ~~ assurn ing I' is constant.

dA [ r 12A)- = 3r 1+-dr 100

dA [ r JC)-=31+-dr 100

0) dA = _3_ r[l + -r-12dr 100 100

dA (r J0)-=300 P1+-dr 100

23) Find the area of the region bounded from above and below by the curves y = sec2 x and y = cos x, 0 s x s :' and

on the right by the line x = :.A) 0.7071 8)0.2929 C) 1.7071 D) 00סס.1

24) The power P (in W) generated by a particular windmill is given by P = 0.015V3 where V is the velocity of thewind (in mph). Find the instantaneous rate at change of power with respect to velocity when the velocity is11.5 mph.A)6.0 W/mph 8)0.5 W/mph C)4S.6 W/mph 0)13.2 W/mph

Write the sum without sigma notation and evaluate it4~ k225) L.J

k = 1 10

12 22 32• 42A)-+-+-+-=3

10 10 10 10

12 22 32 42 3C) "10 +"10 +"10 +"10 ="10

8)£+ 42 =.£10 10 10

D)£.22 • 32 • 42 = 56710 10 10 10 00סס1

Find the values of x for which the geometric series converges.00

26) 2: (-I)n (lOx)2nn=O

A) I x I < 10 B) Ixl < 201

C) Ixl <To 0) l x I < 1

Find the absolute extreme values of each function on the interval.27) f(x) = csc x -..:: < x < ~, 2 - - 2

A) Maximum value: does not exist; minimum value: does not existB) Maximum value otO at x = -x; minimum value of -1 at x = xC) Maximum value of -1 at x = 11:; minimum value of 1 at x = 0

0) Maximum value of 1 at x = x; minimum value of -1 at x = x

tffBll'SC' MATREMA TICS DEPARTMENT <M') 5'7~ FAX<"5') 514-66)7 SPRING2004 PAGE4

OISD MATIIEMA TICS MEET

Graph the (unction and give the period.

28) Y = CO{ x + ~)

A) [Jj

I r 1 r

. -z . -2

Period: 211: IIPeriod:2

C) 0)z ,. 2 r .

• -2 '~

Period: I Period: 211:

Craph the equation over the given interval.29) x; 3 cas t - cas 3t, y == 3 sin t + sin 3t, over 0 s t s 2Jt

A) B)5 ,

, I , I

-s -sC) 0)

s , 5 1

, I• I

-s

UTBII'SC MATHEMArtcs DUARTMENT <"'> 57"""" FAX (956) 57~7 SPRING 2004

CALCULUS

PAGES

RISO MAT IIF.MATICS ~H:~" CALCULUS

Express the vector in the torm Ii+ bj.

30) The unit vector obtained by rotating j clockwise : radians about the origin

A) ~j6

B) 1. ~.-1+-12 2

C) i O)~· -n.-1+-12 2

Evaluate the integral by multiplying by I form of 1and using a substitution (if necessary) to reduce it to standard form

31) f 1 dx1 - sin x

A) tan x - see x + C B) - cot x - csc x + C C) cot x + csc x + C 0) tan x + see x :- c

Express the vector in the form ai + bj.

32) pQ where P is the point (5,9) and Q is the point (-3, 6)

A)4i+9j B)-8i-3j C}8i + 3j 0) 2i + 15j

Find the volume of the solid generated by revolving the shaded region about the given axis.33) About the y-axis .

6 Y

5

4

3+-----~~~---------

2 3 4 5 6 It

B)18x C)3x

Give answers to 3 significant digits of precision. [A non-graphing calculator is needed for these problems.]34) Express 6i + 5; as a product of its length and direction.

A) 0.0909 [f i + f j) B)0.128 (6i + 5j) q7.81 (0.768 i + 0.640 j) 0) 11 (0.545 i + 0.455 j)

find the derivative.. 35) 5 = t7 <:OS t - 9t sin t - 9c:os t

A) ds = _ t7 sin t + 7t6 cos t - 9t cos tdt

C) ~: = t7 sin t - 7t6 cos t + 9t cos t

B) ds = - t7 sin t + 7t6 cos t - 9t <:OS t - 18 sin tdt

O} ~~ = - 7t6 sin t - 9 cos t + 9 sin t

Find the indicated value.

36) Find the amplitude for += (-28 - ~).

A)l 8)l2

C)2x D)2

UTBITSC MATHEMA TICS DEPARTMENT <"') S7~ FAX <"') S744637 SPRING 2004 PAGE 6

OISO MATHEMATIC'S MEET CALCULl"S

Use the shell method to find the volume ot the solid generated by revolvi th . b .lines about the x-axis. 109 e region ounded by the given curves and

37) x = 4y2, x = - 4y, Y = 4A) 2048 It

3B) 512 It

3

{

0, X :5 0

38) Let f(x) = x cas ~, x > a 050.4

0.30.2

0.1

..o.~.4-0.~.J:S.\-4.l..0.3~4..0.5

Find lim f(x) if it existsx-aA) 1 B)Does not exist

Find the derivative of y with respect to O.39) y = ge9(sin e - cos e}

A) 18et1 sin eC)O

cio

C)..!.£@.1t3

B) get1(sin e - cas e) + get1

D) 18et1(sin e - cos e)

Express the given quantity in tenns of sin x or cos x.40) sin (5,.; + x)

A) sin x B)cos x - sin x

D) 0.0007

C) -sin x 0) cos x + sin x

Compute the vector projection of u onto v. Give answers to 4 5ignificant digits.41) u = 2i - 6j and v = -7i + 5;

A) -0.5946v B)-44v C) 6.325v D)-14i-30j

Evaluate each integral by reducing or separating the fraction and using a substitution (if necessary) to reduce it tostandard form. .-

f 4x42) 4x + 7 dx

7A) x + 4'ln I4x + 71 + C B) x - In 14x + 71 + C

7C) x - -In 14x + 71 + C

4

Set up an integral for the length of the curve.43) x = yl /7, a ~ y s 2

A) ~ ~ 49yi2 /7 dy

2

C) ~ 149y:2/7 dy

2B)f

o2

O)fa

VTBfTSCMATnEMA ncs DEPARTMENT (9S6)S7~

7t/7 + 17y6/7 dy

FAX (9S(j)574-6637 SPRING %004

49y12/7 + 149y12/7 dy

D) In 14x + 71 + C

PAGE 7

IJ/SD ,t,f,H/I EM" TICS MEET CALCULUSEstimate velocity at the given point using the given d.tta.

44)r /l

~ Ir• /J

/ sI

~I .......-1 '.)

) .Use the graph to find the slopes of UQ UR US, and UT to estimate velocity at x=3,

A) 6 B)2 C) 4 D) 0

Find the length of the curve.

45) y = ~(x4/3 - 2x2/3) from x = 1 to x = 278

A)93 B) 36 D) 1534

Find the limit and determine if the function is continuous at the point being approached.

46) lim sin (921tcas (tan x»

x--2Jt

A) Does not exist; yes B) Does not exist; no C) 1; no D) 1; yes

Find the value of..s!I. for the curve at the point defined by the given value of t.dx2

47) x = t + cas t, Y = 2 - sin t, t = ~

A)-4 B)-2 0)- .J3

Solve the problem by integration.

48) Find the volume generated by rotating the area bounded by y = 3 2 ,x = 4, x = 8, and y = 0 about thex + 9x + 8x

1

y-axis.2 9

A)7"1t In 16 1 278),1t In 20

2 27C)-:;" In 20

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves andlines about the y-axls.

49) y = 4x2, y = ~

A)E.1t5

B)31t3C)-1t5

6D)-It5

Replace the polar equation with an equivalent Cartesian equation.SO) r=S

A) x + Y = 2S B) x2 + y2 = 2S C) x + y = 5 D) x2 - y2 = 25

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