Math Assistance Center - YSU...MATH 1572 Final Exam Practice Problems - Math Assistance Center ____...

Name: ______________________ Class: _________________ Date: _________ ID: A

1

MATH 1572 Final Exam Practice Problems - Math Assistance Center

____ 1. Differentiate the function.

y = xe7x

a. y′ = xe7x7 ln x +

1

x

Ê

Ë

ÁÁÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜̃̃˜

b. y′ = e7xxe7x7 ln x +

1

x

Ê

Ë


ˆ

¯

˜̃̃˜̃̃˜̃̃˜

c. y′ = e7xxe7x7 ln x +

1

x 2

Ê

Ë


ˆ

¯

˜̃̃˜̃̃˜̃̃˜

d. y′ = 7x7xe 7ln x +1

x

Ê

Ë


ˆ

¯

˜̃̃˜̃̃˜̃̃˜

e. y′ = e7xxe7x7 ln x −

1

x

Ê

Ë


ˆ

¯

˜̃̃˜̃̃˜̃̃˜

Name: ______________________ ID: A

2

____ 2. Differentiate the function.

y = ln x3 sin2 xÊ

ËÁÁÁÁ

ˆ

¯˜̃̃˜

a. y′ =3 cos x + 2x sin x

x cos x

b. y′ =3x2 + cos x sin x

cos x

c. y′ =3 sin x + x cos x

x3 sin2 x

d. y′ =3 sin x − 2x

x sin x

e. y′ =3 sin x + 2x cos x

x sin x

____ 3. Find the limit.

limx → 0 +

ln 5x

x

a. ∞b. πc. 0d. 5e. −∞

Name: ______________________ ID: A

3

____ 4. Evaluate the integral.

8sin θ cos θ dθ∫

a.8sin θ

ln 8( )+ C

b.8sin θ

ln sin θ( )+ C

c.8cos θ

ln 8( )+ C

d.8cos θ

ln sin θ( )

e. none of these

Name: ______________________ ID: A

4

____ 5. Find f ′(x).

f(x) = 5ex − 3x2 arctan x( )

a. f ′(x) = 5ex −3x2

1 + x2− 6x arcsin x( )Ê

ËÁÁ ˆ

¯˜̃

b. f ′(x) = 5ex −3x2

1 + x2− 2x arctan x( )Ê

ËÁÁ ˆ

¯˜̃

c. f ′(x) = 5ex −3x2

1 + x2+ 6x arctan x( )Ê

ËÁÁ ˆ

¯˜̃

d. f ′(x) = 5ex −3x2

1 + x2− 6x arctan x( )Ê

ËÁÁ ˆ

¯˜̃

e. none of these

____ 6. Use logarithmic differentiation to find the derivative of the function.

y = x6x

a. y′ = 6x6x 6 ln x + 1( )

b. y′ = −6x6x ln x + 6( )

c. y′ = 6 ln x + 1( )

d. y′ = xx ln 6x + 1( )

e. y′ = 6x6x ln x + 1( )

Name: ______________________ ID: A

5

____ 7. Find the derivative of the function.

y = 3 sin−1 x2ÊËÁÁÁÁ

ˆ¯˜̃̃˜

a. y′ =6x

1 + x4

b. y′ =x

1 − x2

c. y′ =6x

1 − x4

d. y′ =6x

1 − x2

e. None of these

____ 8. Find the derivative of the function.

y = (4x2 + 1) tan–1 2x

a. 21 + 4x2

b. 4x2 tan–1 2x + 2

c. 2x1 + 4x2

d. 8x tan–1 2x + 2

Name: ______________________ ID: A

6

____ 9. Find the limit.

limx → 0

ex − 1

sin 2x

a. ∞

b.1

2

c. −2

d. −∞

e. 0

____ 10. Evaluate the indefinite integral.

x cos 9xdx∫

a.181

sin 9x +x9

cos 9x + C

b.19

cos 9x +x9

sin 9x + C

c.181

cos 9x +x9

sin 9x + C

d.x81

cos 9x +x9

sin 9x + C

e. None of these

Name: ______________________ ID: A

7


et 25 − e2t dt∫

a. 252

arcsinet

5

Ê

Ë


ˆ

¯

˜̃̃˜̃̃˜̃̃˜+

12

et 25 − e2t + C

b. arcsinet

5

Ê

Ë


ˆ

¯

˜̃̃˜̃̃˜̃̃˜−

12

25 − e2t + C

c. arcsinet

5

Ê

Ë


ˆ

¯

˜̃̃˜̃̃˜̃̃˜+ 25et 25 − e2t + C

d.252

arcsine2t

5

Ê

Ë


ˆ

¯

˜̃̃˜̃̃˜̃̃˜+

12

5 − et + C

e. 25 arcsinet

5

Ê

Ë


ˆ

¯

˜̃̃˜̃̃˜̃̃˜+

12

25 − e2t + C

Name: ______________________ ID: A

8

____ 12. Find the integral.

cos 5 x sin2 x dx∫

a.15

sin5 x −23

sin3 x + sin3 x + C

b.17

sin7 x +25

sin5 x −13

sin3 x + C

c.17

sin7 x −25

sin5 x +13

sin3 x + C

d.15

sin5 x +23

sin3 x − sin3 x + C


sin3 x cos 6 x dx∫

a. −19

cos 9 x +17

cos 7 x + C

b. −17

cos 7 x +15

cos 5 x + C

c.19

cos 9 x −17

cos 7 x + C

d.17

cos 7 x −15

cos 5 x + C

Name: ______________________ ID: A

9


tan2 x sec6 x dx∫

a.15

tan5 x +23

tan3 x + tan x + C

b.17

tan7 x +25

tan5 x +13

tan3 x + C

c.15

tan5 x +23

tan3 x − tan x + C

d.17

tan7 x +25

tan5 x −13

tan3 x + C

____ 15. Find the integral using an appropriate trigonometric substitution.

1

x2 x2 + 25∫ dx

a.x2 + 25

5x+ C

b. −x2 + 2525x

+ C

c. −x2 + 25

5x+ C

d.x2 + 2525x

+ C

Name: ______________________ ID: A

10

____ 16. Evaluate the integral using the indicated trigonometric substitution.

x3

x2 + 16dx; x = 4tanθ∫

a.13

x2 + 16ÊËÁÁÁÁ

ˆ¯˜̃̃˜

3 2− x2 + 16 + C

b. x2 + 16ÊËÁÁÁÁ

ˆ¯˜̃̃˜

3 2− 4 x2 + 16 + C

c.13

x2 + 16ÊËÁÁÁÁ

ˆ¯˜̃̃˜

3 2+ 16 x2 + 16 + C

d.32

x + 16( )3 2 − 16 x + 16 + C

e. x2 + 16ÊËÁÁÁÁ

ˆ¯˜̃̃˜

3 2− x2 + 16 + C

____ 17. Find the integral using an appropriate trigonometric substitution.

x 9 − x2∫ dx

a.13

x2 (9 − x2 ) 3 2 + C

b. −13

x2 (9 − x2 ) 3 2 + C

c.13

(9 − x2 ) 3 2 + C

d. −13

(9 − x2 ) 3 2 + C

Name: ______________________ ID: A

11

____ 18. Use long division to evaluate the integral.

x2

x + 3dx∫

a.12

x − 9( ) x + 3( ) + 9 ln x + 3|| || + C

b.12

x + 9( ) x + 3( ) − 9 ln x + 3|| || + C

c.x2

2− 6x − 27 + ln x + 3|| || + C

d.x2

2− 6x + 27 + 9 ln x + 3|| || + C

e.x2

2+ 2x + ln x + 9|| || + C

Name: ______________________ ID: A

12


7dx

x2 + 2x + 2ÊËÁÁÁÁ

ˆ¯˜̃̃˜

2∫

a.12

tan−1 x + 7( ) +1

x2 + 2x + 2

Ê

Ë

ÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜̃+ C

b.12

tan−1 x + 2( ) +7

x2 + 2

Ê

Ë

ÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜̃+ C

c.12

tan x + 1( ) +17

x2 + 2x + 2

Ê

Ë

ÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜̃+ C

d.72

tan−1 x + 1( ) +x + 1

x2 + 2x + 2

Ê

Ë

ÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜̃+ C

e.72

tan x + 2( ) +x + 1

x2 + 2x + 2

Ê

Ë

ÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜̃+ C


3x − 5x2 − 2x − 3

dx∫

a. ln (x − 1)(x + 3)2|||

||| + C

b. ln (x + 3)(x − 1)2|||

||| + C

c. ln (x − 3)(x + 1)2|||

||| + C

d. ln (x + 1)(x − 3)2|||

||| + C

Name: ______________________ ID: A

13

____ 21. Use long division to evaluate the integral.

x3 + 4x2 − 12x + 1x2 + 4x − 12

dx0

1

∫

The choices are rounded to 3 decimal places.

a. 0.394b. −4.606c. 10.394d. 5.394e. −9.606

Name: ______________________ ID: A

14


1−e−5x + e5x dx∫

a. − lne5x − 1|||

|||

e5x

Ê

Ë

ÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜̃̃˜̃̃˜̃̃+ C

b. lne5x − 1|||

|||

e5x + 1

Ê

Ë


ˆ

¯

˜̃̃˜̃̃˜̃̃˜̃̃˜̃̃+ C

c. −110

lne5x − 1|||

|||

e5x + 1

Ê

Ë


ˆ

¯

˜̃̃˜̃̃˜̃̃˜̃̃˜̃̃+ C

d.110

lne5x − 1|||

|||

e5x + 1

Ê

Ë


ˆ

¯

˜̃̃˜̃̃˜̃̃˜̃̃˜̃̃+ C

e. − lne5x − 1|||

|||

e5x + 1

Ê

Ë


ˆ

¯

˜̃̃˜̃̃˜̃̃˜̃̃˜̃̃+ C

Name: ______________________ ID: A

15

____ 23. Evaluate the integral or show that it is divergent.

5dx4x2 + 4x + 5−∞

∞

∫

a.π8

b. 54π

c. −π10

d.π5

e. divergent

____ 24. Determine whether the improper integral converges or diverges, and if it converges, find its value.1x33

∞

∫ dx

a.19

b. Diverges

c. 0

d.118

Name: ______________________ ID: A

16

____ 25. Determine whether the improper integral converges or diverges, and if it converges, find its value.3ex

3 + e2x−∞

∞

∫ dx

a. Diverges

b. 3

c.π 3

3

d.π 3

2

____ 26. Find the length of the curve.

y =16

x2 + 4ÊËÁÁÁÁ

ˆ¯˜̃̃˜

3 2, 0 ≤ x ≤ 3

a. 5.5000b. 6.5000c. 7.5000d. 4.5000e. 8.5000

Name: ______________________ ID: A

17


y = 2 ln sinx2

Ê

Ë

ÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜̃,

π5

≤ x ≤ π

a. ln 5( )

b. ln 2 + 5Ê

ËÁÁÁÁ

ˆ

¯˜̃̃˜

c. ln 2 5Ê

ËÁÁÁÁ

ˆ

¯˜̃̃˜

d. ln 5Ê

ËÁÁÁÁ

ˆ

¯˜̃̃˜

e. None of these

____ 28. A gate in an irrigation canal is constructed in the form of a trapezoid 6 ft wide at the bottom, 12 ft wide at the top, and 2 ft high. It is placed vertically in the canal, with the water extending to its top. Find the hydrostatic force on one side of the gate..

a. 1015.8lbb. 973.8 lbc. 998.4 lbd. 1009.8 lbe. None of these

____ 29. You are given the shape of the vertical ends of a trough that is completely filled with water. Find the force exerted by the water on one end of the trough. (The weight density of water is 62.4 lb/ft3.)

a. 332.8 lbb. 1331.2 lbc. 2662.4 lbd. 665.6 lb

Name: ______________________ ID: A

18

____ 30. Find the centroid of the region bounded by the graphs of the given equations.

y = 15 − x2 , y = 3 − x

a.375

,12

Ê

Ë

ÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜̃

b.12

,52

Ê

Ë

ÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜̃

c.52

,12

Ê

Ë

ÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜̃

d.12

,375

Ê

Ë

ÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜̃

Name: ______________________ ID: A

19

____ 31. The masses m i are located at the point Pi . Find the moments Mx and My and the center of mass of the

system.

m 1 = 3, m 2 = 7, m 3 = 113;

P 1 1,5ÊËÁÁ

ˆ¯̃̃, P 2 3,− 2Ê

ËÁÁˆ¯̃̃, P 3 −2,− 1Ê

ËÁÁˆ¯̃̃

a. Mx = −50, My = 22,5023

, −2223

Ê

Ë

ÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜̃

b. Mx = 50, My = −22, −5023

,2223

Ê

Ë

ÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜̃

c. Mx = 22, My = 50,2223

,5023

Ê

Ë

ÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜̃

d. Mx = 50, My = 22,5023

,2223

Ê

Ë

ÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜̃

e. Mx = 22, My = 50,5023

,2223

Ê

Ë

ÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜̃

____ 32. Which equation does the function y = e−6t satisfy?

a. y″ − y′ +42y = 0b. y″ − y′ −42y = 0c. y″ + y′ +42y = 0d. y″ + y′ −42y = 0e. y″ − 3y′ +42y = 0

Name: ______________________ ID: A

20

____ 33. Solve the initial-value problem.

drdt

+ 2tr − r = 0, r 0( ) = 10

a. r t( ) = e10t− t 2

b. r t( ) = 2et− t 2

c. r t( ) = 10et− 10t 2

d. r t( ) = 10et 2

e. none of these

____ 34. Solve the differential equation.

3yy′ = 7x

a. 7x2 − 3y2 = Cb. 7x2 + 3y2 = 10c. 3x2 − 7y2 = Cd. 3x2 + 7y2 = Ce. 7x2 + 3y2 = C

____ 35. Find the point(s) on the curve where the tangent is horizontal.

x = t3 − 3t + 4, y = t3 − 3t2 + 4

a. 0,0ÊËÁÁ

ˆ¯̃̃ , 4, −4Ê

ËÁÁˆ¯̃̃

b. −1,1ÊËÁÁ

ˆ¯̃̃ , −4, −4Ê

ËÁÁˆ¯̃̃

c. 4,4ÊËÁÁ

ˆ¯̃̃ , 6, 0Ê

ËÁÁˆ¯̃̃

d. 4, −4ÊËÁÁ

ˆ¯̃̃

e. None of these

Name: ______________________ ID: A

21


x = 3t2 + 8, y = 2t3 + 8, 0 ≤ t ≤ 1

a. 4 2 − 1b. 2 2 − 2c. 4 2 − 2d. 4 2e. None of these

____ 37. Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter.

x = cos θ + sin 2θ+ 8, y = sin θ + cos 2θ + 8 , θ = π

a. y =x2+

32

b. y =2x

c. y =x2+ 2

d. y =252

−x2

e. None of these

____ 38. Find a polar equation for the curve represented by the given Cartesian equation.

x2 = 3y

a. r = 3 sin θb. r = 3 tan θ secθc. r = 3 tan θd. r = 3 cos θ sin θe. r = 3 tan θ cscθ

Name: ______________________ ID: A

22

____ 39. Find the polar equation for the curve represented by the given Cartesian equation.

x + y = 2

a. r =2

cos θ − sin θ

b. r = 2 cos θ + sin θ( )

c. r =1

cos θ − sin θ

d. r = 1 cos θ + sin θ( )

e. r =2

cos θ + sin θ

Name: ______________________ ID: A

23

____ 40. Sketch the polar curve with the given equation.

r = sin 2θ; − π ≤ x ≤ π

a. c.

b. d.

Name: ______________________ ID: A

24

____ 41. Determine whether the sequence defined by an =n2 − 56n2 + 1

converges or diverges. If it converges,

find its limit.

a.16

b. −56

c. Diverges

d. –5

____ 42. Determine whether the sequence defined by an =sin 2n

9n converges or diverges. If it converges, find

its limit.

a. 0

b. 1

c. Diverges

d.29

Name: ______________________ ID: A

25

____ 43. Determine whether the series is convergent or divergent by expressing Sn as a telescoping sum. If it is convergent, find its sum.

3

n n2 − 1ÊËÁÁÁÁ

ˆ¯˜̃̃˜n = 2

∞

∑ .

a. 1

b.34

c.14

d. diverges

e. 2

____ 44. Determine whether the geometric series converges or diverges. If it converges, find its sum.

5n 6−n + 1

n = 0

∞

∑

a. Divergesb. 36c. 5d. 30

____ 45. Determine whether the given series converges or diverges. If it converges, find its sum.

1 +8n

Ê

Ë

ÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜̃

n

n = 1

∞

∑

a. Divergesb. 1c. e− 8

d. e8

Name: ______________________ ID: A

26

____ 46. Determine whether the geometric series converges or diverges. If it converges, find its sum.

7n 8−n + 1

n = 0

∞

∑

a. 7b. 64c. 56d. Diverges

____ 47. Find an approximation of the sum of the series accurate to two decimal places.

−1( )n

n3n = 1

∞

∑

a. –0.83b. –1.02c. –0.96d. –0.90

____ 48. Find the values of p for which the series is convergent.

−1( )n

ln n6ÊËÁÁÁÁ

ˆ¯˜̃̃˜

Ê

ËÁÁÁÁÁ

ˆ

¯˜̃̃˜̃

pn = 2

∞

∑

a. p > 1b. p < 1c. p < 0d. p > 0

____ 49. Determine whether the sequence convergent or divergent.

5n2 + 5n = 1

∞

∑

a. convergesb. diverges

Name: ______________________ ID: A

27

____ 50. Test the series for convergence or divergence.

−6( )m+ 1

48mm = 1

∞

∑

a. The series is convergent.b. The series is divergent.

____ 51. For which positive integers k is the series n!( )

4

kn!( )n = 1

∞

∑ convergent?

a. k ≥ 0b. k ≤ −4c. k ≥ 1d. k ≥ 4e. k ≤ 0

____ 52. Determine whether the series is absolutely convergent, conditionally convergent, or divergent.

4n2 + 33n2 + 4

Ê

Ë


ˆ

¯

˜̃̃˜̃̃˜̃̃˜

n

n = 1

∞

∑

a. absolutely convergentb. conditionally convergentc. divergent

____ 53. Which of the given series are absolutely convergent?

a.sin 5n

nn = 1

∞

∑

b.cos

πn8

n nn = 1

∞

∑

Name: ______________________ ID: A

28

____ 54. Find the radius of convergence and the interval of convergence of the power series.

(7x)n

n!n = 0

∞

∑

a. R = 7, I = [−7, 7]b. R = 0, I = {0}c. R = ∞, I = (−∞, ∞)d. R = 7, I = (−7, 7)


(−1)n (x − 8)n

nn = 1

∞

∑

a. R = 8, I = [−8, 8)b. R = 1, I = (7, 9]c. R = 8, I = (−8, 8)d. R = 1, I = [7, 9)


xn

n + 2n = 0

∞

∑

a. R = 2, I = (−2, 2)b. R = 1, I = (−1, 1)c. R = 1, I = [−1, 1)d. R = 2, I = [−2, 2)

Name: ______________________ ID: A

29

____ 57. Find a power series representation for

f t( ) = ln 14 − t( )

a.tn

n14nn = 0

∞

∑

b. ln 14 −tn

14nn = 1

∞

∑

c. ln 14 +t2n

14nn = 1

∞

∑

d.14tn

nnn = 1

∞

∑

e. ln 14 −tn

n14nn = 1

∞

∑

Name: ______________________ ID: A

30

____ 58. Find the Maclaurin series for f (x) using the definition of the Maclaurin series.

f x( ) = x cos 4x( )

a.−1( )

n 52n x2n

2n( )!n = 0

∞

∑

b.−1( )

n 5n x2n + 1

2n( )!n = 0

∞

∑

c.−1( )

n 52n x2n + 1

n!n = 0

∞

∑

d.−1( )

n 52n x2n + 1

2n( )!n = 0

∞

∑

e.−1( )

n + 1 52n x2n + 1

2n( )!n = 0

∞

∑

____ 59. Which of the following functions are the constant solutions of the equation

dy

dt= y4 − 6y3 + 8y2

a. y t( ) = −4b. y t( ) = −2c. y t( ) = 0d. y t( ) = et

e. y t( ) = 5

60. If f(x) = 2 + 9x + ex , find f − 1Ê

ËÁÁÁÁ

ˆ

¯˜̃̃˜′

3( ) .

Name: ______________________ ID: A

31

61. Find the limit.

limx → 0

1 − 4x( ) 1/x

62. Evaluate the integral.

2 cos x1 + sin2 x0

π3∫ dx


12x cos πxdx0

1 2

∫


2 sin3θ cos 2θ dθ0

π 2

∫


cos x16 + sin2 x

∫ dx


1 + 8ex

1 − ex dx∫

67. Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve.

x = cos θ, y = 5 secθ, 0 ≤ θ <π2

68. Eliminate the parameter to find a Cartesian equation of the curve.

x t( ) = 2 cos 2 t, y t( ) = 7 sin2 t

Name: ______________________ ID: A

32

69. Find the area that the curve encloses.

r = 13 sin θ

70. Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues.

−1

2,

16

3, −

81

4,

256

5, −

625

6, . . . .

Ï

ÌÓ

ÔÔÔÔÔÔÔÔÔÔ

¸

˝˛

ÔÔÔÔÔÔÔÔÔÔ

71. Use the Integral Test to determine whether the series is convergent or divergent.

18n + 2n = 1

∞

∑

72. Test the series for convergence or divergence.

−1( )n

n = 2

∞

∑ n5 ln n


4mm 3

m!m = 1

∞

∑

Name: ______________________ ID: A

33


−4( )m

m = 1

∞

∑ ln m

m

75. Write the expression in algebraic form.

sec(sin–1 8x)

76. Find the limit limx → 0

cos 3x − 1

sin 9x, if it exists.

77. Use the Comparison Test to determine whether the series is convergent or divergent.

7n

2n − 3n = 3

∞

∑

78. Determine whether the series is convergent or divergent.

1

ln n9ÊËÁÁÁÁ

ˆ¯˜̃̃˜n = 2

∞

∑

79. Use the Comparison Test to determine whether the series is convergent or divergent.

5 + sin 2n4n

n = 1

∞

∑

80. Find a power series representation for the indefinite integral.

sin 9xx

dx∫

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