MATH 140/140E FINAL EXAM SAMPLE A

MATH 140/140E

FINAL EXAM SAMPLE A

1. Compute

5. Compute

x?1

.

(x + 1)2

lim

x¡ú?1+

d

dx

a) ¡Þ

a)

1

cos(x) ? 1

b)

cos(x) ? cos(2x)

(1 ? cos(x))2

c)

1 ? cos(x)

sin2 (x)



sin(x)

1 ? cos(x)



.

b) ?¡Þ

c) 0

d) 1

e) ?1

d) cot(x) +

2. Compute

lim

x¡ú3

x |3 ? x|

.

x?3

e)

a) ?3

1

sin(x)

1

1 ? cos(x)

q

6. Compute the derivative of the function f (x) =

cos

¡Ì

2x.

b) 3

c) 0

a)

d) 1

1

¡Ì p

¡Ì

2 2x cos 2x

¡Ì

? sin 2x

c) ¡Ì p

¡Ì

2x cos 2x

¡Ì

? sin 2x

d) ¡Ì p

¡Ì

2 x cos 2x

¡Ì

sin 2x

¡Ì

e)

2x

b)

e) The limit does not exist

3. Compute

lim

x¡ú0

¡Ì

? sin 2x

¡Ì p

¡Ì

2 2x cos 2x

2 sec(x)

.

1 ? tan(x)

a) 0

b) 1

c) 2

d) ?2

e) The limit does not exist

7. Find the equation of the tangent line to the curve y 2 +4xy +x2 = 13

at the point (2, 1).

4. Compute

6x3

lim ¡Ì

x¡ú¡Þ 3 13

+x

+ x9

.

a) y = x ? 11

a) ?¡Þ

b) y = x + 13

6

b)

13

c)

c) y = ?x + 15

7

5

d) y = ? x +

4

2

6

¡Ì

3

13

4

13

e) y = ? x +

5

5

d) 6

e) +¡Þ

1

MATH 140/140E

FINAL EXAM SAMPLE A

Z

p

8. Suppose oil spills from a ruptured tanker and spreads in a circular

3 3

4 ? x2 dx

1 12. By using an appropriate substitution, the indefinite integral x

pattern. If the radius of the oil increases at a constant rate of

can be transformed to which one of the following integrals?

2

m/s, how fast is the area of the spill increasing when the radius is

Z

30 meters?

4

1

1

(u 3 ? 4u 3 )du

a)

2

Z

a) 30¦Ð m2 /s

1

4

b)

(4u 3 ? u 3 )du

2

b) 2¦Ð m /s

Z

4

1

c) 60¦Ð m2 /s

c) ?

u 3 du

2

Z

d) 60 m2 /s

4

d) 2 u 3 du

2

e) 90¦Ð m /s

Z

4

1

e) 2 (u 3 ? 4u 3 )du

2

9. Find the x-values at which all local extrema occur for f (x) = 3x 3 ?

x.

13. Compute

Z

¡Ì

a) minimum at x = 2 only

1

b) maximum at x = 8 only

1

dx.

¡Ì

x(1 + x)2

a) 6

c) minimum at x = 0 and maximum at x = 2

b) ?

d) maximum at x = 2 only

c)

e) minimum at x = 0 and maximum at x = 8

10. How many asymptotes does the graph of the following function have?

(13x2 ? 6x)x

(13x ? 6)(x ? 1)

f (x) =

4

3

4

d) ?

e)

1

6

1

3

1

3

14. Evaluate the integral

a) One vertical asymptote and one slant asymptote

2

Z

b) Two vertical asymptotes and one slant asymptote

1

c) One horizontal asymptote and one slant asymptote

a)

7

4

b)

15

4

d) Two vertical asymptotes only

e) One slant asymptote only

2x4 + 2

dx.

x3

c) 0

11. A particle moves in a straight line and has acceleration given by

a(t)

= cos t. Find

position of the particle, s(t), given that

¦Ð

 ¦Ð the ¦Ð

v

= 2 and s

= .

2

3

3

a) s(t) = cos t + t ?

1

4

e) ?

1

2

¡Ì

15

4

15. Find the derivative of the function

Z sin(x)

F (x) =

sec t dt.

3

2

b) s(t) = ? cos t + t +

c) s(t) = ? cos t + t +

d)

¦Ð

1

2

a) sec(sin x)

d) s(t) = ? cos t + t

e) s(t) = ? cos t + 2t +

b) 1

1

¦Ð

?

2

3

c) sec(x) sin(x)

d) sec(sin(x)) + 1

e) sec(sin x) cos(x)

2

MATH 140/140E

FINAL EXAM SAMPLE A

2

Z

18. Which one of the following curves is the graph of the function f (x) =

x3

?

2

x ?4

(x + |x|) dx.

16. Compute

?1

a) 4

b) 3

c) 8

d)

3

2

e) 5

17. Find the volume of the solid obtained

¡Ì by rotating about the x-axis

the region bounded by the curve y = x ? 1 and the x-axis between

x = 2 and x = 5 .

a)

5¦Ð

2

b)

7¦Ð

2

c)

11¦Ð

2

d)

15¦Ð

2

e)

19¦Ð

2

(b)

(a)

(d)

(c)

(e)

3

MATH 140/140E

FINAL EXAM SAMPLE A

Questions 19 through 21 are true/false type. On your scantron

mark A for true, B for false. Each true/false question is worth

4 points.

19. If

a.

lim f (x) = lim f (x), then f is a continuous function at

x¡úa?

x¡úa+

a) True

b) False

b

Z

20. If a function f is continuous on [a, b], then

Z b

2

f (x) dx .

f 2 (x) dx =

a

a

a) True

b) False

21. If f is a continuous function on [a, b], then

d

dx

Z

b



f (t) dt

= 0.

a

a) True

b) False

4

22. (8 points) For which x-value(s) on the curve of y = 1+40x3 ?3x5

does the tangent line (not y) have the largest slope?

MATH 140/140E

FINAL EXAM SAMPLE A

23. (10 points) Find the area between the curves y = x and y = x2

for 0 ¡Ü x ¡Ü 2.

24. (12 pts total) A region R in the first quadrant is bounded by

8

, y = x2 , and x = 1.

x

y=

a) (4 pts) Sketch the region R. You must find and label any points

of intersection.

b) (4 pts) Write an integral expression for the volume when R is

revolved around the x-axis. (Do not evaluate the integral. Just set

it up.)

c) (4 pts) Write an integral expression for the volume when R is

revolved around the y-axis. (Do not evaluate the integral. Just set

it up.)

FINAL EXAM- VERSION A

1. B 2. E 3. C 4. D 5. A 6. A 7. E 8. A 9. E 10. A 11. C 12. A 13.

E 14. B 15. E 16. A 17. D 18. A 19. B 20. B 21. A 22. x = ?2, 2;

23. 1; 24. a) pic with intersection points: (1, 8), (2, 4), (1, 1); b)

Z 2

Z 2

8

8

V =

¦Ð[( )2 ? (x2 )2 ]dx; c) V =

2¦Ðx( ? x2 )dx

x

x

1

1

5

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