1. Find the area between the curves y = cos2x and y = sin2x for x ...

MATH 104 ? Makeup Final Exam - Fall 2006

1. Find the area between the curves y = cos 2x and y = sin 2x for x between 0 and

the smallest positive value of x for which the two curves intersect.

(A)

2 4

(B)

2 2

(C)

21 2 -2

(D)

21 4 -4

(E)

21 2 -4

(F)

21 4 -2

2. Find the volume of the solid obtained by rotating the region bounded by the curves

y = x5/2 , y = 0 , x = 2

about the x-axis.

(A)

8 3

(B)

32 3

(C)

64 3

(D)

128 3

(E)

256 3

(F)

1024 2

3. What is the volume of a solid of revolution generated by rotating around the y-axis

the and

regio?n

enclosed

by

the

graph

of

y

=

sin(x2)

and

the

x-axis,

for

x

between

0

(A)

(B)

3 2

(C)

(D) 2

(E)

2

(F)

2

4. Evaluate the integral.

(A)

e2 -4

+

e

-

1 4

(B)

e 6

1

x2e2xdx

0

(C)

e2 6

(D)

e2 2

-e

(E)

e2 2

-

e 2

(F)

e2 4

-

1 4

5. Evaluate the integral.

(A)

1 8

(B)

31 4 -8

2 dx

0 (8 - x2)3/2

(C)

31 36 - 32

(D)

3 12

(E)

1 6

-

3 16

(F)

31 2 -4

6. Evaluate the following integral.

(A)

3 2

(B)

31 2 -4

dx

4 x2 + 16

(C)

3 2

+

1 4

(D)

4

(E)

16

(F) diverges

7. Calculate the volume of the solid obtained by rotating the region between the

graphs

of

y

=

x2

1 - 5x + 6

and

y

=

0

for

4

x

5

around

the

y-axis.

(A) 2 ln(10/9)

(B) 2 ln(32/9)

(C) 4 ln(5/3)

(D) 4 ln(7/3)

(E) 2 ln(16/5)

(F) 2 ln(16/3)

8. Find the area of the surface obtained by rotating the curve y = 1 - x2 about the

x-axis for 0 x 1/2.

(A)

2

(B)

3 4

(C)

2 3

(D)

5 6

(E)

(F)

3 2)

9. Evaluate the improper integral if possible:

35

1 x1.2

dx

(A) integral diverges

(B) 5

(C)

5 3

(D)

5 6

(E)

6 5

(F) 15

10. Find the equation for the line tangent to the curve defined by the parameterization

x

=

t

+

1 t2

,

y

=

t3

+

3

(for

t

>

0)

at

the

point

(x,

y)

=

(2,

4).

(A) y = -3x + 10

(B) y = -3x + 14

(C) y = 3x - 2

(D) y = 3x - 8

(E)

y

=

-

x 3

+

14 3

(F)

y

=

x -3

+

10 3

11. The curve given parametrically by

x = t3 - 4t

y = 2 3(t2 - 4)

passes through the origin for two different values of t, and hence contains a loop.

What is the arc length of the loop?

(A) 4

(B) 4 2

(C) 8 2

(D) 16

(E) 16 2

(F) 32

12. Find the area inside one leaf (i.e., one loop) of the graph of r = 3 cos 3.

(A)

3 4

(B)

4

(C)

3 2

(D) 3

(E)

9 4

(F)

9 2

13. Find the limit. (A) 1/e (B) 1/2

lim

x0

cos x - 1 ex2 - 1

(C) -1 (D) 0 (E) -1/2

(F) no finite limit

14. Find the limit of the sequence

ln(2n

+

1)

-

1 2

ln(n2

+

1)

.

(A) -2 (B) - ln 2 (C) 0 (D) ln 2 (E) 2 (F) sequence diverges

15. Determine whether the series is convergent or divergent. If it is convergent, find

its sum.

1 + 3n

n=1 5n

(A) 1/4 (B) 1/2 (C) 7/4 (D) 3/2 (E) 15/4 (F) divergent

16. How many of the following series converge?

1

n=1 ( 2)n

1 n=1 n

n n=1 e

(A) none of these series converge

(C) two series converge

(E) four series converge

3n - 1 n=1 4n

e1/n

n=1

(B) just one series converges

(D) three series converge

(F) all five series converge

17. Which statement below is true about the following series?

?

(I)

(-1)n+1n2 n=1 1 + n2

=

1 2

-

4 5

+

9 10

-...

?

(II)

(-1)n+1n n=1 2 + n3

=

1 3

-

2 10

+

3 29

-...

?

(III)

(-1)nn ln n n=2 1 + n2

=

2 ln 2 5

-

3 ln 3 10

+

4 ln 4 17

-...

(A) (I) diverges, (II) converges conditionally, (III) converges absolutely (B) (I) diverges, (II) converges absolutely, (III) converges conditionally (C) (I) and (III) converge conditionally, (II) converges absolutely (D) (I) and (II) converge absolutely, (III) converges conditionally (E) (I) and (III) converge absolutely, (II) converges conditionally (F) (I) and (III) converge conditionally, (II) diverges

18. Find the precise interval of convergence of the series

n=1

(2x - 5)n n24n

.

(A) (-1, 1]

(D)

1 2

,

9 2

(B) [-1, 1] (E) [0, 4]

(C)

1 2

,

9 2

(F) (0, 4]

19. Which of the following is the beginning of the Maclaurin series for arctan (x2)?

(A)

x2

-

x4 3

+

x6 5

-

x8 7

+

...

(B)

x2 3

-

x6 6

+

x10 9

-

x14 12

+...

(C) x2 - 2x4 + 3x6 - 4x8 + . . .

(D) 1 + x4 + 3x8 + 4x12 + . . .

(E)

x2 3

+

x6 6

+

x10 9

+

x14 12

+

...

(F)

x2

-

x6 3

+

x10 5

-

x14 7

+

...

x 20. Let F (x) = cos t dt. Which of the following is the beginning of the Maclaurin

0

series for F ?

(A)

1

-

x 2

+

x2 24

-

x3 720

+

.

.

.

(B)

x

-

x2 4

+

x3 72

-

x4 2880

+

.

.

.

(C)

x

-

x2 2

+

x4 6

-

x6 24

+

...

(D)

1

-

x2 2

+

x4 24

-

x6 720

+

...

(E)

x

-

x2 3

+

x4 15

-

x5 105

+

...

(F)

x

+

x2 3

+

x4 15

+

x5 105

+

...

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