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Math 31

Final Exam

May 22nd, 2021

NAME & ID:

(Please Print)

DIRECTIONS:

• Do not open Exam until instructed to do so.

• Do each of the problems and show all work.

• NO WORK MEANS NO POINTS!

• Box or circle and LABEL your final solution.

• You will have 135 minutes to complete this Exam.

• No part of this document can be shared without the expressed written consent of Dr. Dashiell

Fryer

SCORES:

1.

/5

2.

/10

3.

/10

4.

/10

5.

/15

6.

/10

7.

/20

8.

/10

Total:

/90

1. 5 points Find derivatives of the following functions with respect to x.

Z xp

(a) 3 points

1 + t3 dt

0

Z

(b) 2 points

3×2

p

1 + t3 dt

0

1

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2

2. 10 points Write the following integral as the limit of Riemann sums using either left or right

endpoints.

Z

5

(7 + x3 )1/3 dx

2

You do not need to find the limit.

3

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4

3. 10 points Compute the following integrals.

Z

(a) (ln x)2 dx

Z

(b)

3x + 1

dx

x3 − x2

5

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6

4. 10 points Let f (x) =

√

x and g(x) = x2 .

(a) Find the area of the region entirely bounded by f and g.

(b) Rotate the region about the y-axis and find the solid’s volume.

7

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8

5. 15 points Let f (x) = x2

(a) 3 points Set up the integral for the arc length of f from x = 0 to x = 1.

(b) 2 points Set up the integral for the surface area of the surface obtained by rotating

f (x) about the x-axis.

(c) 10 points Find the arc length and surface areas exactly.

9

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10

6. 10 points Does the following integral converge or diverge?

Z

0

−∞

x4

Justify your answer in either case.

11

x

dx.

+ 16

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12

7. 20 points The hyperbolic cosine and hyperbolic sine functions are defined as

cosh x =

(a) 2 points Show that

ex + e−x

ex − e−x

, sinh x =

2

2

d

d

cosh x = sinh x, and

sinh x = cosh x.

dx

dx

(b) 2 points What are cosh 0 and sinh 0?

(c) 10 points Derive the Taylor series for f (x) = cosh x centered at c = 0, using only the

derivatives of f (x). Do not use a Taylor Series for ex .

(d) 6 points What is the interval of convergence for the Taylor series for f (x) = cosh x

centered at x0 = 0.

13

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14

8. 10 points

Approximate

1

Z

2

e−x dx

0

accurately to two decimal places.

You must describe the method used, eg Taylor Series, Simpson’s Rule, etc, in full detail.

Clearly explain how you know your approximation is correct to the specified accuracy.

15

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16

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