Math 182- Spring 2014Name ________________________

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Final (28 pts total)

Multiple Choice - Choose 7 out of the following 8 questions to complete. Put an X through the problem you do not want graded. (7 pts total, 1 pt each)Identify the choice that best completes the statement or answers the question.

1. Use separation of variables to find the general solution to the differential equation.

y dx − xdy = 0

a. y = x2

3

Ê

Ë

ÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜̃

2 / 3

+ C

b. y = x2

3+ C

Ê

Ë

ÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜̃

2 / 3

c. y = x2 + C

d. y = ln| x|+ CÊËÁÁ

ˆ¯̃̃

2

e. y =ln| x|

2+ C

Ê

Ë

ÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜

2

2. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations

y = x , y = 0, and x = 2 about the x-axis.

a.π2

b. π

c. 2π

d.8π3

e.2π 2

3

2

3. Choose a valid method to evaluate the given integral. dx

x2 (9x2 − 1)a. Use trig substitution with x = 3tanx .

b. Use trig substitution with x = 13

tanx .

c. Use partial fraction decomposition with 1

x2 (9x2 − 1)= A

x2+ B

3x − 1+ C

3x + 1.

d. Use partial fraction decomposition with 1

x2 (9x2 − 1)= A

x + Bx2

+ C3x − 1

+ D3x + 1

.

e. Use integration by parts with u = x2 and dv = dx9x2 − 1

.

4. Select the limit to which L’Hopital’s rule CANNOT be applied. (There is only one correct answer.)

a. limx → ∞

xex d. lim

x → 0

sinxx

b. limx → 0

x lnx e. limx → 0

cos xx

c. limx → ∞

lnxx

5. Let f(x) = xn

nn = 0

∞

. Find the interval of convergence of f (x) dx .

a. (−1,1)b. [−1,1)c. (−1,1]d. [−1,1]e. None of the above

6. Find the first three terms of the taylor series for f(x) = x centered at x = 1.

a. 1 + 12

x − 18

x2

b. 1 + 12

x − 14

x2

c. (x − 1) + 12

(x − 1)2 − 14

(x − 1)3

d. 1 + 12

(x − 1) − 14

(x − 1)2

e. 1 + 12

(x − 1) − 18

(x − 1)2

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7. Choose the set of parametric equations that correspond to a circle with radius 3 and center (1,2).a. x = −1 + 3cos t , y = −2 + 3sin tb. x = 1 + 3cos t , y = 2 + 3sin tc. x = 3 + cos t , y = 3 + 2sin td. x = −1 + 3tan t , y = −2 + 3sec te. x = 1 + 3tan t , y = 2 + 3sec t

8. Find the slope of the given curve at the point 0,3ÊËÁÁ

ˆ¯̃̃ .

x = t2 − t, y = t2 + 2t .

a. 4 b. 2 c. 1 d.12

e.14

Short Answer - Choose 7 out of the following 8 questions to complete. Put an X through the problem you do not want graded. (21 pts total, 3 pts each)***Answer the question in as much detail as necessary. ***Partial credit will be awarded for correct work. ***No credit will be awarded if no work or incorrect work is shown.

1. SET UP an integral expression for the volume of the solid generated by revolving the region, R, about

the y-axis, where R is the region bounded by y = 132

x3 , x = y2 .

(1 pt bonus): Set up an integral expression for the same volume using the other method. That is, if you used the Washer Method above, now use the Shell Method. If you used the Shell Method above, now use the Washer Method.

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2. SET UP an integral expression for the volume of the solid generated by revolving the region R bounded by y = x + 1 and y = ex − 1 (shaded below) about the line x = 4.

(1 pt bonus): R is the base of a solid that has square cross sections perpendicular to the x-axis. Setup (but do not integrate) an expression for the volume of the solid.

3. Consider the differential equation dydx = x2 + 6

y − 2. Let y = f(x) be the particular solution to this differential

equation with f (−1) = −4.

Use two steps of Euler’s method, starting at x = −1, to approximate f(0).

(1 pt bonus): At what points will the function y = f(x) be tangent to the x-axis.

5

4. Evaluate the indefinite integral. xe5x dx

(1 pt bonus): Evaluate the improper integral: xe5xdx−∞

0

5. The power series for f(x) = 11 − x is given by xn

n = 0

∞

.

a. Find f ′(x) and its power series.

b. Find a power series for g(x) = 2x(1 − 2x)2

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6. The Maclaurin series for f (x) = ex is given by xn

n!n = 0

∞

a. Apply the ratio test to find the interval of convergence for the series.

b. Approximate e−1 using the first 5 terms of the Maclaurin series for ex .

c. What does the alternating series remainder threorem say about the accuracy of the estimate in part b?

7. Find the area of the smaller region bounded by r = 4cos θ and θ = π6

(shaded below).

(1 pt bonus): Find the area of the larger region bounded by r = 4cos θ and θ = π6

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8. Let x = cos t and y = cos t + sin t for 0 ≤ t ≤ 2π . The resulting curve is sketched below.

a. Find all values of t for which the tangent line to the curve is either horizontal or vertical.

b. What is the domain and range of the curve?

(1 pt bonus): Eliminate the parameter for the above parametric equations.

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Extra Credit [2 pts each]You can put work/answers on the back of the test, or on another sheet of paper.

1. Consider the infinite region in the first quadrant bounded by y = 1x , x = 1, and y = 0 (shaded below).

a. Show that the area of the shaded region is infinite.

b. Show that the volume of the solid obtained by revolving the region around the x-axis is finite.

2. Evaluate the indefinite integral. sin47x dx