Math 132, Midterm 3

College of the Holy Cross, Spring Semester, 2005

Math 132, Makeup Midterm 3 Monday, May 2

Name:

Instructions Please write your answers in the spaces provided, and show work on the test itself. For possible partial credit, you must show work. Use the back of the preceding page if you need more space for scratch work.

Please do not write in the space below

Problem Points/Poss

1  / 15

2  / 10

3  / 10

4  / 15

5  / 10

6  / 15

7  / 15

8  / 10

Total  /100

1. [5 points each] Suppose f(0) = 0.5, f′(0) = −1 and f′′(0) = −2. (a) Write down the Taylor polynomial of degree 2 for f near a = 0.

Answer:

(b) Use your answer to part (a) to estimate f(0.2).

Answer:

(c) Which (if any) of the following could be the graph of f? (Recall that f(0) = 0.5, f′(0) = −1 and f′′(0) = −2. More than one answer may be correct.) ◯ (i) ◯ (ii) ◯ (iii) ◯ (iv) ◯ None

(i)^{−1}  ^{1}_{−1}^{1}  (ii)^{−1}  ^{1}_{−1}^{1}  (iii)^{−1}  ^{1}_{−1}^{1}  (iv)^{−1}  ^{1}_{−1}^{1}

2. [10 points] Evaluate (find the sum of) the series 4 − \frac{8}{3} + \frac{16}{9} − \frac{32}{27} + \frac{64}{81} − ⋯

Answer:

3. [10 points] Use the power series for e^{x} about a = 0 to find the power series for z^{2}e^{−z} about a = 0. Express your answer both in summation form, and by writing out the first four nonzero terms.

Summation form:

First four terms:

4. The distribution of length in a certain population of inchworms is described by the pdf p(x) = c(3x^{2} − x^{3}) for 0 ≤ x ≤ 3.

(a) [8 points] Find the value of the constant c.

Answer:

(b) [7 points] Find the proportion of inchworms with length at betweem 1 cm and 2 cm.

Answer:

Math 132, Midterm 3

5. [5 points each] Use the indicated test to determine whether the given series converges or diverges. Justify your answers.

(a) ∑_{n=1}^{∞} \frac{n}{n^{2} + 4}; integral test ◯ Converges ◯ Diverges

(b) ∑_{n=1}^{∞} \frac{n}{3n^{3} + n^{2} + 1}; comparison test ◯ Converges ◯ Diverges

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Math 132, Midterm 3

6. [15 points] Let f(x) = \sqrt{x}. Use the definition to calculate its Taylor polynomial of degree 3 at a = 1.

Math 132, Midterm 3

7. Both parts refer to the power series ∑_{n=1}^{∞} \frac{(x − 2)^{n}}{n · 7^{n}}. (a) [10 points] Use the ratio test to find the radius of convergence.

Answer:

(b) [5 points] Investigate the endpoint behavior, and determine the interval of convergence.

Math 132, Midterm 3

8. [5 points each] Suppose

p(x) = \frac{4}{π} · \frac{1}{1 + x^{2}}

is the probability density function for the quantity x, for 0 ≤ x ≤ 1.

(a) Find the mean of x.

Answer:

(b) Find the median of x.

Answer:

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