Math 132, Midterm 3

College of the Holy Cross, Spring Semester, 2005

Math 132, Midterm 3 (All Sections) Wednesday, April 27, 6 PM

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  / 14

2  / 12

3  / 12

4  / 10

5  / 12

6  / 14

7  / 14

8  / 12

Total  /100

1. [7 points each] Suppose that the length x (in cm) of a certain type of inchworm has density p(x) = c(3x^{2} − x^{3}) for 0 ≤ x ≤ 3.

(a) Find the value of the constant c.

c =

(b) Find the proportion of inchworms with length between 1 and 2 cm.

Fraction of population between 1 and 2 cm in length

Math 132, Midterm 3

2. [6 points each] Suppose

P (x) = \frac{4}{π} arctan(x)

is the cumulative distribution function for the quantity x, for 0 ≤ x ≤ 1.

(a) Find the median of x.

Median of x

(b) Find the mean of x.

Mean of x

Page 3 of 9 3. [6 points each] Use the indicated test to determine whether the given series converges

or diverges. Justify your answers, but do not attempt to compute the sum of either series.

(a) ∑_{n=1}^{∞} \frac{n^{2}}{n^{3} + 1000000}; integral test

◯ Converges

◯ Diverges

(b) ∑_{n=1}^{∞} \frac{n}{4n^{3} + 3n^{2} + 5}; comparison test

◯ Converges

◯ Diverges

Math 132, Midterm 3

4. [5 points each] Use the geometric series formula to evaluate (find the sum of) the following:

(a) 5 − \frac{10}{3} + \frac{20}{9} − \frac{40}{27} + \frac{80}{81} − ⋯

Sum

(b) ∑_{n=1}^{∞} \frac{3^{n} + 7}{4^{n}}

Sum

Page 5 of 9 5. Suppose f(0) = 0.5, f′(0) = −1 and f′′(0) = −2.

(a) [9 points] Write down the Taylor polynomial of degree 2 for f near a = 0.

Taylor polynomial:

(b) [3 points] Which (if any) of the following could be the graph of f? (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}

Math 132, Midterm 3

6. [14 points] Let f(x) = \sqrt{x}. Use the definition to calculate the Taylor polynomial of degree 3 for f centered at a = 4.

Math 132, Midterm 3

7. [7 points each] Both parts refer to the power series

∑_{n=1}^{∞} \frac{(x − 1)^{n}}{n · 5^{n}}.

(a) Use the ratio test to find the radius of convergence.

Radius:

(b) Investigate the endpoint behavior, and determine the interval of convergence.

Interval of Convergence:

Page 8 of 9 8. (a) [6 points] Use the power series for sin x about a = 0 to find the power series

for sin(x^{2}) about a = 0. Express your answer both in summation form, and by writing out the first four nonzero terms.

Summation form:

First four terms:

(b) [6 points] Use the first four nonzero terms in the result from part (a) to estimate the value of the integral

∫_{0}^{1} sin(x^{2}) dx.

Estimate: