College of the Holy Cross, Fall Semester, 2004 Math 132, Midterm 1 (All Sections) Wednesday, February 23, 6 PM

Name:

Instructions Please write your answers in the spaces provided, and show work on the test itself. 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  / 20

2  / 10

3  / 10

4  / 20

5  / 20

6  / 10

7  / 10

Total  /100

Math 132, Midterm 1 2

1. [5 points each] Compute the following: (a) ∫ 2x(x^{2} + 5)^{7/2} dx

(b) ∫ sin(3θ) dθ

(c) ∫ \frac{dx}{2x + 1}

(d) F′(x), where F (x) = ∫_{π}^{x} \frac{t}{1 + sin^{2} t} dt 2. [10 points] The graph y = f(x) is shown. Let F be the antiderivative of f satisfying

F (2) = 0. Find the indicated values, and carefully sketch the graph y = F (x) in the grid provided.

F (1) =  F (3) =  F (4) =  F (6) =

1  2  3  4  5  6  7 −10 −505 10  y = f(x) Area = 6.25 Area = 6.25

0  1  2  3  4  5  6  7 −10 −8 −6 −4 −202468 10  y = F (x)

Math 132, Midterm 1 4

3. [5 points each] A drag racer accelerates at a(t) = (20 + t) ft/sec^{2}.

(a) How fast is the car traveling after 6 seconds? (The car is initially at rest.) Speed:

(b) How far has the car traveled after 6 seconds?

Distance traveled:

4. [10 points each] Find the integrals. Use the indicated method to get started. (a) ∫_{1}^{4} \begin{matrix}√ & \overline{1 + \sqrt{x}} \\ √ & \overline{x} \end{matrix} dx (Substitution)

(b) ∫ t sec^{2} t dt (Integrate by parts)

Math 132, Midterm 1 6

5. [10 points each] Compute the integrals

(a) ∫ \frac{dx}{(4 − x^{2})^{3/2}}

(b) ∫ \frac{dx}{x^{2} + 4x + 8}

Math 132, Midterm 1 7 6. [10 points] Find the indefinite integral ∫ \frac{x^{2} + 1}{x(x + 2)(x − 1)} dx Math 132, Midterm 1 8

7. [10 points] Compute any two of the following; clearly mark your choices and indicate the method of integration.

(i) ∫ \frac{1}{\sqrt{1 − x^{4}}} dx (ii) ∫ \frac{x}{\sqrt{1 − x^{4}}} dx (iii) ∫ \frac{x^{2}}{\sqrt{1 − x^{4}}} dx (iv) ∫ \frac{x^{3}}{\sqrt{1 − x^{4}}} dx