Math 141H.1, Honors Calculus II

Midterm Exam 2

Stephen G. Simpson February 20, 2002

The exam consists of eight problems. Calculators are not allowed.

1. Is the improper integral

∫_{1}^{∞} \frac{5 + 2 tan^{−1} x}{x^{2}} dx

convergent or divergent? Justify your answer.

2. Let y be the amount of money in your bank account. Assume that y is $1000 on January 1, 2002 and grows at the rate of 6 percent per year compounded continuously. What is y on January 1, 2052? Is it more than $10,000 ? Why or why not?

3. Evaluate the improper integral

∫_{0}^{∞} \frac{dx}{(x + 1)(x + 2)} .

4. Find the indefinite integral

∫ \frac{x^{3}}{x^{2} + 1} dx .

5. Find the indefinite integral ∫ (x − 1)^{2} cos 2x dx .

6. Evaluate the limit

lim_{x→0} \frac{x − sin x}{x^{3}} .

7. Evaluate the definite integral

∫_{0}^{1} \frac{x dx}{(x^{2} + 1)^{5}} .

8. Simpson’s Rule may be written as

∫_{a}^{b} f(x) dx ≈ \frac{h}{3} (y_{0} + 4y_{1} + 2y_{2} + 4y_{3} + ⋯ + 2y_{n−2} + 4y_{n−1} + y_{n}) .

What values of h and n are needed, in order to approximate

∫_{2}^{20} cos x dx

to within 10^{−5} ?