Math 106 Exam 1, October 3, 2003, Exams of Calculus

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Name Math 106 October 3, 2003 Exam 1

1. (20) 3. (16) 5. (16)
2. (24) 4. (24) Total
3. (i) Evaluate

x^2 cos

x^3

dx. (ii) Evaluate

x^3 cos

x^2

dx.

2. (a) Evaluate

xex^ dx. (b) What is the derivative of xex^ (with respect to x)?

(c) Evaluate

xex (x + 1)^2 dx.

4. (i) Calculate

1

dx 1 + ex^ if it converges. Hint: multiply by e−x e−x^ inside the integral.

(ii) Does

∑^ ∞

n=

1 + en^ converge, or does it diverge? Explain.

5. Suppose we make a slight modification in the trains and hummingbird problem we did in class. Again assume that the hummingbird flies 60 miles per hour (or 1 mile per minute) and can turn around instanta- neously, and assume the two trains are 1 mile apart when the bird starts flying between them. This time let’s assume that the first train, where the hummingbird starts, is going a constant 40 miles per hour, and the second train is going 20 miles an hour in the opposite direction. If the bird flies back and forth between these trains until they collide head on, the distance that it flies equals the infinite series

3 4

(a) Is this a geometric series? Explain. (b) Can you find its sum? Explain.