Name: Final Exam

Write your name on this paper. Explain all your answers. Use common sense. Do not use calculators/book/notes. There are twelve parts all equally weighted. Here is the first one:

1. Evaluate the following limits.

(a) lim_{x→∞} 2x − \sqrt{4x^{2} + 9} (b) lim_{x→∞} \frac{x^{3}}{3^{x}}

2. Evaluate the following limits.

(a) lim_{x→0} \frac{tan(x) + 3x}{sin(x)} (b) lim_{x→0} \frac{x}{\sqrt{x^{2} + x^{3}}}

3. Differentiate the following functions.

(a) f(x) = \frac{ax + b}{cx + d} (b) g(x) = x sin(x) cos(x)

4. True or false^{†}: the function

f(x) = \{ \begin{matrix}x^{2}, & x < 0, \\ 0, & x ≥ 0 \end{matrix}

(a) is continuous at 0.

(b) is differentiable at 0.

† Explain your answer as always.

5. Suppose f is continuous and differentiable on all of R. Suppose that f′(x) > 0 for all x ∈ [0, 1] and that f(0) = 0. Is f(1) positive of negative? Explain your answer.

6. Theory. (a) State the fundamental theorem of calculus.

(b) True or false: The following calculation is correct

∫_{−1}^{3} \frac{1}{x^{2}} dx = [\frac{−1}{x}]_{−1}^{3} = \frac{−1}{3} − \frac{−1}{−1} = \frac{−4}{3}

7. Examples.

(a) Give an example of a function f on [0, 2] with an inflection point at (1, 1). No explanation necessary for this part.

(b) Give an example of a function defined on R with a horizontal asymptote at y = 1. No explanation necessary for this part.

8. Find the following indefinite integrals.

(a) ∫ cos(x)(sin(x))^{2008}dx (b) ∫ \frac{1}{1 + 3x^{2}} dx 9. Compute the following definite integrals.

(a) ∫_{0}^{√\overline{π}} x sin(x^{2})dx (b) ∫_{−π/4}^{π/4} tan(x)dx 10. Find the area enclosed by the line y = 2x+1 and the curve y = x^{4} +2x^{3} +x^{2} +2x+1. Please explain briefly what steps you are taking (as always).

11. Integrals.

(a) Suppose that f is a continuous function on [0, 10] with x + 1 ≤ f(x) ≤ x + 3 for all x ∈ [0, 10]. What can you say about ∫_{0}^{10} f(x)dx?

(b) Suppose that f is a continuous function on [0, 10] with ∫_{0}^{5} f(x)dx = 7 and ∫_{5}^{10} f(x)dx = −2. What can you say about ∫_{0}^{10} f(x)dx?

12. A man is standing at point A on the bank of a 0.5 mile wide river that whose current is 1 mph. He needs to traverse the river and go to point B directly across the river from him. His speed in the water is 3 mph and his speed on land is 5 mph. He decides to swim at an angle α with the line AB. If he does so it takes him

t = \frac{0.5}{3 cos(α)}

hours to get to the other side, and he ends up at a point C which is

BC = t(3 sin(α) + 1)

miles away from point B.

(a) Give a formula for the total time it takes to get from A to B in terms of α.

(b) What is the angle α that minimizes his time to get from A to B?