Name: Final Exam

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

1. Evaluate the following limits.

(a) lim_{x→0} \frac{x + \sqrt{1 + x} − 1}{1 − \sqrt{1 − x}} (b) lim_{x→0} \frac{sin(x^{2})}{x^{2}} (c) lim_{x→∞} (ln(x) − ln(x + 1))

2. Differentiate the following functions.

(a) \sqrt{e^{x} + 1} (b) \frac{x^{2} + 1}{x + 1} (c) x^{1+sin(x)}

3. Consider the following function

f(x) = \{ \begin{matrix}5x, & 0 ≤ x ≤ 1, \\ x^{2} − 6x + 10, & 1 < x ≤ 4 \end{matrix}

(a) Explain why f is continuous on [0, 4].

(b) Find the local and global maxima and minima of f on [0, 4].

4. Graphs. (a) Find all the horizontal and vertical asymptotes of the function f(x) = \frac{x+1}{2x+10}.

(b) What are the intervals where f is concave upwards or downwards and what are the inflection points of the function f(x) = e^{\frac{2}{3} x^{3}}.

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

(b) Suppose a continuous function f on [0, 10] has values given by the following table

\begin{matrix}x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ f(x) & 1 & 1 & 2 & 2 & 3 & 3 & 4 & 4 & 5 & 5 & 6 \end{matrix}

Evaluate the Riemann sum for f with 10 subintervals using as sample points the right endpoints.

6. Examples.

(a) Give an example of a function f on [0, 4] with an absolute maximum at 1 and an absolute minimum at 3. No explanation necessary for this part.

(b) Give an example of a function on [0, 1] that is not integrable. No explanation necessary for this part.

(c) Give an example of a continuous function f on [0, 10] such that ∫_{0}^{10} f(x)dx = 1/2.

7. Find the following indefinite integrals.

(a) ∫ (x^{2} + sin(2x))dx (b) ∫ cos(x)e^{sin(x)}dx (c) ∫ \frac{1 + 2x}{\sqrt{1 − x^{2}}} dx 8. Compute the following definite integrals.

(a) ∫_{1}^{2}(3x + 3)^{1/3}dx (b) ∫_{−10}^{10}(1 + x^{111})dx (c) ∫_{0}^{1} x\sqrt{1 − x^{2}}dx 9. Area and Volume.

(a) Find the area enclosed by the line y = x + 1 and the parabola y = x^{2} + 1.

(b) Find the volume of the solid obtained by rotating the region x^{2} + 5y^{2} ≤ 1 around the x-axis. (Hint: Draw a picture of the region, and set x^{2} + 5y^{2} equal to 1 and solve for y.)