Sample Midterm Exam 2

Question 1 (10). Find the derivative of the following functions:
a. $f(x) = x^8 \sin 5x$

b. $f(x) = (x + \sin x)^{23}$

c. $f(x) = \frac{x^2 - x}{ \tan x}$

d. $f(x) = \frac{1 + (1/x)}{1 - (1/x)}$

e. $f(x) = (2x+1)^7 (3x+1)^5$

Question 2 (15). a. State the definition of the derivative of a function $f$ at a point $x$.

b. Use the definition of the derivative to compute $f'(x)$ for $f(x) = \frac{2}{x}$

Question 3 (10). Find all the vertical and horizontal asymptotes of the graph of $f(x) = \frac{x^2 - 2x + 1}{x^2 - 1}$

Question 4 (20). For each of the following, either find the limit or state that "no limit exists" and briefly explain why. Show work used to get your answer.

a. $\lim_{x \to 0} \frac{2 + 3\sin x} {x^3 + 1}$

b. $\lim_{x \to \infty} \cos x$

c. $\lim_{x \to \infty} \frac{\cos x}{x^2 - \sin x}$

d. $\lim_{x \to 2} \frac {x^2 - 4} { x-2}$.

Question 5 (10). For the function $f(x) = x^2 + 2 \tan x - 2$

a. Find the equation of the tangent line to the graph of $f(x)$ at the point (0,-2).

b. Show that $f(x) = 0$ at some point.

Question 6 (10) a. State the precise definition of what is meant by $\lim_{x \to a} f(x) = L$.

Use the precise definition of the limit to prove that $\lim_{x \to 0} 5 x^2 - 4 = -4$.

Question 7 (5) Give an example of a function $f(x)$ which is continuous at $x=1$ but not differentiable at $x=1$.

Question 8 (5) Suppose $f$ and $g$ are functions and $f(3) = 2,\ f'(3) = 4,\ g(5) = 3,\ g'(5) = 7.$ Where can you calculate the derivative of $f \circ g$? What is it equal to?

Question 9 (5) Let $f(x) = \sqrt[4] {x^5}$. Find $f'(16)$.

Question 10 (10). Find an anti-derivative of the following functions:
a. $f(x) = 5/x^2$

b. $f(x) = 3 \sin x$

c. $f(x) = \csc x \cot x$