Name: Math 241: Calculus & Analytic Geometry A
Exam 1
4 October 2002

Always the beautiful answer who asks a more beautiful question.
-E. E. Cummings

Instructions: Show all work to receive full or partial credit. All University rules and guidelines for student conduct are applicable.

1.

[20 pts] Find the equation of the hyperbola that passes through the points (-1,0) and (1,0), and has asymptotes of y=2x and y=-2x, as shown.

2.

[20 pts] For what value of c is f(x) continuous on the whole real line? Why?

$$f(x) = \begin{cases} \frac{\sin(2x)}{x} & x<0 \\ x + c & x \geq 0 \end{cases}$$

3.

[20 pts] Determine an equation for the line tangent to the curve

$$f(x) = x^2 + \sin x$$

at $x=\frac{\pi}{3}$. Simplify your solution as much as possible.

4.

[10 pts] Evaluate the following limit:

$$\lim_{x \rightarrow 2} \frac{(x^2+2x-8)^2}{x^2-4}.$$

5.

[10 pts] Evaluate the following limit:

$$\lim_{x \rightarrow 0} \frac{2 \sin x-\sin(2 x)}{x^3}.$$

6.

[10 pts] Evaluate the derivative of f(x) where

$$f(x) = \frac{\tan(x)}{x^2 - 1/x}.$$

7.

[10 pts] Evaluate the derivative of f(x) where

$$f(x) = (x^2 - 1/x)\cos(x).$$