Math 161 §2, Calculus 1 , Fall 97
Exam 2

Name:

Answer all questions. 10 points each.

1.

Use the definition of the derivative as a limit to find $${D_x(\frac{1}{\sqrt{x}})}$$.

2.

What three kinds of practical problems give rise to the concept of the derivative?

3.

Find the following derivatives:

(a)

$${D_x\left( x\sin(\frac1x) \right)}$$

(b)

$${D_x\left( \frac{x^3 \sin(2x-1)}{x^3-2x} \right)}$$

(c)

$${\frac{d}{dx}\left( x\tan(x)\sqrt{x^2+1} \right)}$$

(d)

$${y' \mbox{ where } y=\left( \cos(\sec(3x+\frac1x))) \right)}$$

(e)

$${D^2_x\left( \frac{x}{x^3-1} \right)}$$

4.

Prove the product rule for derivatives.

5.

Use implicit differentiation to find the equation of the line tangent to the curve $xy = \sin(x+y)$ at the point $( 0, \pi)$.

6.

A spherical rain drop falling through a cloud gains volume at a rate directly proportional to its surface area. How is the radius changing?