Gateway 2 Practice Test Fall Semester 1997-1998

NO CALCULATORS MAY BE USED ON THIS TEST. SHOW YOUR WORK. DO NOT SIMPLIFY YOUR ANSWERS. EACH PROBLEM IS WORTH 10 POINTS.

1.

Suppose that f(x) = 1/x. Use the definition of derivative to show that the derivative of f(x) at x = 4 is -1/16.

2.

Find the derivative. $${s=\frac{2}{t^3}-\frac{1}{t}+7 + 8t^2}$$

3.

Find the derivative. $${f(u) = \frac{1}{\sqrt{u}}-3\sqrt{u}+\pi}$$

4.

Find the derivative. $${r=\theta^3(\cos(\theta))}$$

5.

Find the derivative. $${x=\frac{2+t-t^2}{t^3-3t+1}}$$

6.

Find the derivative. $${y = \sqrt{x^2+3x-1}}$$

7.

Find the derivative. $${v=\cos^3(u)}$$

8.

Suppose that the point (4,5) is on the graph of y = f(x) and that the derivative of f(x) at x = 4 is 11. Give an equation of the tangent line to y = f(x) at the point (4,5).

9.

Find q''. $${q=3\sin\left(\frac{t-1}{\pi}\right)}$$

10.

Suppose that $${x^2y-xy^2=2x}$$. Find $y^\prime$ at the point (x,y) = (3,1).