Mathematics 2171-01
Calculus I
Spring 04
Dr. A. Shlapentokh
Test #2

Each problem below is worth 5 points. Show your work !!!
1. State the definition of derivative.
2. Suppose a function is differentiable at a point . Does it mean it is continuous at this point? Explain.
3. Suppose a function is continuous at a point . Does it mean it is differentiable at this point? Explain.
4. What are possible formulas for and possible values for in the following equations? Justify your answer.
(a) .
(b) .
(c) .
(d) .
5. Compute the derivative of the following functions using the definition of derivative:
(a) .
(b) .
(c) .
6. Compute the derivatives of the following functions using the appropriate differentiation rules:
(a) .
(b) .
(c) .
(d) .
(e) .
(f) .
(g) .
(h) .
(i) .
(j) .
(k) .
(l) .
7. Suppose . Find:
(a) if .
(b) if .
(c) if .
(d) if .
(e) if .
(f) if .
8. Find the tangent line to at .
9. Find the average rate of change of the function between and .
10. Find the instantaneous rate of change of the function a .
11. Suppose the position of a particle moving in a straight line is given by .
(a) Find the velocity of the particle as a function of time.
(b) Find the average velocity of the particle between and ? and ?
(c) When is the particle at rest?
(d) When is the particle moving in a positive direction?
(e) What is the total distance travelled by the particle between and ?
12. Suppose is a differentiable function of . Find if
(a) and .
(b) and .