Midterm review, MATH 9A

The midterm is a closed book test. Calculators, cellular phones and helpful friends are NOT allowed.

For full credit you must justify your answers and show all steps.

You should be familiar with limits, limits from the right and left, with continuity of functions. You should be able to find the derivative by applying the various rules (e.g. power rule, derivative of a sum, quotient rule, chain rule) and to find the equation for the tangent line and to the line normal to the tangent line at a given point of a function. Furthermore, you should be familiar with velocity of a falling object (e.g. Problem 71, Section 2.2). Related rates and implicit differentiation are important.

Familiarity with the material of Chapter 3, Section 3.1-3.3, is required.

Example questions:

1. (Limits, limits from the right and left, vertical asymptotes)

   (Review exercises for Chapter 1, p. 87) Problems 21,25,29,31,33

2. (Continuity and Differentiability)
   1. Prove that if a function is differentiable at a point $c$ then it is continuous at $c$. (Section 2.1, Theorem 2.1)
   2. Use the definition of derivatives to compute the derivatives of $\sin(x)$ and $x^2$.
   3. Problems 66 and 73 in Section 2.1

3. (Derivatives) (Review exercises for Chapter 2, pp. 150-152)

   Problems 17,19,29,33,37,41,47

4. (Tangent lines and normal lines)
   1. (Exercises for Section 2.5) 43
   2. (Review exercises for Chapter 2, pp. 150-152) Problems 59, 63
5. (Implicit differentation) (Review exercises for Chapter 2) Problems 53, 55, 57

6. (Related rates) (Exercises for Section 2.6) Problems 11, 13, 17

7. (Rolle's Theorem and the Mean Value Theorem)

   (Exercises for Section 3.2) Problems 46, 53, 56

8. (Extrema and relative extrema, First Derivative Test)

   (Exercises for Section 3.3) Problems 15, 19, 27, 31