MATH 255A, Review for Midterm Test 2

Test topics

1. Definition of the derivative using limits. Finding the derivative using the definition (3.4)
2. Geometrical interpretation of the derivative. Equation of the tangent line. Increasing and decreasing functions and the derivative. Examples when the derivative does not exist (3.4).
3. Rules for finding derivatives: sums, differences, constant multiple, power rule (4.1); product and quotient rules (4.2); the rule for the composition (the chain rule) (4.3)
4. Exponential (4.4), logarithmic (4.5) and trigonometric functions (4.6)
5. Increasing and decreasing functions. Finding the intervals on which the function increases/decreases, using the first derivative (5.1).
6. Critical points. Relative (local) extrema. First and second derivative tests. Finding the locations and the values of the local extrema (max and min) (5.2, 5.3).
7. The higher derivatives. The second derivative; convex and concave functions. Inflection points. Finding the intervals of convexity/concavity, and the inflection points (5.3).
8. Curve sketching, using the first and the second derivative. For the steps see lecture notes or page 306 in the book. Type of problem: sketch the graph of a function (5.4).
9. Absolute maxima and minima. Algorithm for finding the absolute maximum/minimum, on bounded intervals (blue box on p. 324) and unbounded intervals (cf. problems 31, 34, 6.1).

Formulas/facts you need to know for the test

• Different notations for the derivative (4.1)
• All the formulas in blue boxes in 4.1, 4.2, 4.3 (sum/difference, power, product/quotient, chain rule...) except the one on p. 227 (we did not discuss it in class, and it is a consequence of the chain rule).
• The formula for the derivative of e^mx (the particular case of the blue box for e^g(x) with g(x)=mx)
• The blue box on p. 224 and the derivative for ln |x| from the blue box on p. 245
• Definitions of trig. functions (sin, cos, tan, cot, sec, csc) their graphs, values for special angles (0, π/6, π/4, π/3, π/2, etc.); derivatives of sin and cos.
• Equation for the tangent line (blue box on p. 183)
• Definitions of critical points, inflection points, maximum and minimum values (y-values) and points of max. and min. (x-values)
• First and second derivative tests (particularly, the diagram for the sign of f ' )

Preparation for the test

Problems on the test will not be taken from the homework/practice problems but they will be generally similar. The best way to prepare is to study examples in the lectures and in the book and to solve practice problems (think about how to solve it, rather than look in the answer). Old exams are available at http://www.csun.edu/~panferov/math255a_f07/ and http://www.csun.edu/~panferov/math255a_s08/ (see Tests 2 and 3).

Exercises in 5.4 (Curve sketching) provide for good review problems, since you have to go through all the steps...

Format of the test

Problems on the test will not be taken from the homework but they will generally be similar. You will need to provide a complete solution for each problem, showing all steps. You may get partial credit for solutions that are not completely correct as long as you have the right ideas and explain them. The test will be closed books/notes. A simple scientific calculator (example: TI-30XII) is allowed, but it is not required to answer the questions. Graphing calculators (example: TI-83, 84, 89) may not be used.