Math 141                          Mark-->'s Guide To Life, Volume 2                         Fall 02004

The second exam will be calculator-active, but your calculator will be inspected before the exam and any illegal programs will be deleted. I decide what’s “illegal”, and my judgment is final.

Section 2.4: The product and quotient rules for differentiation. The derivative of a product is not the product of the derivative, and similarly for the quotient.

Section 2.5: Derivatives of trigonometric functions. These are 6 formulas that you should have memorized. Limits that are useful in these calculations. Product and quotient rule differentiation incorporating these derivatives.

Section 2.6: Derivatives of exponential and logarithmic functions–here again are more rules to know.

Section 2.7: The Chain Rule–a critically important derivative rule. We saw a couple of shortcut formulas for derivatives of complicated exponentials and logarithms which are derived from the chain rule.

Section 2.8: Implicit differentiation–an application of the chin rule, somewhat undercover. Remember when differentiating implicitly that y is a function of x, and as such must be hit with the chain rule. Applications include related rates problems–see p. 226 for the outline of a simple strategy for handling these.

Section 2.9: The Mean Value Theorem and its relative, Rolle’s Theorem. What the hypotheses mean and what the conclusions say. Applications.

Section 3.1: Linear approximation of a function and L’Hôpital’s Rule for indeterminate forms in limits of the form 0/0. How to construct a general linear approximation and how to use it to get a numerical estimate for a function at a point. For L’H’s Rule: Remember that you must confirm that the limit is of the form 0/0 before differentiating and that it’s the quotient of the derivatives, not the derivative of the quotient, that matters.

Section 3.2: Newton’s Method for approximating solutions to equations. The sequence of values generated by this process and how it converges (or doesn’t) on a solution. The calculator program NEWTON is acceptable for use on the exam, but you will be expected to document your work by:

1. Setting up the recurrent sequence (see p. 252, formula #2.2).

2. Giving the initial value x₀ and the first two derived members of the sequence x₁ and x₂. 3. Providing the final answer to a specified level of accuracy.

Section 3.3: Maximum and minimum values of a function and how the derivative reveals them to us. Absolute and local extrema. Extreme Value theorem for continuous functions on closed, bounded intervals. Critical numbers of a function. Theorem 3.3 on the location of local extrema (and hence also absolute extrema).

Section 3.4: Increasing and decreasing functions–and, once again, how the derivative points these out for us. Strictly increasing and decreasing functions and how they differ from increasing and decreasing functions. Connection between the sign of the derivative and the behavior of the function. The First Derivative Test for classifying extrema.