Mathematics 133 – Intensive Calculus for Science 1 Things to Know – Exam 2 October 23, 2000

The second full-period exam of the semester will be given in class on Friday, October 27. The format will be similar to that of the first exam. We will review in class on Thursday, October 26. If there is interest, Jodi Marani (our TA) has volunteered to run an informal review/question session on Thursday afternoon or evening as well. You should know the following material from Chapters 2 and 4 of the text:

1) The Meanings of the First and Second Derivatives of f(x)

a) Know how the sign of f′(x) is related to whether the function is increasing or decreasing at x, and

b) how the sign of f′′(x) is related to concavity.

c) Know how to sketch the graphs y = f′(x) and y = f′′(x) qualitatively, given the graph y = f(x) (or y = f′(x)).

2) Computing the derivative f′(x) from the definition

a) Know how to estimate a numerical value of f′(a) from a graph of y = f(x), given scales on x and y axes

b) Know how to compute f′(a) from the limit definition

f′(a) = lim_{h→0} \frac{f(a + h) − f(a)}{h}

for “simple” functions f(x) like linear or quadratic polynomials.

c) Know how to estimate f′(a) numerically by finding \frac{f(a+h)−f(a)}{h} for h tending to 0.

d) Know how to compute f′(x) from the limit definition

f′(x) = lim_{h→0} \frac{f(x + h) − f(x)}{h}

for “simple” functions f(x) like linear or quadratic polynomials.

3) Derivative Formulas (Sections 1-3 from Chapter 4)

a) f′(x) = nx^{n−1} for the power function f(x) = x^{n}

b) f′(x) = ln(a)a^{x} for the exponential function f(x) = a^{x}

c) Sum and constant multiple rules

d) Product rule – \frac{d}{dx}(f(x)g(x)) = f(x)g′(x) + g(x)f′(x)

e) Quotient rule:

\frac{d}{dx} (\frac{f(x)}{g(x)}) = \frac{g(x)f′(x) − f(x)g′(x)}{(g(x))^{2}}

Good Review Problems:

Review problems at the end of Chapter 2; Pages 237-238 of the text have a large collection of practice differentiation problems. Not all of the functions there can be differentiated using only the rules covered on this exam (for instance, any function involving the trigonometric, inverse trigonometric, or logarithm functions), but many can. Try a good selection of these.