Study Guide for Exam 3

Exam 3 will focus on material from Chapters 6-7: Sections 6.1-6.5 and 7.1-7.5, specifically. Of course, we have continued in these chapters to call upon knowledge that came earlier.

Antiderivatives

1. Some rules
1. the antiderivative of a sum (resp. difference) is the sum (resp. difference) of antiderivatives
2. the antiderivative of a constant multiple of some function is that constant times the antiderivative of the function
3. given one antiderivative of a function, any constant added to that antiderivative produces another antiderivative
2. Notation of antidifferentiation
looks very much like that of the definite integral (Can you guess why?)
3. specific antiderivatives to know
1. antiderivative of xⁿ (Note the difference in result when n equals (-1) vs. when it does not.)
2. antiderivative of e^mx+b (see Exercises 13-16, p. 315)
3. antiderivative of (mx+b)ⁿ (see Exercises 17-22, p. 315; note the case n=-1 is special, yielding antiderivatives involving the natural log fn.)

Definite integrals

1. Polygonal approximation
1. regions above/under “polygonal” f may be divided easily into triangles and rectangles to get exact “area” (see class handout; also Example 1, p. 328)
2. when f not polygonal, rectangles are what we used (see Riemann sum approximations)
2. Riemann sum (rectangle) approximations (uniform partition case)
1. take the form [f(x₁) + f(x₂) + ... + f(x_n)](Δx), where f is the integrand
2. “area” (geometric) interpretation of integral
1. truly approximates area if function being integrated is nonnegative
2. interpretation must be adapted to a “difference of areas” when function is negative
3. Fundamental Theorem of Calculus
   1. gives rise to non-geometric interpretation of definite integral:
   If the function being integrated is continuous and is the rate of change (derivative) of some quantity on a ≤ x ≤ b then its definite integral from a to b gives the total change in that quantity between a and b.
   2. allows for exact computation of definite integrals in cases where an antiderivative may be found
4. Applications of definite integrals
   1. gives total change in “the” antiderivative
   doesn't matter which antiderivative
   2. areas in the plane: regions bounded by various “curves”
   1. sketch region, determining which function serves as upper boundary, which as lower
   2. solve for limits of integration if not specified
   3. set up integral and evaluate (remember to subtract lower-boundary function from upper)
   3. average value of the function being integrated
   remember that it is not the definite integral itself, but this integral divided by (b-a), where a and b are the left and right endpoints respectively of the interval in which the average value is desired
   4. consumers' surplus
   integrate f(x) - B from 0 to A, where A is a specified sales level, f(x) is the demand function (sales level is input, price is output), and B = f(A).
   5. volumes of rotation
   note similarity to the area-of-circle formula A = π r²

Functions of multiple variables

1. General knowledge
1. evaluating them (require multiple inputs to produce ones output)
2. domain and range (How do these ideas change from functions of 1 variable?)
3. viewing them
1. graph requires one more dimension than the number of inputs (so a function of 3 or more variables cannot be graphed)
2. level sets
1. also called level curves when there are 2 inputs; level surfaces when there are 3
2. a different one for each possible value in the range of the function
3. for a fixed output, a level set consists of all input pairs/tuples that produce that output
4. use one less dimension than the graph, yet help us see the “terrain” (analogy of a topographical map)
3. Be able to associate collections of level sets with corresponding graphs (as in Exercises 23-26, p. 391)
2. Partial derivatives
   1. Notations used to denote them
   2. Definition as limits: f_x(x,y) = lim_{h → 0} [f(x+h, y) - f(x,y)] / h, and f_y(x,y) = lim_{h → 0} [f(x, y+h) - f(x,y)] / h
   3. Understand what they tell you about the surface z = f(x,y)
   4. Know how to find them for various f using the usual derivative rules
   5. How are these derivatives different than implicit differentiation?
3. Optimization
   1. Finding possible sites of relative extrema (i.e., critical points) in unconstrained problems
   An application: Least squares regression (Be able to carry out exercises like those assigned in Section 7.5)
   2. Methods for finding absolute extrema under a constraint
   1. Solve the constraint equation for one of its variables, then substitute into objective function
   2. Lagrange multiplier method

Back to Math 132A Home Page

This page maintained by: Thomas L. Scofield
Department of Mathematics and Statistics, Calvin College