Mt 217 Accelerated Calculus - Fall 2003

Study Guide for Test #2

This test covers material through the first part of Chapter 5 of our text. Although you will be expected to do some computations during the test, this test will focus on conceptual understanding. As I make up problems for this test, I will be thinking of several major themes which have been part of our work this semester:

• mathematical modeling (i.e., using mathematics as a problem-solving tool),
• functions,
• limits and continuity,
• difference quotients and derivatives,
• Riemann sums and definite integrals.

  This will be a two-phase test. Phase I is to be done in-class with no books, notes, or computational aids. After turning in Phase I (at the end of the class period), you are to take the same questions home as a take-home test. During Phase II, you are allowed to use any books, internet resources, and computation devices to which you have access. You are on your honor not to work together, consult other friends or experts, or to seek help from anyone (except possibly the instructor) during Phase II. Phase II will be due at the beginning of the following class period. Phase II is an opportunity to re-think some of the problems outside of the constraints of an in-class test. Think of this as an opportunity to use all the personal resources you have -- your mind, your calculator or computer, your text -- to solve problems.

Mathematical modeling:

Given a situation expressed as a word problem, you should be able to:

• make a sketch which illustrates the situation,
• identify variables in the situation, and give an appropriate domain for the independent variable,
• describe a strategy for solving the problem (using some combination of English and mathematical notation),
• construct a function to represent the problem situation,
• sketch a graph showing the relationship between two of the variables in the situation.

  The Highway Slope Design problem involved taking some information about the highway you were designing and developing a mathematical model in the form of a parabolic function to help you solve the problem. This situation could be a source of problems for this test. Newton's Law of Cooling involves working with a function of the form: y = f(t) = A + B exp(-kt), where A, B, and k are parameters depending on the problem situation, and t is the variable. This situation could be a source of problems for this test.

Functions:

A function is a process that takes an input, and converts it to an output. A function may be expressed in algebraic notation, as a table, in a graph, or in a verbal problem situation.

• You should be able to recognize and work with various representations of functions.

• Given a function, f, expressed in any of the above forms, you should be able to
  • calculate or estimate a slope or a rate of change at (near) a particular value,

    How would you estimate the derivative if you are given the function as a table of values? What if the function is presented in a graph?

  • calculate or estimate a Riemann sum over a given range of values.

    How can you use geometry to calculate the exact value of a definite integral if the function is presented in a graph? How would you estimate a Riemann sum if the function is presented in a table of values?

• Given a function represented as an expression or as a graph, you should be able to estimate some of the key values or properties of that function, such as:
  • the basic shape of the graph,
  • the x- and y-intercepts,
  • the zeros of the function,
  • the regions where the function is positive or negative,
  • the regions where the function is increasing or decreasing,
  • the limiting value of the function as x gets very large (that is, the limit as the variable x goes to positive or negative infinity),
  • any vertical and non-vertical asymptotes,
  • an appropriate domain and range for the function.

• In representing a function as a parametric equation, the values of x and y are each expressed as functions of a third variable t. Thus the function is represented as [x(t), y(t), for a < t < b]. Since you will not be using a calculator during the in-class phase of this test, I will not ask you to graph a complicated function given its parametric equation. However, I might give you a function represented in parametric form, and ask some conceptual questions about it.

  If a function is given parametrically, how to you evaluate the derivative dy/dx? How can you find horizontal and vertical asymptotes for the curve? What would this mean graphically?

• Given an expression which represents a function, you should be able to sketch a graph of the function over an appropriate domain showing its general shape, any x- and y-intercepts, regions where it is positive or negative, regions where it is increasing or decreasing, and any vertical or non-vertical asymptotes.

• You should be able to sketch a graph showing the important features of any of the curves in the list below (where a, b, and c denote constants).

  y = a * x + b	y = a * sin(x) + b	y = a * |x| + b
  y = a * x^2 + b	y = a * cos(x) + b	y = |x + c|
  y = a * x^3 + b	y = a * tan(x) + b	y = |sin(x)| + b
  y = a / x	y = a / (x + c)	y = (a/x) + b
  y = a / x^2	y = a / (x^2 + c)	y = (a/x^2) + b
  y = sqrt(x)	y = sqrt(x + c)	y = sqrt(x) + b
  y = 2^x	y = 2^(-x)	y = e^x
  x^2 + y^2 = r^2	y = (x + a) (x + b) (x + c)	y = ln(x)

• Good problems to use for review and study can be found in Section 4.4.

Limits and Continuity:

• Does the graph of the function have a "hole" or a vertical asymptote where x = a? Is the function continuous at this point? Does the function have a jump discontinuity at this point?

• What is the limiting value of the function f(x) as x gets very large? Does the function have any horizontal or oblique asymptotes? Does the function have an asymptotic curve? As the variable x goes to infinity, does the graph of the function come "down" to the asymptote from above, or go "up" to the asymptote from below?

Difference quotients and Derivatives:

• Proving the various short-cut rules for the derivative involves a two-step process. You should be able (1) to set up and simplify a difference quotient, then (2) to take a limit to prove the basic rules of differentiation. You should be able to prove the rules for calculating derivatives of the following functions:
  • f(x) = a x^2
  • f(x) = a x^3
  • f(x) = a x + b
  • f(x) = a / x
  • f(x) = a / (x^2)
  • f(x) = sqrt(x)
  • f(x) = exp(x) = e^x
  • f(x) = a
  • f(x) = a x + b x^2

• You should be able to estimate a rate of change using a difference quotient. You should be able to estimate the derivative of a function that is presented in a table of values.

• You should be able to estimate the slope of a curve presented in a graph. You should be able to estimate a derivative if the function is presented in graphical form.

• You should be able to evaluate the derivative (dy/dx) of a function presented in parametric form. By evaluating this derivative at particular points, you should be able to say something about the graph of the curve at these points. By solving for when dy = 0 or dx = 0, you should be able to say something about horizontal or vertical tangents. -- See Example 10, page 231, and some of the problems at the end of Section 3.5.

Riemann Sums and Definite integrals

• Given a function as an expression or as a graph, you should be able to use basic geometry formulas to calculate the exact value of the definite integral.

• You should be able to set up a Riemann sum over a small number of intervals (say, n <= 6) estimate the value of a definite integral.

• Good problems to use for review and study can be found in Sections 5.2 and 5.2.