Mt 217 Accelerated Calculus - Fall 2003

Department of Mathematics & Computer Science
CARDINAL STRITCH UNIVERSITY
Sr. Barbara E. Reynolds, Ph.D.

Study Guide for Benchmark Test #3

Your average score on three benchmark tests will contribute 15% to your final course grade. You have already taken the first two benchmark tests. The window of opportunity for the third benchmark is Monday December 3 through Monday December 8.

Benchmark testing is the Department's way of assuring that students have achieved minimum levels of calculational competency. It is generally expected that students who have successfully taken a course in calculus can do certain calculations by hand. Although we have been using computers and calculators throughout this course, you are still expected to learn to do these hand calculations.

This test will have fifteen problems. Four problems will ask you to calculate a limit, four problems will ask you to evaluate a derivative, and seven problems will be antiderivatives and definite integrals.

You will find some good study problems among the following exercises:

Limits: Exercises 1 - 21, page 182. You should be able to calculate limits as the variable approaches a finite value, and as the variable approaches infinity. These two situations require different rules.

Derivatives: Exercises 1 - 33, page 259. You can check your work with Maple.

Antiderivatives and definite integrals: Exercises 1 - 75, on the handout which you have for extra credit problems; also exercises 9 - 34, page 439 - 440. You can check your work with Maple. Some of these problems on are more difficult than those on the benchmark test.

Derivatives

You should be able to:

calculate the derivative of a constant function, a linear function, a function defined as x^n (for an integer n), a function defined as a square root or cube root and a function defined as x^p (for an rational number p);
calculate derivatives for polynomials, and for the six trigonometric functions;
calculate the derivative of e^x, e^u(x), ln(x), ln(u), and exp(x);
use the product rule, the quotient rule, and the chain rule to calculate derivatives for simple combinations of functions.

Antiderivatives

You should be able to:

calculate an antiderivative of a constant function, and a linear function;
use the power rule to integrate an expression of the form x^n, both for n /= 1 and for n = 1;
calculate an antiderivative for polynomials, and for the derivatives of the six trigonometric functions;
use simple substitutions to calculate an antiderivative; (that is, you should be able to find the antiderivative when the derivative was calculated using the chain rule);
use integration by parts (u, dv substitution) to evaluate an antiderivative;
evaluate the antiderivative of an expression of the form e^u du;
evaluate the antiderivative of an expression of the form (1/u) du.

Definite Integrals

You should be able to:

evaluate a definite integral for any function for which you can find the antiderivative;
evaluate an improper integral -- one whose limits are + or - infinity -- by taking a limit,
use elementary geometry to evaluate a definite integral by interpreting it as "area" of a region bounded by the graph of the function.

Additional Sample Questions for the Benchmark Test

Limits

Find the limit as x --> 5 for (x^3 - 25x) / (x - 5).
Find the limit as x --> -5 for (x^3 - 25x) / (x - 5).
Find the limit as x --> 5 for (x - 5) / (x^3 - 25x).
Find the limit as x --> 0+ for (x - 5) / (x^3 - 25x).
Find the limit as x --> 0- for (x - 5) / (x^3 - 25x).
Find the limit as x --> infinity for ((x^3 - 36x) / (x - 6)) (x + 3).
Find the limit as x --> infinity for (x^3 - 36x) / ((x - 6) (x + 3)).
Find the limit as x --> -infinity for ((x^3 - 36x) / (x - 6)) (x + 3).
Find the limit as x --> -infinity for (x^3 - 36x) / ((x - 6) (x + 3)).
Find the limit as x --> infinity for (5x - x^3) / (4 + x^2 + sin(x)).
Find the limit as x --> 0 for sin (36 x) / (6 x).
Find the limit as x --> Pi for sin (36 x) / (6 x).

(For which of these last two problems can you use L'Hopital's Rule? Why does L'Hopital's not apply in the other case?)

(You should also be able to estimate a limit from a graph. This kind of problem is difficult to give on a web-page, but there is such a problem on page 182.)
Derivatives
Find y'(x) if y(x) = 5 x^2 + Pi x + 3. Evaluate this derivative at x = 2.
Find the derivative of x^2 cos(x).
Calculate the derivative of (3x^2 + 2x) / cot(4x).
Find y'(x) if y(x) = 5 x^2 + tan(x) + 3. Where is this derivative equal to 0?
Find the derivative of sec(3x^2) + ln(5x).
Find the derivative of sec(3x^2) * exp(5x).
Find the derivative of sec(3x^2) / exp(5x).
Calculate the derivative of (3x^2 + 2x) + ln(4x).
Calculate the derivative of (3x^2 + 2x) sin(4x).
Calculate the derivative of (3x^2 + 2x) / sin(4x).
Antiderivatives
Find the antiderivative of y(x) = 5 x^2 + Pi x + 3.
Find the antiderivative of x cos(x).
Find the antiderivative of x cos(x^2).
Calculate the antiderivative of (sec(4x))^2 / tan(4x).
Calculate the integral int( (sec(4x))^2 tan(4x), x).
Find int(y,x) if y(x) = 5 x^2 + sin(x) + 3.
Find int(y,x) if y(x) = 5/(x^2) + 3/x.
Calculate int(x * cot(3*x^2) * csc(3*x^2), x).
Definite Integrals
Find the definite integral of y(x) = 5 x^2 + Pi x + 3 for x = 1 to x = 5.
Evaluate int(|x|, x = -2 .. 3).
Evaluate int(sqrt(25 - x^2), x = -5 .. 0).
Find the definite integral of exp(-3x) for x = 0 to x = 5.
Find the improper integral of exp(-3x) for x = 0 to x = infinity.

To pass the benchmark test on the first attempt, you must get thirteen or more of the fifteen problems completely correct; there will be no partial credits. If you pass on the first attempt, your score will be recorded as 100%.
If you do not pass the benchmark test on your first attempt, you may demonstrate that you have done some additional practice, and make an appointment with me to try the test once or twice again. You must complete the re-test by Monday, December 8.
Again, to pass the benchmark test, you must get thirteen or more problems correct with no partial credits. If you pass the test on the re-test, your score will be recorded as the average of your first score and your passing score.
If you do not pass the benchmark test or the re-test by December 8, your score will be recorded as 0%.

Return to Sr. Barbara E. Reynolds Home Page.
Return to course list for 2003 -- 2004.
Go to Mt 217 Accelerated Calculus Syllabus.
Return to the Mt 217 Assignments page.

The easiest way to contact me is to send an email message to Sr. Barbara E. Reynolds.

This page was updated on November 18, 2003.