Study Guide for Test 3

Note:  For any problem on the test dealing with critical numbers, you NEED to use either the 1st or 2nd derivative test to see if the critical number yields a local minimum or maximum.  You MUST include this step in your work to receive full credit for such a problem.  For example, if a test problem asks you to minimize the volume of a solid subject to certain constraints, then you NEED  justification that the critical number found in doing the problem does, in fact, yield a local minimum.  This justification is met by using either the 1st or 2nd derivative tests.           

Question #1        Be able to distinguish graphically between an absolute maximum and a local maximum.  Sometimes they occur at the same value of x and other times they do not.  The same holds true for absolute minimums and local minimums.   Other times a critical number will not yield a local extreme value at all; review the possibilities for such a graph.  Be prepared to sketch the graph of a function that has specified local and absolute extreme values along with information about continuity and differentiability at various values of x.  The first fourteen problems in section 4.1 cover these ideas thoroughly.

Question #2        You need to know how to use the closed interval method for finding absolute extreme values of continuous functions such as polynomials and trigonometric functions involving sine and cosine.  You can review the closed interval method outlined in the book at the bottom of page 227.  To receive full credit you must show your work for each step of this method.  The steps will not be given on the test so make sure you know them by heart.  As part of this method you should review the definition of critical number and get practice finding critical numbers of a wide range of functions from polynomial and root functions to rational and trigonometric functions.  

Question #3        Be prepared to find horizontal asymptotes for functions that have them.  Many rational functions have such asymptotes but root functions also can have horizontal asymptotes.  Examples for both types of functions are found in section 4.4.  The procedure for finding horizontal asymptotes is an algebraic one involving limits.  Basically, to find a horizontal asymptote you compute the limit of the function as x approaches either negative or positive infinity.  Please review your notes and examples from section 4.4 to see how this procedure is done.  Remember that there are THREE cases to consider when finding horizontal asymptotes for rational functions.  The first is when the degree of the top equals the degree of the bottom.  The second is when the degree on the bottom is bigger than the degree on the top.  The third is when the degree on the top is bigger than the degree on the bottom.  Also, I had you underline an important sentence in your textbook on page 253; you may need to use the advice contained in this sentence on the test.  Know, for sure, how to evaluate limits at infinity for functions that are NOT rational.  In particular, review examples 4 and 5 on page 254 -255.  

Question #4        You will need to be able to use the various tests to see how derivatives affect the shape of a graph from section 4.3.  More specifically, you will need to know how to find the intervals where a given function is increasing and decreasing.  Be able to use the first derivative test to determine what kind of local extreme values, if any, there are at each critical number for a given function.  Also know how to apply the second derivative test to find the intervals where a function is concave up or down.  Know how to locate inflection points.  The increasing/decreasing test, the first derivative test, and the concavity test can ALL be done by using a sign chart.  Refer to your notes to see how sign charts work using test points.

Question #5        Section 4.7 consists of word problems involving optimization.  Review the five step procedure in solving these types of word problems found on page 278.  Click here to go to a great website that gives problems and solutions to common types of optimization problems.  This should help you to gain additional insight into the process for solving them as well as helping you to sharpen your skills.  One problem that will appear on Test 3 will be taken verbatim from your homework from section 4.7.  If you are able to independently do all the homework problems from this section you will have no problem with this question on the test.  There will also be a second question on the test dealing with optimization that will not be taken from the HW.  Read examples 1, 2, and 4 on pages 278 - 282. 

Question #6        Your knowledge of the Mean Value Theorem (MVT) will be tested.  As with any theorem in calculus, know by heart the hypotheses and conclusion of the MVT.  You will need to know how to estimate the values of c that satisfy the conclusion of the theorem based solely from a graph see problem #7 in section 4.2.  Be able to find the exact values of c that satisfy the conclusion of the theorem based on a given function on a specified interval.  Examples of both types are found in section 4.2.  See problems #7, #11, #12, #13, #14.

Question #7        For this question, you will be asked to sketch the graph of a function that satisfies certain conditions.  The conditions themselves will be information given to you in function notation.  This information might include intervals where the first and second derivatives are positive and/or negative.  Information about horizontal asymptotes via limits at infinity might be given as well as specific function values for certain values of x.  For this problem, you will use these pieces of information to construct the graph of a function that obeys each of the given criteria.  You should know the definitions of odd and even functions.  Look in the index of your textbook to find where you can review these definitions.