Study Guide for Test 2

Below is a study guide that will aid you in your preparation for this test.  Use this guide to study the specific kinds of questions that will appear on the test. 

IMPORTANT: Before you begin studying for the Test 2, please click here to read a short a short article on how to study mathematics.  In particular, read the last two sections that describe how to review for and take tests.

Question #1            Know in depth the relationships between the graph of a given function and the graph of its derivative.  You should be able to match the graph of a function with the graph of its derivative.  You should also be able to sketch the graph of the derivative function given the graph of the original function.  Section 3.2 problems #4 - 12 is a good resource for this.

Question #2            Memorize the various rules for differentiation and know how to use them.  The rules are the power rule, product rule, quotient rule, chain rule, constant multiple rule, and sum/difference rule as well as the derivatives of constants.  Each of these rules is found in section 3.3.  You need to know when it is appropriate to simply a function after differentiating it and be able to follow directions for specific types of simplification i.e. leave no negative exponents in your final answer.  Review the HW from this section to get practice with each one of the rules. 

Question #3            Be able to answer various questions that relate to the motion of a particle described by an equation or formula.  Be prepared to answer questions about the particle's velocity, and position at various times as well as sketch a diagram illustrating the motion of the particle.  The substitute teacher did a problem like this with you in class so review your notes from 10/02/2006.  Page 166 problems #1 - 6 are a good resource for this.  Also, know how to find the time intervals where a particle is speeding up/slowing down.  Example 2 on the bottom of page 191 will guide you through this.      

Question #4            Get practice finding the equation of tangent lines to different functions at various points.  Examples of this are scattered throughout the book.  This is a standard type of calculus problem and will most assuredly be on the test at least once.  See Example 10 on page 153 to see how these types of problems are solved.  Scour the book for more examples and problems of this type for extra practice.

Question #5            Practice some related rates problems.  Specifically, you should know how to do the type that involve triangles.  By this I mean, you should be able to do any related rates problem where a triangle might be used to illustrate the objects in motion for a given problem.  When dealing with right triangles remember to use the Pythagorean Theorem to set up an equation in which to differentiate.  Many of your HW problems from this section are of this kind.  Review selected HW problem solutions for section 3.9 by clicking here.  You can go to a great website by clicking here that shows you the ins and outs of related rates problems.  This website has the basic steps to go about solving any type of related rates problem.  Be sure to read these steps.

Question #6            Be able to use the rules for differentiation as they relate to the graph of a function without a specific formula given.  An example of what I mean is problem #57 on page 182.  This one specifically uses the chain rule.  Be ready to do the same thing on the test for other rules of differentiation too.  Look in the exercise set of section 3.3 to find problems of this type using the product and quotient rules.     

Question #7            Be able to implicitly differentiate an equation involving both x and y and then solve for y'.  Problems #1 - 14 on page 188 provide more than adequate practice for these type of problems.  For some students, implicit differentiation is a difficult hurdle to cross.  Be sure to come see me during office hours if you are having trouble with this concept and I will help you to thoroughly understand the process of implicit differentiation.

Question #8            Know how to find higher derivatives of functions.  Specifically, you should be able to find the second derivative of almost any function that you can find the first derivative of.  Problems #5 - 20 on page 195 from section 3.8 provide more than adequate preparation for this.

Question #9            You need to know how to use the chain rule in combination with any of the other rules such as the power rule, product rule, or quotient rule.  You should also be able to use the chain rule twice in succession (i.e. chain/chain rule combo) when differentiating a function.  Practice problems can be found scattered among the exercises in section 3.6 and 3.8.  See if you can try to find exercises in the text where there is a chain/product rule combo or a quotient/chain rule combo as well as the chain/chain rule combo. 

Question #10          Know by heart, the derivatives of all 6 trigonometric functions.  There is a note in the left margin on page 172 that will help you when you memorize these derivatives.  You need to be able to find derivatives involving simple combinations of trigonometric and algebraic functions using the various rules of differentiation.  For example, you should be able to use the product or quotient rule when the component functions are a mixture of both trigonometric and algebraic expressions.