Study Guide for Test 1

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. 

Question #1            Using only the graph of a given function (not the formula), be able to find the function's limiting values for various values of x.  Know how to read actual function values from a graph and know the various ways a function can be undefined at various points.  Know the notation for one-sided limits.  Be able to tell using a graph when the limit of a function does not exist.  This happens when the one-sided limits do not agree.  Practice problems that relate to this question are #4, 5, and 8 in section 2.2.

Question #2            Be able to USE definition #1 on page 113 to find the slope of the tangent line to parabolas at specified points.  In addition, be able to find the equation of the tangent line, itself, using the point-slope form of a line.  A practice problem that relates to this question is #29 on page 123.  You should be able to do this type of problem even when the middle term (involving x) is not missing in the equation of the parabola.

Question #3            For this question, be ready to graph a function that satisfies a set of given conditions.  Such conditions may include specified function values such as f(3)=2 or limiting values such as "the limit of f(x) as x approaches 6 from the left equals -1".  A practice problem that relates to this question is #13 on page 81.

Question #4            Given a function's formula, be ready to recognize how the limit of the function does not exist or is equal to positive or negative infinity.  Next, be ready to explain (in a sentence) why is doesn't exist.  Two such reasons are: the graph has a vertical asymptote at the x value in question.  Another is that the the graph of the function has a "jump" discontinuity at the x value.  A practice problem that relates to this question is #13 on page 90.

Question #5            Know how to algebraically compute one-sided limits of piecewise defined functions and be able to graph one.  You need to know that the  limit of a function exists if and only if the one-sided limits exist AND are equal.   A practice problem that relates to this question is #12 on page 81.

Question #6            On this problem, know how to algebraically compute the limit of a quotient by first factoring and then canceling out the common factors in both the numerator and denominator.  You need to know how to use the conjugate trick when computing limits involving square roots.  Practice problems that relates to this question are #22 and #23 and #27 on page 90.

Question #7            You should be able to interpret what the slope means in the context of a particular "real world" problem.  An example of what I mean is problem #27 on page 132.  Sometimes "real world" data is given in a table of values like problem #31 on page 133.  You should be able to use such a table and calculate the slope between two data points in the table and interpret it.  Also, know to find the "best" possible estimate for the slope of the tangent line by averaging the two "closest" secant lines to a given point.  An example of this is problem #25 on page 121.    

Question #8            Be able to USE definition #2 at the bottom of page 114 to find the slopes of tangent lines to rational functions where the numerator contains only a constant and the denominator contains only one linear factor of x.  For these types of rational functions, you specifically need to know how to use this definition to find the slopes of tangent lines at "arbitrary values of x"; not just specified values of x such as x=3.  A practice problem that relates to this question is #30 on page 123.  In this practice problem, instead of finding the slope at x=0 or x=1 be able to use definition #2 to find the slope at x = a.  This is what I meant when I said "arbitrary values of x".  Another problem that relates to this is #11 (part a) on page 120.  As before, be ready to find the equation of the tangent line at a specified point.

Question #9            This problem involves the Intermediate Value Theorem (IVT).  You need to know verbatim the three hypothesis and conclusion of this theorem as given in class.  You will be given a third degree polynomial function, f(x), in which you must use the IVT to show the existence of a number c such that f(c) = N for a given value of N.  The procedure for how to do this will be outlined for you in multiple parts.  In the first part, you will need to find appropriate values for a and b such that f(b)>N and f(a)<N.  For the second part, you need to know that all polynomial are continuous.  Continuity is the crucial property that functions must have in order for the IVT to work.  In other words, if a given function is not continuous on the closed interval [a, b] then there is no guarantee that the IVT will work.  For the last part to this question, be able to find the value of  'c'  either through algebraic or graphical methods.  To be fully prepared for this question, you need to review #43 on page 112.  In this problem c = 2.365018995.  Click here to view the solution to #43 on page 112.  

Question #10          Know how to algebraically compute limits like those that appear in example #5 on page 86.  Notice that you had to FOIL out the numerator and combine like terms first.  You should be able to do a problem similar to this where (a+h) is cubed instead of squared.  Search the problems in section 2.3 for such an example.

Question #11          Review HW problem #39 on page 112 as well as #58 in the supplementary problem packet to be successful in completing this problem on the test.  To do this problem, you need to know the limit definition of continuity as it appears at the top of the page on 102.  Keep in mind that for the limit of a function to exist, the one-sided limits must be equal.  There will be a piecewise defined function given in this problem.  You need to know how to compute the one-sided limits for it and find a value for the number c so that the one-sided limits are equal.  Remember that piecewise defined functions whose components are polynomial functions are continuous everywhere except possibly at the points where the domain is split up.  It is at these x-values that you need to know how to algebraically compute the one-sided limits and find the value of c so that these one-sided limits are equal. 

Question #12         If you are given the graph of a function y= f(x), be able to look at it and sketch, by hand, the graph of y= f '(x)  i.e. the derivative function.  Examples of this are found in problems #5 through #12 on page 143.