Chapter 12 Review Important Terms, Symbols, Concepts 12.1. First Derivatives and Graphs Increasing and decreasing properties of a function can be determined by examining a sign chart for the derivative. A number c is a partition number for f ’(x) if f ’(c) = 0 or f ’(x) is discontinuous at c. If c is also in the domain of f (x), then c is a critical value. The first derivative test is used to locate extrema of a function. Barnett/Ziegler/Byleen Business Calculus 11e

Chapter 12 Review 12.2. Second Derivative and Graphs The second derivative of a function can be used to determine the concavity of the graph of f. Inflection points on a graph are points where the concavity changes. Concavity of the graph of f (x) can also be determined by examining the graph of f ’(x). A graphing strategy is used to organize the information obtained from the first and second derivatives. Barnett/Ziegler/Byleen Business Calculus 11e

Chapter 12 Review 12.3. L’Hôpital’s Rule Limits at infinity and infinite limits involving powers of (x-c), ex, and ln x are reviewed. The first version of L’Hôpital’s rule applies to limits involving the indeterminate form 0/0 as x  c. You must always check that L’Hôpital’s rule can be applied. L’Hôpital’s rule can be applied more than once. L’Hôpital’s rule applies to one-sided limits. L’Hôpital’s rule applies to limits at infinity. L’Hôpital’s rule applies to limits involving the indeterminate form /. Barnett/Ziegler/Byleen Business Calculus 11e

Chapter 12 Review 12.4. Curve Sketching Techniques The graphing strategy used previously is expanded to include horizontal and vertical asymptotes. If f (x) = n(x)/d(x) is a rational function with the degree of n(x) one more than the degree of d(x), then the graph of f (x) has an oblique asymptote of the form y = mx + b. Barnett/Ziegler/Byleen Business Calculus 11e

Chapter 12 Review 12.5. Absolute Maxima and Minima The steps involved in finding the absolute maximum and absolute minimum values of a continuous function on a closed interval are listed in a procedure. The second derivative test for local extrema can be used to test critical values, but it does not work in all cases. If a function is continuous on an interval I and has only one critical value in I, the second derivative test for absolute extrema can be used to find the absolute extremum, but it does not work in all cases. Barnett/Ziegler/Byleen Business Calculus 11e

Chapter 12 Review 12.6. Optimization The methods used to solve optimization problems are summarized and illustrated by examples. Barnett/Ziegler/Byleen Business Calculus 11e