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Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 39

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 39. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | 10.3 Principles in Action 361 10.3 PRINCIPLES IN ACTION In this section we put the principles discussed in Sections 10.1 and 10.2 into action. EXAMPLE 10.6 Below is a graph of f the derivative of f. The questions that follow refer to f. The domain of f is all real numbers. Figure 10.21 a Identify all critical points of f. Which of these critical points are also stationary points of f b On a number line plot all the critical points of f. Indicate the sign of f and indicate where f is increasing and where f is decreasing. Is this enough information to classify all the extrema Does f have a global maximum If so can we determine where it is attained Does f have a global minimum If so can we determine where it is attained c Where is the graph of f increasing and concave down Where is the graph of f decreasing and concave up d Determine all points of inflection of f. SOLUTION a Critical points of f are points at which either f 0 f is undefined or endpoints of the domain. The critical points in this example are x 0 and x 5. Both are points at which f 0 so they are both stationary points. b graph off sign of f 0 We can see that f has a local maximum at x 0 because f changes from increasing to decreasing at x 0. We can tell that x 0 is also the global maximum point. f is increasing for all negative x and decreasing for all positive x. x 5 is not an extreme point because f is decreasing both before and after x 5. f -1 for x 7 consequently f is decreasing with a slope of -1 on 7 rc . Therefore there is no global minimum. c f is increasing and concave down on -2 0 where f is positive and decreasing. Check the graph of f . f is decreasing and concave up where f is negative and increasing. That is on 2 5 . 362 CHAPTER 10 Optimization d Points of inflection of f are points at which the concavity of f changes. That is points of inflection of f are points at which f changes sign. f is the slope of f so f changes sign at x -2 x 2 and x 5. Below is a sketch6 of f. It is drawn without