2 Chapter Outline Describing Graphs of Functions The First and Second Derivative RulesThe First and Second Derivative Tests and Curve SketchingCurve Sketching (Conclusion)Optimization ProblemsFurther Optimization ProblemsApplications of Derivatives to Business and Economics
9 Changing SlopeEXAMPLEDraw the graph of a function y = f (T) with the stated properties.In certain professions the average annual income has been rising at an increasing rate. Let f (T) denote the average annual income at year T for persons in one of these professions and sketch a graph that could represent f (T).SOLUTIONSince f (T) is rising at an increasing rate, this means that the slope of the graph of f (T) will continually increase. The following is a possible example.Notice that the slope becomes continually steeper.
11 Inflection PointsNotice that an inflection point is not where a graph changes from an increasing to a decreasing slope, but where the graph changes its concavity.
12 Intercepts Definition x-Intercept: A point at which a graph crosses the x-axis.y-Intercept: A point at which a graph crosses the y-axis.
13 Asymptotes Definition Horizontal Asymptotes: A straight, horizontal line that a graph follows indefinitely as x increases without bound.Vertical Asymptotes: A straight, vertical line that a graph follows indefinitely as y increases without bound.Horizontal asymptotes occur whenexists, in which case the asymptote is:If a function is undefined at x = a, a vertical asymptote occurs when a denominator equals zero, in which case the asymptote is:x = a.
15 Describing GraphsEXAMPLEUse the 6 categories previously mentioned to describe the graph.SOLUTION1) The function is increasing over the intervals The function is decreasing over the intervals Relative maxima are at x = -1 and at x = Relative minima is at x = 3 and at x = -3.
16 Describing GraphsCONTINUED2) The function has a (absolute) maximum value at x = -1. The function has a (absolute) minimum value at x = -3.3) The function is concave up over the interval The function is concave down over the interval This function has exactly one inflection point, located at x =1.4) The function has three x-intercepts, located at x = -2.5, x = 1.25, and x = The function has one y-intercept at y = 3.5.5) Over the function’s domain, , the function is not undefined for any value of x.6) The function does not appear to have any asymptotes, horizontal or vertical.