Curve Sketching ALWAYS LEARNING Slide 1 ALWAYS LEARNING

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Local maximum Local minimum ALWAYS LEARNING Slide 7

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x 1 & x 3 are local maximums x 2 & x 4 are local minimums ALWAYS LEARNING Slide 10

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Example: Solution: Find the critical numbers of the given function. Critical numbers are numbers in the domain of f where do not exist. We have so for every x. Setting shows that or exists Therefore − 3 and 4 are the critical numbers of f; these are the only places where local extrema could occur. The graph at the right shows that there is a local maximum at and a local minimum at ALWAYS LEARNING Slide 13

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Example: Find the second derivative of the given function. (a) Solution: Here, of ALWAYS LEARNING The second derivative is the derivative Slide 17

Example: Find the second derivative of the given function. (b) Solution: Use the quotient rule to find Use the quotient rule again to find ALWAYS LEARNING Slide 18

Example: Find the second derivative of the given function. (c) Solution: Using the product rule gives Differentiate this result to get ALWAYS LEARNING Slide 19

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Example: Graph Solution: Step 1 The y-intercept is Step 2 To find the x-intercepts, we must solve the equation Step 3 There is no easy way to do this by hand, so skip this step. Since f(x) is a polynomial function, the graph has no asymptotes, so we can also skip Step 3. Step 4 The first derivative is Step 5 The first derivative is defined for all x, so the only critical numbers are the solutions of and the second Divide both sides by 6. Factor. ALWAYS LEARNING Slide 32

Example: Graph Solution: Step 5 Using the second-derivative test on the critical number have, Hence, there is a local maximum when Similarly, so there is a local minimum when ). we that is, at the point (at the point Next, we determine the intervals on which f is increasing or decreasing by solving the inequalities ALWAYS LEARNING Slide 33

Example: Graph Solution: Step 5 The critical numbers divide the x-axis into three regions. Testing a number from each region, as indicated below, we conclude that f is increasing on the intervals and decreasing on Figure 12. 46 Step 6 The second derivative is defined for all x, so the possible points of inflection are determined by the solutions of ALWAYS LEARNING Slide 34

Example: Graph Solution: Step 6 Determine the concavity of the graph by solving Therefore, f is concave upward on the interval and concave downward on Consequently, the only point of inflection is Step 7 Since f is a third-degree polynomial function, we know that when x is very large in absolute value, its graph must resemble the graph of its highest degree term, that is, the graph must rise sharply on the right side and fall sharply on the left. ALWAYS LEARNING Slide 35

Example: Graph Solution: Step 7 Combining this fact with the information in the preceding steps, we see that the graph of f must have the general shape shown below. ALWAYS LEARNING Slide 36

Example: Graph Solution: Step 8 Now we plot the points determined in Steps 1, 5, and 6, together with a few additional points to obtain the graph below. ALWAYS LEARNING Slide 37