Copyright © Cengage Learning. All rights reserved. 12 Further Applications of the Derivative

Copyright © Cengage Learning. All rights reserved Analyzing Graphs

33 Features of a Graph 1. The x- and y-intercepts: If y = f (x), find the x-intercept(s) by setting y = 0 and solving for x; find the y-intercept by setting x = 0 and solving for y:

44 2. Extrema: Use previous techniques to locate the maxima and minima: 3. Points of inflection: Use previous techniques to locate the points of inflection: Analyzing Graphs

55 4. Behavior near points where the function is not defined: If f (x) is not defined at x = a, consider and to see how the graph of f behaves as x approaches a:

66 Analyzing Graphs 5. Behavior at infinity: Consider and if appropriate, to see how the graph of f behaves far to the left and right:

77 Figure 34 – 50 ≤ x ≤ 50, – 20 ≤ y ≤ 20 – 10 ≤ x ≤ 10, – 3 ≤ y ≤ 1 Example 1 – Analyzing a Graph Analyze the graph of f (x) = Solution: The graph, as drawn using graphing technology, is shown in Figure 34, using two different viewing windows. (Note that x = 0 is not in the domain of f.) The second window in Figure 34 seems to show the features of the graph better than the first.

88 Example 1 – Solution Does the second viewing window include all the interesting features of the graph? Or are there perhaps some interesting features to the right of x = 10 or to the left of x = –10? Also, where exactly do features like maxima, minima, and points of inflection occur? In our five-step process of analyzing the interesting features of the graph, we will be able to sketch the curve by hand, and also answer these questions. 1. The x- and y-intercepts: We consider y = To find the x-intercept(s), we set y = 0 and solve for x: cont’d

99 Example 1 – Solution Multiplying both sides by x 2 (we know that x cannot be zero, so we are not multiplying both sides by 0) gives x = 1. Thus, there is one x-intercept at x = 1. For the y-intercept, we would substitute x = 0 and solve for y. cont’d

10 Example 1 – Solution However, we cannot substitute x = 0; because f (0) is not defined, the graph does not meet the y-axis. We add features to our freehand sketch as we go. Figure 35 shows what we have so far. cont’d Figure 35

11 Example 1 – Solution 2. Relative extrema: We calculate f (x) = To find any stationary points, we set the derivative equal to 0 and solve for x: Thus, there is one stationary point, at x = 2. cont’d

12 Example 1 – Solution We can use a test point to the right to determine that this stationary point is a relative maximum: The only possible singular point is at x = 0 because f (0) is not defined. However, f (0) is not defined either, so there are no singular points. cont’d

13 Example 1 – Solution Figure 36 shows our graph so far. 3. Points of inflection: We calculate f  (x) = To find points of inflection, we set the second derivative equal to 0 and solve for x: cont’d Figure 36

14 Example 1 – Solution 2x = 6 x = 3. Figure 34 confirms that the graph of f changes from being concave down to being concave up at x = 3, so this is a point of inflection. cont’d Figure 34 – 50 ≤ x ≤ 50, – 20 ≤ y ≤ 20 – 10 ≤ x ≤ 10, – 3 ≤ y ≤ 1

15 Example 1 – Solution f  (x) is not defined at x = 0, but that is not in the domain, so there are no other points of inflection. In particular, the graph must be concave down in the whole region (, 0), as we can see by calculating the second derivative at any one point in that interval: f  (–1) = –8 < 0. Figure 37 shows our graph so far (we extended the curve near x = 3 to suggest a point of inflection at x = 3). cont’d Figure 37

16 Example 1 – Solution 4. Behavior near points where f is not defined: The only point where f (x) is not defined is x = 0. From the graph, f (x) appears to go to as x approaches 0 from either side. To calculate these limits, we rewrite f (x): Now, if x is close to 0 (on either side), the numerator x – 1 is close to –1 and the denominator is a very small but positive number. The quotient is therefore a negative number of very large magnitude. cont’d

17 Example 1 – Solution Therefore, and From these limits, we see the following: (1) Immediately to the left of x = 0, the graph plunges down toward. (2) Immediately to the right of x = 0, the graph also plunges down toward. cont’d

18 Example 1 – Solution Figure 38 shows our graph with these features added. We say that f has a vertical asymptote at x = 0, meaning that the points on the graph of f get closer and closer to points on a vertical line (the y-axis in this case) further and further from the origin. cont’d Figure 38

19 Example 1 – Solution 5. Behavior at infinity: Both 1/x and 1/x 2 go to 0 as x goes to or ; that is, and Thus, on the extreme left and right of our picture, the height of the curve levels off toward zero. Figure 39 shows the completed freehand sketch of the graph. cont’d Figure 39

20 Example 1 – Solution cont’d Figure 39 We say that f has a horizontal asymptote at y = 0. In summary, there is one x-intercept at x = 1; there is one relative maximum (which, we can now see, is also an absolute maximum) at x = 2; there is one point of inflection at x = 3, where the graph changes from being concave down to concave up. There is a vertical asymptote at x = 0, on both sides of which the graph goes down toward, and a horizontal asymptote at y = 0.