Can you please help with 54, 65, and 68. Thanks so much! :)

\fAT&T 4:15 PM 31% SECTION 4.4 The Second Derivative and Concavity 245 67. An infectious flu spreads slowly at the beginning of an epi- demic. The infection process accelerates until a majority of the susceptible individuals are infected. at which point the process slows down. (a) If R() is the number of individuals infected at time , describe the concavity of the graph of R near the beginning and end of the epidemic. (b) Describe the status of the epidemic on the day that R has a point of inflection. Both: Library of Congress Prints and Photographs Division 68. Water is pumped into a sphere at a constant rate (Figure 20), Let Inv be the water level at time ?. Sketch the graph of h (approximately. but with the correct concavity). Where does the point of inflection occur? 69. Water is pumped into a sphere of radius R at a variable rate in Original Sigmoidal correction such a way that the water level rises at a constant rate (Figure 20). Let V() be the volume of water in the tank at time . Sketch the graph V (approxi- FIGURE 21 mately. but with the correct concavity). Where does the point of inflection occur? Figure 22 shows that g(w) reduces the intensity of low-intensity pixels [where g(a) 0 and use this to show that g(u) increases from 0 to 1 for 0 S u s 1. (b) Where does g(u) have a point of inflection? FIGURE 20 70. (Continuation of Exercise 69) If the sphere has radius R, the volume of water is 0.4 V = * (RH] - -43). 0.2 where h is the water level. Assume the level rises at a constant rate of 1 (i.e., h = (). 0.2 0.4 0.6 0.8 1.0 (a) Find the inflection point of V. Does this agree with your conclusion In Exercise 69? FIGURE 22 Sigmoidal correction with (b) (GU Plot V for R = 1. a = 0.47, b = 12. 71. Image Processing The intensity of a pixel in a digital image is mea- sured by a number a between 0 and I. Often, images can be enhanced 72. Use graphical reasoning to determine whether the following by rescaling intensities, as in the images of Amelia Earhart in Figure 21. statements are true or false. If false. modify the statement to make it When rescaling, pixels of intensity u are displayed with intensity g() for correct. a suitable function g. One common choice is the sigmoidal correction, (a) If f is increasing, then f- is decreasing. defined for constants a, b by (b) If f is decreasing, then f-is decreasing. g(1) = f(u) - f(0) where f(u) = (1 + (a-=))-1 (c) If f is concave up, then fis concave up. f(1) - f(0)' (d) If f is concave down, then fis concave up. Further Insights and Challenges In Exercises 73-75, assume that f is differentiable. (b) Show that G(c) = G'(c) = 0 and G"(x) > 0 for all x. Conclude that 73. Proof of the Second Derivative Test Let c be a critical point such G'(x) 0 for x > c. Then deduce, using the that f"(c) > 0 [ the case f"(c) G(c) for x # c. (a) Show that /"(c) = lim S'(e + 4) h 75. Assume that f" exists and let c be a point of inflection (b) Use (a) to show that there exists an open interval (a. b) containing c of f such that f'(x) 0ife 0 for all x, then the graph at x = c. of / "sits above" its tangent lines. (a) For any c, set G(x) = f(x) - f'(e)(x - c) - f(c). It is sufficient to (b) GU Verify this conclusion for f (x) = by graphing / and 3x2 + 1 prove that G(x) 2 0 for all c. Explain why with a sketch. the tangent line at each inflection point on the same set of axes. 4/4