in problems 57 - 64, use a graphing utility to graph each function over the indicated interval and…

in problems 57 - 64, use a graphing utility to graph each function over the indicated interval and approximate any local maximum values and local minimum values. determine where the function is increasing and where it is decreasing. round answers to two decimal places. 57. f(x)=x^3 - 3x + 2 -2,2 58. f(x)=x^3 - 3x^2 + 5 -1,3 59. f(x)=x^5 - x^3 -2,2 60. f(x)=x^4 - x^2 -2,2 61. f(x)=-0.2x^3 - 0.6x^2 + 4x - 6 -6,4 62. f(x)=-0.4x^3 + 0.6x^2 + 3x - 2 -4,5 63. f(x)=0.25x^4 + 0.3x^3 - 0.9x^2 + 3 -3,2 64. f(x)=-0.4x^4 - 0.5x^3 + 0.8x^2 - 2 -3,2 65. find the average rate of change of f(x)=-2x^2 + 4: (a) from 0 to 2 (b) from 1 to 3 (c) from 1 to 4 (d) by hand, graph f and illustrate the average rate of change found in parts (a), (b), and (c). 66. find the average rate of change of f(x)=-x^3 + 1: (a) from 0 to 2 (b) from 1 to 3 (c) from -1 to 1 (d) by hand, graph f and illustrate the average rate of change found in parts (a), (b), and (c). 67. find the average rate of change of g(x)=x^3 - 2x + 1: (a) from -3 to -2 (b) from -1 to 1 (c) from 1 to 3 68. find the average rate of change of h(x)=x^2 - 2x + 3: (a) from -1 to 1 (b) from 0 to 2 (c) from 2 to 5 71. g(x)=x^2 - 2 (a) find the average rate of change from -2 to 1. (b) find an equation of the secant line containing (-2,g(-2)) and (1,g(1)). (c) using a graphing utility, draw the graph of g and the secant line obtained in part (b) on the same screen. 72. g(x)=x^2 + 1 (a) find the average rate of change from -1 to 2. (b) find an equation of the secant line containing (-1,g(-1)) and (2,g(2)). (c) using a graphing utility, draw the graph of g and the secant line obtained in part (b) on the same screen. 73. h(x)=x^2 - 2x (a) find the average rate of change from 2 to 4. (b) find an equation of the secant line containing (2,h(2)) and (4,h(4)). (c) using a graphing utility, draw the graph of h and the secant line obtained in part (b) on the same screen.

in problems 57 - 64, use a graphing utility to graph each function over the indicated interval and approximate any local maximum values and local minimum values. determine where the function is increasing and where it is decreasing. round answers to two decimal places. 57. f(x)=x^3 - 3x + 2 -2,2 58. f(x)=x^3 - 3x^2 + 5 -1,3 59. f(x)=x^5 - x^3 -2,2 60. f(x)=x^4 - x^2 -2,2 61. f(x)=-0.2x^3 - 0.6x^2 + 4x - 6 -6,4 62. f(x)=-0.4x^3 + 0.6x^2 + 3x - 2 -4,5 63. f(x)=0.25x^4 + 0.3x^3 - 0.9x^2 + 3 -3,2 64. f(x)=-0.4x^4 - 0.5x^3 + 0.8x^2 - 2 -3,2 65. find the average rate of change of f(x)=-2x^2 + 4: (a) from 0 to 2 (b) from 1 to 3 (c) from 1 to 4 (d) by hand, graph f and illustrate the average rate of change found in parts (a), (b), and (c). 66. find the average rate of change of f(x)=-x^3 + 1: (a) from 0 to 2 (b) from 1 to 3 (c) from -1 to 1 (d) by hand, graph f and illustrate the average rate of change found in parts (a), (b), and (c). 67. find the average rate of change of g(x)=x^3 - 2x + 1: (a) from -3 to -2 (b) from -1 to 1 (c) from 1 to 3 68. find the average rate of change of h(x)=x^2 - 2x + 3: (a) from -1 to 1 (b) from 0 to 2 (c) from 2 to 5 71. g(x)=x^2 - 2 (a) find the average rate of change from -2 to 1. (b) find an equation of the secant line containing (-2,g(-2)) and (1,g(1)). (c) using a graphing utility, draw the graph of g and the secant line obtained in part (b) on the same screen. 72. g(x)=x^2 + 1 (a) find the average rate of change from -1 to 2. (b) find an equation of the secant line containing (-1,g(-1)) and (2,g(2)). (c) using a graphing utility, draw the graph of g and the secant line obtained in part (b) on the same screen. 73. h(x)=x^2 - 2x (a) find the average rate of change from 2 to 4. (b) find an equation of the secant line containing (2,h(2)) and (4,h(4)). (c) using a graphing utility, draw the graph of h and the secant line obtained in part (b) on the same screen.

Answer

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is given by $\frac{f(b)-f(a)}{b - a}$.

Step2: Solve problem 65(a)

For $f(x)=-2x^{2}+4$ and the interval $[0,2]$, first find $f(0)$ and $f(2)$. $f(0)=-2(0)^{2}+4 = 4$. $f(2)=-2(2)^{2}+4=-2\times4 + 4=-8 + 4=-4$. Then the average rate of change is $\frac{f(2)-f(0)}{2 - 0}=\frac{-4 - 4}{2}=\frac{-8}{2}=-4$.

Step3: Solve problem 65(b)

For the interval $[1,3]$, find $f(1)$ and $f(3)$. $f(1)=-2(1)^{2}+4=-2 + 4 = 2$. $f(3)=-2(3)^{2}+4=-2\times9+4=-18 + 4=-14$. The average rate of change is $\frac{f(3)-f(1)}{3 - 1}=\frac{-14 - 2}{2}=\frac{-16}{2}=-8$.

Step4: Solve problem 65(c)

For the interval $[1,4]$, find $f(1)$ and $f(4)$. $f(1)=2$ (from above), $f(4)=-2(4)^{2}+4=-2\times16+4=-32 + 4=-28$. The average rate of change is $\frac{f(4)-f(1)}{4 - 1}=\frac{-28 - 2}{3}=\frac{-30}{3}=-10$.

Answer:

65(a): -4 65(b): -8 65(c): -10