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Finding a Derivative In Exercises 13–32, find
the derivative of the function.
y = 5(2 − x3)4
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- Section 2.4: Chain Rule In Exercises 9–34, find the derivative of the function.Finding a Derivative In Exercises 13–32, findthe derivative of the function. y=\frac{1}{x-2}Finding a Derivative In Exercises 7–26, usethe rules of differentiation to find the derivative ofthe function. \text { 16. } g(x)=6 x+3
- Piecewise-Defined FunctionsGraph the functions in Exercises 25–28.Using the definition, calculate the derivatives of the functions in Exercises 1–6. Then find the values of the derivatives as specified. 1. ƒ(x) = 4 - x2; ƒ′(-3), ƒ′(0), ƒ′(1) 2. F(x) = (x - 1)2 + 1; F′(-1), F′(0), F′(2) 3. g(t) = 1 /t2 ; g′(-1), g′(2), g′(sqrt(3)) 4. k(z) = (1 - z )/2z ; k′(-1), k′(1), k′(sqrt(2)) 5. p(u) = sqrt(3u) ; p′(1), p′(3), p′(2/3) 6. r (s) = sqrt(2s + 1) ; r′(0), r′(1), r′(1/2)Graphing Inverse Functions Each of Exercises 11–16 shows the graph of a function y = ƒ(x).Copy the graph and draw in the line y = x. Then use symmetry withrespect to the line y = x to add the graph of ƒ -1 to your sketch. (It isnot necessary to find a formula for ƒ -1.) Identify the domain andrange of ƒ -1.
- Inventing Graphs and FunctionsIn Exercises 75–78, sketch the graph of a function y = ƒ(x) that satisfies the given conditions. No formulas are required—just label the coordinate axes and sketch an appropriate graph. (The answers are not unique, so your graphs may not be exactly like those in the answer section.)The figure shows the graphs of the cost and revenue functions for a company that manufactures and sells small radios. Use the information in the figure to solve Exercises 67–72. 35,000 30,000 C(x) = 10,000 + 30x 25,000 20,000 15,000 R(x) = 50x 10,000 5000 100 200 300 400 500 600 700 Radios Produced and Sold 67. How many radios must be produced and sold for the company to break even? 68. More than how many radios must be produced and sold for the company to have a profit? 69. Use the formulas shown in the voice balloons to find R(200) – C(200). Describe what this means for the company. 70. Use the formulas shown in the voice balloons to find R(300) – C(300). Describe what this means for the company. 71. a. Use the formulas shown in the voice balloons to write the company's profit function, P, from producing and selling x radios. b. Find the company's profit if 10,000 radios are produced and sold. 72. a. Use the formulas shown in the voice balloons to write the company's profit function,…In Exercises 11–18, graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. 11. f(x) = 4" 13. g(x) = ()* 15. h(x) = (})* 17. f(x) = (0.6) 12. f(x) = 5" 14. g(x) = () 16. h(x) = (})* 18. f(x) = (0.8)* %3!
- Each of Exercises 81–84 shows the graphs of the first and second derivatives of a function y = f(x). Copy the picture and add to it a sketch of the approximate graph of f, given that the graph passes through the point P.In Exercises 65–68, find and sketch the domain of ƒ. Then find an equation for the level curve or surface of the function passing through the given point.In Exercises 63–65, find the domain and range of each composite function. Then graph the composition of the two functions on separate screens. Do the graphs make sense in each case? Give reasons for your answers. Comment on any differences you see. 63. a. y = tan-1 (tan x) b. y = tan (tan-1 x) 64. a. y = sin-1 (sin x) b. y = sin (sin-1 x) 65. a. y = cos-1 (cos x) b. y = cos (cos-1 x)