Section Exercises
1. Can the average rate of change of a function be constant? 2. If a function [latex]f[/latex] is increasing on [latex]\left(a,b\right)[/latex] and decreasing on [latex]\left(b,c\right)[/latex], then what can be said about the local extremum of [latex]f[/latex] on [latex]\left(a,c\right)?[/latex] 3. How are the absolute maximum and minimum similar to and different from the local extrema? 4. How does the graph of the absolute value function compare to the graph of the quadratic function, [latex]y={x}^{2}[/latex], in terms of increasing and decreasing intervals? For exercises 5–15, find the average rate of change of each function on the interval specified for real numbers [latex]b[/latex] or [latex]h[/latex]. 5. [latex]f\left(x\right)=4{x}^{2}-7[/latex] on [latex]\left[1,\text{ }b\right][/latex] 6. [latex]g\left(x\right)=2{x}^{2}-9[/latex] on [latex]\left[4,\text{ }b\right][/latex] 7. [latex]p\left(x\right)=3x+4[/latex] on [latex]\left[2,\text{ }2+h\right][/latex] 8. [latex]k\left(x\right)=4x - 2[/latex] on [latex]\left[3,\text{ }3+h\right][/latex] 9. [latex]f\left(x\right)=2{x}^{2}+1[/latex] on [latex]\left[x,x+h\right][/latex] 10. [latex]g\left(x\right)=3{x}^{2}-2[/latex] on [latex]\left[x,x+h\right][/latex] 11. [latex]a\left(t\right)=\frac{1}{t+4}[/latex] on [latex]\left[9,9+h\right][/latex] 12. [latex]b\left(x\right)=\frac{1}{x+3}[/latex] on [latex]\left[1,1+h\right][/latex] 13. [latex]j\left(x\right)=3{x}^{3}[/latex] on [latex]\left[1,1+h\right][/latex] 14. [latex]r\left(t\right)=4{t}^{3}[/latex] on [latex]\left[2,2+h\right][/latex] 15. [latex]\frac{f\left(x+h\right)-f\left(x\right)}{h}[/latex] given [latex]f\left(x\right)=2{x}^{2}-3x[/latex] on [latex]\left[x,x+h\right][/latex] For exercises 16–17, consider the graph of [latex]f[/latex]. 16. Estimate the average rate of change from [latex]x=1[/latex] to [latex]x=4[/latex]. 17. Estimate the average rate of change from [latex]x=2[/latex] to [latex]x=5[/latex]. For exercises 18–21, use the graph of each function to estimate the intervals on which the function is increasing or decreasing. 18. 19. 20. 21. For exercises 22–23, consider the graph shown below. 22. Estimate the intervals where the function is increasing or decreasing. 23. Estimate the point(s) at which the graph of [latex]f[/latex] has a local maximum or a local minimum. For exercises 24–25, consider the graph below. 24. If the complete graph of the function is shown, estimate the intervals where the function is increasing or decreasing. 25. If the complete graph of the function is shown, estimate the absolute maximum and absolute minimum. 26. The table below gives the annual sales (in millions of dollars) of a product from 1998 to 2006. What was the average rate of change of annual sales (a) between 2001 and 2002, and (b) between 2001 and 2004?Year | Sales (millions of dollars) |
1998 | 201 |
1999 | 219 |
2000 | 233 |
2001 | 243 |
2002 | 249 |
2003 | 251 |
2004 | 249 |
2005 | 243 |
2006 | 233 |
Year | Population (thousands) |
2000 | 87 |
2001 | 84 |
2002 | 83 |
2003 | 80 |
2004 | 77 |
2005 | 76 |
2006 | 78 |
2007 | 81 |
2008 | 85 |
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