Solutions 3: Rates of Change and Behavior of Graphs

Solutions to Try Its

1. [latex]\frac{$2.84-$2.31}{5\text{ years}}=\frac{$0.53}{5\text{ years}}=$0.106[/latex] per year.

2. [latex]\frac{1}{2}[/latex]

3. [latex]a+7[/latex]

4. The local maximum appears to occur at [latex]\left(-1,28\right)[/latex], and the local minimum occurs at [latex]\left(5,-80\right)[/latex]. The function is increasing on [latex]\left(-\infty ,-1\right)\cup \left(5,\infty \right)[/latex] and decreasing on [latex]\left(-1,5\right)[/latex].

Graph of a polynomial with a local maximum at (-1, 28) and local minimum at (5, -80).

Solution to Odd-Numbered Exercises

1. Yes, the average rate of change of all linear functions is constant.

3. The absolute maximum and minimum relate to the entire graph, whereas the local extrema relate only to a specific region around an open interval.

5. [latex]4\left(b+1\right)[/latex]

7. 3

9. [latex]4x+2h[/latex]

11. [latex]\frac{-1}{13\left(13+h\right)}[/latex]

13. [latex]3{h}^{2}+9h+9[/latex]

15. [latex]4x+2h - 3[/latex]

17. [latex]\frac{4}{3}[/latex]

19. increasing on [latex]\left(-\infty ,-2.5\right)\cup \left(1,\infty \right)[/latex], decreasing on [latex]\left(-2.5,\text{ }1\right)[/latex]

21. increasing on [latex]\left(-\infty ,1\right)\cup \left(3,4\right)[/latex], decreasing on [latex]\left(1,3\right)\cup \left(4,\infty \right)[/latex]

23. local maximum: [latex]\left(-3,\text{ }60\right)[/latex], local minimum: [latex]\left(3,\text{ }-60\right)[/latex]

25. absolute maximum at approximately [latex]\left(7,\text{ }150\right)[/latex], absolute minimum at approximately [latex]\left(-7.5,\text{ }-220\right)[/latex]

27. a. –3000; b. –1250

29. –4

31. 27

33. –0.167

35. Local minimum at [latex]\left(3,-22\right)[/latex], decreasing on [latex]\left(-\infty ,\text{ }3\right)[/latex], increasing on [latex]\left(3,\text{ }\infty \right)[/latex]

37. Local minimum at [latex]\left(-2,-2\right)[/latex], decreasing on [latex]\left(-3,-2\right)[/latex], increasing on [latex]\left(-2,\text{ }\infty \right)[/latex]

39. Local maximum at [latex]\left(-0.5,\text{ }6\right)[/latex], local minima at [latex]\left(-3.25,-47\right)[/latex] and [latex]\left(2.1,-32\right)[/latex], decreasing on [latex]\left(-\infty ,-3.25\right)[/latex] and [latex]\left(-0.5,\text{ }2.1\right)[/latex], increasing on [latex]\left(-3.25,\text{ }-0.5\right)[/latex] and [latex]\left(2.1,\text{ }\infty \right)[/latex]

41. A) a relative (local) maximum of the function

43. [latex]b=5[/latex]

45. 2.7 gallons per minute

47. approximately –0.6 milligrams per day