1.3E: Rates of Change and Behavior of Graphs

SECTION 1.3 EXERCISE

1. The table below gives the annual sales (in millions of dollars) of a product. What was the average rate of change of annual sales…
   
   a) Between 2001 and 2002?
   b) Between 2001 and 2004?

   year	1998	1999	2000	2001	2002	2003	2004	2005	2006
   sales	201	219	233	243	249	251	249	243	233

2. The table below gives the population of a town, in thousands. What was the average rate of change of population…
   
   a) Between 2002 and 2004?
   b) Between 2002 and 2006?

   year	2000	2001	2002	2003	2004	2005	2006	2007	2008
   population	87	84	83	80	77	76	75	78	81

3. Based on the graph shown, estimate the average rate of change from \(x = 1\) to \(x = 4\).
4. Based on the graph shown, estimate the average rate of change from \(x = 2\) to \(x = 5\).
   
   
   
   
   
   
   
   Find the average rate of change of each function on the interval specified.
5. \(f(x)=x^{2}\) on [1, 5]
6. \(q(x)=x^{3}\) on [-4, 2]
7. \(g(x)=3x^{3} -1\) on [-3, 3]
8. \(h(x)=5 - 2x^{2}\) on [-2, 4]
9. \(k(t)=6t^{2} +\dfrac{4}{t^{3} }\) on [-1, 3]
10. \(p(t)=\dfrac{t^{2} - 4t + 1}{t^{2} + 3}\) on [-3, 1]
    
    Find the average rate of change of each function on the interval specified. Your answers will be expressions involving a parameter (\(b\) or \(h\)).
11. \(f(x)= 4x^{2} -7\) on [1, \(b\)]
12. \(g(x)= 2x^{2} -9\) on [4, \(b\)]
13. \(h(x)= 3x + 4\) on [2, 2 + \(h\)]
14. \(k(x)= 4x - 2\) on [3, 3 + \(h\)]
15. \(a(t)=\dfrac{1}{t + 4}\) on [9, 9 + \(h\)]
16. \(b(x)=\dfrac{1}{x + 3}\) on [1, 1 + \(h\)]
17. \(j(x)=3x^{3}\) on [1, 1 + \(h\)]
18. \(r(t)=4t^{3}\) on [2, 2 + \(h\)]
19. \(f(x)=2x^{2} + 1\) on [\(x\), \(x + h\)]
20. \(g(x)=3x^{2} - 2\) on [\(x\), \(x + h\)]
    
    For each function graphed, estimate the intervals on which the function is increasing and decreasing.
21. 
22. 
23. 
24. 

For each table below, select whether the table represents a function that is increasing or decreasing, and whether the function is concave up or concave down.

For each function graphed, estimate the intervals on which the function is concave up and concave down, and the location of any inflection points.

33.

34.

35.

36.

Use a graph to estimate the local extrema and inflection points of each function, and to estimate the intervals on which the function is increasing, decreasing, concave up, and concave down.

37. \(f(x) = x^4 - 4x^3 + 5\)

38. \(h(x) = x^5 + 5x^4 + 10x^3 + 10x^2 - 1\)

39. \(g(t) = t \sqrt{t + 3}\)

40. \(k(t) = 3t^{\dfrac{2}{3}} - t\)

41. \(m(x) = x^4 + 2x^3 - 12x^2 - 10x + 4\)

42. \(n(x) = x^4 - 8x^3 + 18x^2 - 6x + 2\)

Answer

1. a) 6 million dollars per year
b) 2 million dollars per year

3. \(\dfrac{4 - 5}{4 - 1} = -\dfrac{1}{3}\)

5. 6

7. 27

9. \(\dfrac{352}{27}\)

11. \(4b + 4\)

13. 3

15. \(-\dfrac{1}{13h + 169}\)

17. \(9 + 9h + 3h^2\)

19. \(4x + 2h\)

21. Increasing: (-1.5, 2). Decreasing: \((-\infty, -1.4) \cup (2, \infty)\)

23. Increasing: \((-\infty, 1) \cup (3,4)\). Decreasing: \((1, 3) \cup (4, \infty)\)

25. Increasing, concave up

27. Decreasing, concave down

29. Decreasing, concave up

31. Increasing, concave down

33. Concave up \((-\infty, 1)\). Concave down \((1, \infty)\). Inflection point at (1, 2)

35. Concave down \(-\infty, 3) \cup (3, \infty)\)

37. Local minimum at (3, -22).
Inflection points at (0, 5) and (2, -11).
Increasing on \(3, \infty)\). Decreasing \((-\infty, 3)\)
Concave up \((-\infty, 0) \cup (2, \infty)\). Concave down (0, 2)

39. Local minimum at \((-2, -2)\)
Decreasing (-3, -2)
Increasing \((-2, \infty)\)
Concave up \((-3, \infty)\)

41. Local minimums at (-3.152, -47.626) and (2.041, -32.041)
Local maximum at (-0.389, 5.979)
Inflection points at (-2, -24) and (1, -15)
Increasing \((-3.152, -0.389) \cup (2.041, \infty)\)
Decreasing \((-\infty, -3.152) \cup (-0.389, 2.041)\)
Concave up \((-\infty, -2) \cup (1, \infty)\)
Concave down (-2, 1