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Mathematics LibreTexts

1.3E: Exercises

  • Page ID
    17337
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    Verbal

    Verbal

    Exercise 1.3.1

    Can the average rate of change of a function be constant?

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

    Exercise 1.3.2

    If a function \(f\) is increasing on \((a,b)\) and decreasing on \((b,c)\), then what can be said about the local extremum of \(f\) on \((a,c)\)?

    Exercise 1.3.3

    How are the absolute maximum and minimum similar to and different from the local extrema?

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

    Exercise 1.3.4

    How does the graph of the absolute value function compare to the graph of the quadratic function, \(y=x^2\), in terms of increasing and decreasing intervals?

    Algebraic

    For the following exercises, find the average rate of change of each function on the interval specified for real numbers b or h.

    Exercise 1.3.5

    \(f(x)=4x^2−7\) on \([1, b]\)

    Answer:
    \(4(b+1)\)

    Exercise 1.3.6

    \(g(x)=2x^2−9\) on \([4, b]\)

    Exercise 1.3.7

    \(p(x)=3x+4\) on \([2, 2+h]\)

    Answer:
    3

    Exercise 1.3.8

    \(k(x)=4x−2\) on \([3, 3+h]\)

    Exercise 1.3.9

    \(f(x)=2x^2+1\) on \([x,x+h]\)

    Answer:
    \(4x+2h\)

    Exercise 1.3.10

    \(g(x)=3x^2−2\) on \([x,x+h]\)

    Exercise 1.3.11

    \(a(t)=\dfrac{1}{t+4}\) on \([9,9+h]\)

    Answer:
    \(\dfrac{−1}{13(13+h)}\)

    Exercise 1.3.12

    \(b(x)=1x+3\) on \([1,1+h]\)

    Exercise 1.3.13

    \(j(x)=3x^3\) on \([1,1+h]\)

    Answer:
    \(3h^2+9h+9\)

    Exercise 1.3.14

    \(r(t)=4t^3\) on \([2,2+h]\)

    Exercise 1.3.15

    \(\frac{f(x+h)−f(x)}{h}\) given \(f(x)=2x^2−3x\) on \([x,x+h]\)

    Answer:
    \(4x+2h−3\)

    Graphical

    For the following exercises, consider the graph of f shown in Figure 1.3.15.

    Graph of a polynomial.

    Figure 1.3.15: Graph of a polynomial.

    Exercise 1.3.16

    Estimate the average rate of change from \(x=1\) to \(x=4\).

    Exercise 1.3.17

    Estimate the average rate of change from \(x=2\) to \(x=5\).

    Answer:
    \(\dfrac{4}{3}\)

    Exercise 1.3.18

    Graph of an absolute function.

    Exercise 1.3.19

    Graph of a cubic function.

    Answer:
    increasing on \((−\infty,−2.5)\cup(1,\infty)\), decreasing on \((−2.5, 1)\)

    Exercise 1.3.20

    Graph of a cubic function

    Exercise 1.3.21

    Graph of a reciprocal function.

    Answer:
    increasing on \((−\infty,1)\cup(3,4)\), decreasing on \((1,3)\cup(4,\infty)\)

    For the following exercises, consider the graph shown in Figure 16.

    Figure 16: Graph of a cubic function.

    Exercise 1.3.22

    Estimate the intervals where the function is increasing or decreasing.

    Exercise 1.3.23

    Estimate the point(s) at which the graph of f has a local maximum or a local minimum.

    Answer:
    local maximum: \((−3, 60)\), local minimum: \((3, −60)\)

    For the following exercises, consider the graph in Figure 17.

    Graph of a cubic function.

    Figure 17: Graph of a cubic function.

    Exercise 1.3.24

    If the complete graph of the function is shown, estimate the intervals where the function is increasing or decreasing.

    Exercise 1.3.25

    If the complete graph of the function is shown, estimate the absolute maximum and absolute minimum.

    Answer:
    absolute maximum at approximately \((7, 150)\), absolute minimum at approximately \((−7.5, −220)\)

    Numeric

    Exercise 1.3.26

    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

    Exercise 1.3.27

    The table below gives the population of a town (in thousands) from 2000 to 2008. What was the average rate of change of population (a) between 2002 and 2004, and (b) between 2002 and 2006?

    Year Population (thousands)
    2000 87
    2001 84
    2002 83
    2003 80
    2004 77
    2005 76
    2006 78
    2007 81
    2008 85
    Answer:
    a. –3000
    b. –1250

    For the following exercises, find the average rate of change of each function on the interval specified.

    Exercise 1.3.28

    \(f(x)=x^2\) on \([1, 5]\)

    Exercise 1.3.29

    \(h(x)=5−2x^2\) on \([−2,4]\)

    Answer:
    -4

    Exercise 1.3.30

    \(q(x)=x^3\) on \([−4,2]\)

    Exercise 1.3.31

    \(g(x)=3x^3−1\) on \([−3,3]\)

    Answer:
    27

    Exercise 1.3.32

    \(y=\dfrac{1}{x}\) on \([1, 3]\)

    Exercise 1.3.33

    \(p(t)=\dfrac{(t^2-4)(t+1)}{t^2+3}\) on \([-3,1]\)

    Answer:
    –0.167

    Exercise 1.3.34

    \(k(t)=6t^2+\dfrac{4}{t^3}\) on \([−1,3]\)

    Technology

    For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing.

    Exercise 1.3.35

    \(f(x)=x^4−4x^3+5\)

    Answer:
    Local minimum at \((3,−22)\), decreasing on \((−\infty, 3)\), increasing on \((3, \infty)\)

    Exercise 1.3.36

    \(h(x)=x^5+5x^4+10x^3+10x^2−1\)

    Exercise 1.3.37

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

    Answer:
    Local minimum at \((−2,−2)\), decreasing on \((−3,−2)\), increasing on \((−2, \infty)\)

    Exercise 1.3.38

    \(k(t)=3t^{\frac{2}{3}}-t\)

    Exercise 1.3.39

    \(m(x)=x^4+2x^3−12x^2−10x+4\)

    Answer:
    Local maximum at \((−0.5, 6)\), local minima at \((−3.25,−47)\) and \((2.1,−32)\), decreasing on \((−\infty,−3.25)\) and \((−0.5, 2.1)\), increasing on \((−3.25, −0.5)\) and \((2.1,\infty)\)

    Exercise 1.3.40

    \(n(x)=x^4−8x^3+18x^2−6x+2\)

    Extension

    Exercise 1.3.41

    The graph of the function \(f\) is shown in Figure 18

    Graph of f(x) on a graphing calculator.

    Figure 18: Graph of f(x) on a graphing calculator.

    Based on the calculator screen shot, the point \((1.333, 5.185)\) is which of the following?

    A. a relative (local) maximum of the function
    B. the vertex of the function
    C. the absolute maximum of the function
    D. a zero of the function

    Answer:
    A

    Exercise 1.3.42

    Let \(f(x)=\dfrac{1}{x}\). Find a number \(c\) such that the average rate of change of the function f on the interval \((1,c)\) is \(-\dfrac{1}{4}\).

    Exercise1.3.43

    Let \(f(x)=\dfrac{1}{x}\). Find the number \(b\) such that the average rate of change of f on the interval \((2,b)\) is \(−\dfrac{1}{10}\).

    Answer:
    \(b=5\)

    Real-World Applications

    Exercise 1.3.44

    At the start of a trip, the odometer on a car read 21,395. At the end of the trip, 13.5 hours later, the odometer read 22,125. Assume the scale on the odometer is in miles. What is the average speed the car traveled during this trip?

    Exercise 1.3.45

    A driver of a car stopped at a gas station to fill up his gas tank. He looked at his watch, and the time read exactly 3:40 p.m. At this time, he started pumping gas into the tank. At exactly 3:44, the tank was full and he noticed that he had pumped 10.7 gallons. What is the average rate of flow of the gasoline into the gas tank?

    Answer:
    2.7 gallons per minute

    Exercise 1.3.46

    Near the surface of the moon, the distance that an object falls is a function of time. It is given by \(d(t)=2.6667t^2\), where \(t\) is in seconds and \(d(t)\) is in feet. If an object is dropped from a certain height, find the average velocity of the object from \(t=1\) to \(t=2\).

    Exercise 1.3.47

    The graph in Figure 19 illustrates the decay of a radioactive substance over \(t\) days.

    Graph of an exponential function.

    Figure 19: Graph of an exponential function.

    Use the graph to estimate the average decay rate from \(t=5\) to \(t=15\).

    Answer:
    approximately –0.6 milligrams per day