The graphing calculator and the computer are indispensable tools in helping us to solve complex mathematical problems. In this book we will use them to help us explore ideas and concepts in calculus both graphically and numerically. But the amount and accu- racy of the information obtained by using a graphing utility depend on the experience and sophistication of the user. As you progress through this text, you will see that the more knowledge of calculus you gain, the more effective the graphing utility will prove to be as a tool for problem solving. But there are pitfalls in using the graphing utility, and we will point them out when the opportunity arises.
In this section we will look at some basic capabilities of the graphing calculator and the computer that we will use later.
Finding a Suitable Viewing Window
The first step in plotting the graph of a function with a graphing utility is to select a suitable viewing window that displays the portion of the graph of the function in the rectangular set {(x, y)a x b, c y d}. For example, you might
[a, b][c, d]
EXAMPLE 1 Plot the graph of in the standard viewing window .
Solution The graph of , shown in Figure 1a, is a parabola. Figure 1b shows a typi- cal window screen, and Figure 1c shows a typical equation screen.
f [10, 10][10, 10]
f(x)2x24x5
FIGURE 1
EXAMPLE 2 Let .
a. Plot the graph of in the standard viewing window.
b. Plot the graph of in the window . Solution
a. The graph of in the standard viewing window is shown in Figure 2. Since the graph does not appear to be complete, we need to adjust the viewing window.
b. The graph of in the window , shown in Figure 3, is an improvement over the previous graph. (Later, we will be able to show that the figure does in fact give a rather complete view of the graph of .)
Evaluating a Function
A graphing utility can be used to find the value of a function with minimal effort, as the following example shows.
f [1, 5][40, 40]
f f
[1, 5][40, 40]
f f
f(x)x3(x3)4
first plot the graph using the standard viewing window . If nec- essary, you then might adjust the viewing window by enlarging it, reducing it, or even changing it altogether to obtain a sufficiently complete view of the graph or at least the portion of the graph that is of interest.
[10, 10][10, 10]
10 10
WINDOW Xmin10
Plot1 Plot2 Plot3 Xmax 10
Xsc1 1 Ymin10 Ymax 10 Ysc1 1 Xres 1 10
10
\Y12X^24X5
\Y2
\Y3
\Y4
\Y5
\Y6
\Y7
(a) The graph of f(x) 2x2 4x 5 in [10, 10] [10, 10]
(b) A window screen on a graphing calculator
(c) An equation screen on a graphing calculator
FIGURE 2
An incomplete sketch of on [10, 10][10, 10]
f(x)x3(x3)4
10 10 10
10
FIGURE 3
A more complete sketch of
is shown by using the window [1, 5][40, 40]. f(x)x3(x3)4
1 5 40
40
EXAMPLE 3 Let .
a. Plot the graph of in the standard viewing window.
b. Find using a calculator, and verify your result by direct computation.
c. Find .f(4.215) f(3)
f
f(x)x34x24x2
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EXAMPLE 4 Number of Alzheimer’s Patients The number of patients with Alzhei- mer’s disease in the United States is approximated by
where is measured in millions and is measured in decades, with correspond- ing to the beginning of 1990.
a. Use a graphing utility to plot the graph of in the viewing window .
b. What is the anticipated number of Alzheimer’s patients in the United States at the beginning of 2010 ? At the beginning of 2030 ?
Solution
a. The graph of is shown in Figure 5.
b. Using the evaluation function of the graphing utility and the value 2 for , we see that the anticipated number of Alzheimer’s patients at the beginning of 2010 is given by , or approximately 5 million. The anticipated number of Alzheimer’s patients at the beginning of 2030 is given by , or approximately 7.1 million.
Finding the Zeros of a Function
There will be many occasions when we need to find the zeros of a function. This task is greatly simplified if we use a graphing calculator or a computer algebra system (CAS).
f(4)7.1101 f(2)5.0187
x f
(t4) (t2)
[0, 6][0, 12]
f
t0 t
f(t)
0 t 6
f(t) 0.0277t40.3346t31.1261t21.7575t3.7745
FIGURE 5
The graph of in the viewing window [0, 6][0, 12]
f
EXAMPLE 5 Let . Find the zero of using (a) a graphing calcu- lator and (b) a CAS.
Solution
a. The graph of in the window is shown in Figure 6. Using
TRACEand ZOOMor the function for finding the zero of a function, we find the zero to be approximately .
b. In Maple we use the command
; and in Mathematica we use the command
to obtain the solution x0.682328.
Solve[x^3x10,x]
solve(x^3x10,x) 0.6823278
[2, 2][5, 5]
f
f f(x)x3x1
FIGURE 6
The graph of intersects the -axis at .
x0.6823278
x f
FIGURE 4 The graph of
in the standard viewing window f(x)x34x24x2
Solution
a. The graph of is shown in Figure 4.
b. Using the evaluation function of the graphing utility and the value 3 for , we find . This result is verified by computing
c. Using the evaluation function of the graphing utility and the value 4.215 for , we
find . Thus, . The efficacy of the
graphing utility is clearly demonstrated here!
f(4.215)22.679738375 y22.679738375
x f(3)334(3)24(3)227361225 y5
x f
10 10 10
10
0 6
12
2 2 5
5
Finding the Point(s) of Intersection of Two Graphs
A graphing calculator or a CAS can be used to find the point(s) of intersection of the graphs of two functions. Although the points of intersection of the graphs of the func- tions and can be found by finding the zeros of the function , it is often more illuminating to proceed as in Example 6.
ft f t
EXAMPLE 6 Find the points of intersection of the graphs of and .
Solution The graphs of both and in the standard viewing window are shown in Fig- ure 7a. Using TRACEand ZOOMor the function for finding the points of intersection of two graphs on your graphing utility, we find the point(s) of intersection, accurate to four decimal places, to be (2.4158, 2.1329)(Figure 7b) and (5.5587, 1.5125)(Figure 7c).
f t t(x) 0.4x20.8x6.4
f(x)0.3x21.4x3
FIGURE 7
Constructing Functions from a Set of Data
A graphing calculator or a CAS can often be used to find the function that fits a given set of data points “best” in some sense. For example, if the points corresponding to the given data are scattered about a straight line, then we use linear regression to obtain a function that approximates the data at hand. If the points seem to be scattered about a parabola (the graph of a quadratic function), then we use second-degree polynomial regression, and so on.
We will exploit these capabilities of graphing calculators and computer algebra sys- tems in Section 0.6, where we will see how “mathematical models” are constructed from raw data. The solution to the following example is obtained by using linear regres- sion. (Consult the manual that accompanies your calculator for instructions for using linear regression. If you are using a CAS, consult your HELP menu for instructions.)
10 10 10
10
10 10 10
10
5 14 6
15 (a) The graphs of f and g in the
standard viewing window
(b) An intersection screen Intersection
X= -2.415796 Y= 2.1329353
Intersection
X= 5.5586531 Y= -1.512527
(c) An intersection screen
EXAMPLE 7
a. Use a graphing calculator or computer algebra system to find a linear function whose graph fits the following data “best” in the sense of least squares:
x 1 2 3 4 5
y 3 5 5 7 8
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b. Plot the data points for the values of and given in the table (the graph is called a scatter diagram) and the graph of the least-squares line (called the regression line) on the same set of axes.
Solution
a. We first enter the data and then use the linear regression function on the calcula- tor or computer to obtain the graph shown in Figure 8. We also find that the equation of the least-squares regression line is .
b. See Figure 8.
y1.2x2 y
x (x, y)
FIGURE 8
The scatter diagram and least-squares line for the data set.
In Exercises 1–4,plot the graph of the function in (a)the standard viewing window and (b)the indicated window.
1. ;
2. ;
3. ;
4. ;
In Exercises 5–16,plot the graph of the function in an appro- priate viewing window. (Note: The answer is not unique.)
5.
6. 7.
8. 9.
10. 11.
Hint:Stay close to the origin.
12. 13.
14. 15.
16.
In Exercises 17–22,find the zero(s) of the function to five decimal places.
17. 18.
19. f(x)x42x33x1 20. f(x)2x44x21 f(x)x39x4 f(x)2x33x2
f f(x)x20.1x
f(x)x0.01 sin 50x f(x)1
2sin 2xcosx
f(x)sin1x 1x
f(x) 1
2cosx
f(x)x2sin1 f(x) 5x x
x15x
f(x)13x 13x1 f(x)2x43x
x21
f(x) x3 x31 f(x) 2x45x24
f(x)2x43x35x220x40 f [5, 5][5, 5]
f(x) 4
x28
[3, 3][2, 2]
f(x)x24x2
[2, 2][6, 10]
f(x)x42x28
[20, 20][1200, 100]
f(x)x320x28x10
f