nsider the function f(x) = 4x - x² and the point P(2, 4) on the graph of f. Exercise (a) Graph f and the secant lines passing through P(2, 4) and Q(x, f(x)) for x-values of 3, 2.5, 1.5. Step 1 To sketch the graph of f(x) = 4x-x², first construct a table of values by evaluating the function f(x) for various values of x. Consider x = 3. To evaluate f(x) with x = 3, substitute 3 for x in f(x). f(x) = 4x-x² f(3) = 4(3)-(3)2 = 33. Step 2 Similarly, evaluate f(x) for the values below, and complete the table. 2.5 O 4 F(x) 00 3.5 Step 3 Plot the above points and join them with a smooth curve graph the function f(x). Also join P(2, 4) with (3, 3), (2.5, 3.75), and (1.5, 3.75) to obtain the secant lines. Exercise (b) Find the slope of each secant line. f V (x-2) x-2 x-2 3.75 3.75 3.75 3.75 Step 1 Recall that the slope of a non-vertical line is a measure of the number of units the line rises (or falls) vertically for each unit of horizontal change from left to right. Thus, the slope of a l passing through P(2, 4) and Q(x, f(x)) is given by Slope = m = f(x)-4 x-2 Submit p(you cannot come back) 1.5 -4 OD x 2. f Exercise (c) Use the results of part (b) to estimate the slope of the tangent line to the graph of fat P(2, 4). Describe how to improve your approximation of the slope. Step 1 The slope of the tangent line is the limit of the slopes of the secant lines. Thus, to estimate the slope of the tangent line to the graph of f at P(2, 4), substitute 2 for x in the equation m = -(x-2). slope -(2- |) The approximation of the slope can be improved by considering the values of x close to
nsider the function f(x) = 4x - x² and the point P(2, 4) on the graph of f. Exercise (a) Graph f and the secant lines passing through P(2, 4) and Q(x, f(x)) for x-values of 3, 2.5, 1.5. Step 1 To sketch the graph of f(x) = 4x-x², first construct a table of values by evaluating the function f(x) for various values of x. Consider x = 3. To evaluate f(x) with x = 3, substitute 3 for x in f(x). f(x) = 4x-x² f(3) = 4(3)-(3)2 = 33. Step 2 Similarly, evaluate f(x) for the values below, and complete the table. 2.5 O 4 F(x) 00 3.5 Step 3 Plot the above points and join them with a smooth curve graph the function f(x). Also join P(2, 4) with (3, 3), (2.5, 3.75), and (1.5, 3.75) to obtain the secant lines. Exercise (b) Find the slope of each secant line. f V (x-2) x-2 x-2 3.75 3.75 3.75 3.75 Step 1 Recall that the slope of a non-vertical line is a measure of the number of units the line rises (or falls) vertically for each unit of horizontal change from left to right. Thus, the slope of a l passing through P(2, 4) and Q(x, f(x)) is given by Slope = m = f(x)-4 x-2 Submit p(you cannot come back) 1.5 -4 OD x 2. f Exercise (c) Use the results of part (b) to estimate the slope of the tangent line to the graph of fat P(2, 4). Describe how to improve your approximation of the slope. Step 1 The slope of the tangent line is the limit of the slopes of the secant lines. Thus, to estimate the slope of the tangent line to the graph of f at P(2, 4), substitute 2 for x in the equation m = -(x-2). slope -(2- |) The approximation of the slope can be improved by considering the values of x close to
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.5: Graphs Of Functions
Problem 36E
Related questions
Question
Transcribed Image Text:Consider the function f(x) = 4x - x² and the point P(2, 4) on the graph of f.
Exercise (a)
Graph f and the secant lines passing through P(2, 4) and Q(x, f(x)) for x-values of 3, 2.5, 1.5.
Step 1
To sketch the graph of f(x) = 4x - x², first construct a table of values by evaluating the function f(x) for various values of x.
Consider x = 3. To evaluate f(x) with x = 3, substitute 3 for x in f(x).
f(x) = 4x - x²
f(3) = 4(3) - (3)²
Step 2
Similarly, evaluate f(x) for the values below, and complete the table.
f(x)
0
-10 8
4
-10 -8 6
10:
8-
6
2
for
y
-10-
-20 2 4 6 8 10
Step 3
Plot the above points and join them with a smooth curve to graph the function f(x). Also join P(2, 4) with (3, 3), (2.5, 3.75), and (1.5, 3.75) to obtain the secant lines.
101
8
6
-20
-2
3.5 3
1.75 3
-8-
-10
(x - 2)
Exercise (b)
Find the slope of each secant line.
3.75
X -2
x-2
6
Submit Skip (you cannot come back)
2.5
, x 2.
3.75
10
2
3.75
Describe how to improve your approximation of the slope.
1.5
3.75
-10
J
-10-8 -6
-20 2
/6
10
-10-
Step 1
Recall that the slope of a non-vertical line is a measure of the number of units the line rises (or falls) vertically for each unit of horizontal change from left to right. Thus, the slope of a line passing through P(2, 4) and Q(x, f(x)) is given by
f(x) - 4
Slope = m =
x - 2
10-
2
-4 -20
-2
2
6 8 10
Exercise (c)
Use the results of part (b) to estimate the slope of the tangent line to the graph of f at P(2, 4).
Step 1
The slope of the tangent line is the limit of the slopes of the secant lines. Thus, to estimate the slope of the tangent line to the graph of f at P(2, 4), substitute 2 for x in the equation m = -(x - 2).
slope = -(2-
The approximation of the slope can be improved by considering the values of x close to
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