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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Let .f (x) = x2.

(a) Estimate the values of f'(0), f ' ( 1 2 ) , f'(l) , and f'(2) by using a graphing device to zoom in on the graph of f.

(b) Use symmetry to deduce the values of f ' ( 1 2 ) , f'(– 1 ), and f'(–2).

(c) Use the results from pans (a) and (b) to guess a formula for f'(x).

(d) Use the definition of derivative to prove that your guess in part (c) is correct.

(a)

To determine

To estimate: The value of f(0), f(12), f(1) and f(2) by using a graphing device to zoom in on the graph of f.

Explanation

Given:

The function is f(x)=x2.

Estimation:

Obtain the value of f(x) at the point 0 as follows.

Use the online graphing calculator to zoom toward the point (0,0) as shown below in Figure 1.

The calculation of f(0) is as follows,

From Figure 1, the tangent to the curve at x=0 is the x-axis.

Thus, f(0)=0.

Obtain the value of f(x) at the point 12.

Use the online graphing calculator to zoom toward the point (12,14) as shown below in Figure 2.

The calculation of f(12) is as follows.

From Figure 2, the slope of the tangent line to the curve is computed as follows.

m=0.35360.1640.60360.4141

Thus, f(12)=1.

Obtain the value of f(x) at the point 1.

Use the online graphing calculator to zoom toward the point (1,1) as shown below in Figure 3

(b)

To determine

To deduce: The values of f(12),f(1) and f(2) using symmetry.

(c)

To determine

To guess: The formula for f by using part (a) and part (b).

(d)

To determine

To prove: The answer in part (c) is correct by using the definition of derivative.

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