BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 2.7, Problem 17E

(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.

Expert Solution

Answer to Problem 17E

The value of f(0), f(12), f(1) and f(2) are 0, 1, 2 and 4 respectively.

Explanation of Solution

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.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 2.7, Problem 17E , additional homework tip  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.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 2.7, Problem 17E , additional homework tip  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.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 2.7, Problem 17E , additional homework tip  3

The calculation of f(1) is as follows.

From Figure 3, the slope of the tangent line to the curve is computed as follows.

m=1.50.61.250.8=0.90.45=2

Thus, f(1)=2.

Obtain f(x) at the point 2.

Use the online graphing calculator to zoom toward the point (2,4) as shown below in Figure 4.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 2.7, Problem 17E , additional homework tip  4

The calculation of f(2) is as follows.

From Figure 4, the slope of the tangent line is computed as follows.

m=251.52.25=30.75=334=4

Thus, f(2)=4.

(b)

To determine

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

Expert Solution

Answer to Problem 17E

The values of f(12),f(1) and f(2) are 1,2 and 4 respectively.

Explanation of Solution

Result Used:

For an odd function f, f(x)=f(x).

For an even function f, f(x)=f(x).

Calculation:

The function f(x)=x2 is an even function.

Thus, the function f(x) is an odd function.

From part (a), f(12)=1,f(1)=2 and f(2)=4.

Using the symmetry,

f(12)=f(12)=1

Thus, f(12)=1.

Using the symmetry,

f(1)=f(1)=2

Thus, f(1)=2.

Using the symmetry,

f(2)=f(2)=4

Thus, f(2)=4.

Therefore, the values of f(12),f(1) and f(2) are 1,2 and 4 respectively.

(c)

To determine

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

Expert Solution

Answer to Problem 17E

The formula of f is f(x)=2x

Explanation of Solution

From part (a) and part (b),

f(12)=1=2(12)Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 2.7, Problem 17E , additional homework tip  5

f(1)=2=2(1)

f(2)=4=2(2)

From the above calculations, it is observed that the derivative of the function is twice the input value.

Thus, f(x)=2x.

Thus, the formula of f is f(x)=2x.

(d)

To determine

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

Expert Solution

Explanation of Solution

Definition used:

The derivation of a function is given by the formula

f(x)=limh0f(x+h)f(x)h

Proof:

Consider the function f(x)=x2.

Use the definition of derivative to obtain the derivative of f(x)=x2 as follows.

f(x)=limh0f(x+h)f(x)h=limh0(x+h)2(x)2h=limh0x2+h2+2xhx2h

Simplify the terms in numerator,

f(x)=limh0h2+2xhh=limh0h(h+2x)h

Since the limit h approaches zero but is not equal to zero, cancel the common term h from both the numerator and the denominator.

f(x)=limh0(h+2x)=limh0(h)+limh0(2x)=0+2xlimh0(1)=2x

So, f(x)=2x

Thus, the guess in part (c) is correct.

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