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.

Answer

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

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=0.3536−0.1640.6036−0.414     ≈1

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.

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=1.5−0.61.25−0.8     =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.

The calculation of f′(2) is as follows.

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

m=2−51.5−2.25     =−3−0.75   =−3−34   =4

Thus, f′(2)=4.

To determine

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

Answer

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

Explanation

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.

To determine

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

Answer

The formula of f′ is f′(x)=2x

Explanation

From part (a) and part (b),

f′(−12)=−1=2(−12)

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.

To determine

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

Explanation

Definition used:

The derivation of a function is given by the formula

f′(x)=limh→0f(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)=limh→0f(x+h)−f(x)h=limh→0(x+h)2−(x)2h=limh→0x2+h2+2xh−x2h

Simplify the terms in numerator,

f′(x)=limh→0h2+2xhh=limh→0h(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)=limh→0(h+2x)=limh→0(h)+limh→0(2x)=0+2xlimh→0(1)=2x

So, f′(x)=2x

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