   Chapter 2.8, Problem 57E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Let f ( x ) = x 3 . (a) If a ≠ 0, use Equation 2 .7 .5 to find f'(a).(b) Show that f'(0) does not exist.(c) Show that f ( x ) = x 3 . has a vertical tangent line at (0, 0).(Recall the shape of the graph of f. See Figure 1.2.13.)

(a)

To determine

To find: The derivative of the function f(x) at x=a if a0.

Explanation

Given:

The given function is, f(x)=x3.

Result Used:

The derivative of a function f at x=a, denoted by f(x), is

f(a)=limxaf(x)f(a)xa (1)

Difference of cube formula: x3y3=(xa)(x2+y2+xy).

Calculation:

Obtain the derivative of the function f(x) at x=a.

Compute f(a) by using the equation (1),

f(a)=limxaf(x)f(a)xa=limxax3a3xa=limxa(x13a13)xa=limxa(x13a13)xa×(x23+a13x13+a23)(x23+a13x13+a23)

Apply the difference of cube formula in the numerator as follows,

f(a)=limxa(x13)3(a13

(b)

To determine

To Show: The function f(0) does not exist.

(c)

To determine

To show: The y=x3 has a vertical tangent line at (0,0)

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