To evaluate f ( x ) = x 3 at a number, substitute the number for x in the definition of f ( x ) The given function of the values f ( a ) = a 3 , f ( a + h ) = h 3 + 3 h 2 a + 3 h a 2 + a 3 , f ( a + h ) − f ( a ) h = h 2 + 3 h a + 3 a 2 Given information: consider the function f ( x ) = x 3 Calculation: Start by assume x is the variable of a function f ( x ) . Evaluate the values of the function f ( x ) = x 3 for the given values of the variable x Plug the value x = a , in the function f ( x ) = x 3 then the value of f ( a ) . f ( a ) = a 3 Plug the value x = a + h , in the function f ( x ) = x 3 then the value of f ( a + h ) . f ( a + h ) = h 3 + 3 h 2 a + 3 h a 2 + a 3 f ( a + h ) − f ( a ) h = ( a + h ) 3 − a 3 h Since f ( a + h ) = ( a + h ) 3 , f ( a ) = a 3 = h 3 + 3 h 2 a + 3 h a 2 + a 3 − a 3 h = h 3 + 3 h 2 a + 3 h a 2 h = h 2 + 3 h a + 3 a 2 Thus, f ( a ) = a 3 , f ( a + h ) = h 3 + 3 h 2 a + 3 h a 2 + a 3 , f ( a + h ) − f ( a ) h = h 2 + 3 h a + 3 a 2

BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071
BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

Solutions

Chapter 2.1, Problem 42E
Solution

To evaluate f(x)=x3 at a number, substitute the number for x in the definition of f(x)

The given function of the values

  f(a)=a3,f(a+h)=h3+3h2a+3ha2+a3,f(a+h)f(a)h=h2+3ha+3a2

Given information:consider the function f(x)=x3

Calculation:

Start by assume x is the variable of a function f(x) .

Evaluate the values of the function f(x)=x3 for the given values of the variable x

Plug the value x=a, in the function f(x)=x3 then the value of f(a).

  f(a)=a3

Plug the value x=a+h, in the function f(x)=x3 then the value of f(a+h).

  f(a+h)=h3+3h2a+3ha2+a3

  f(a+h)f(a)h=(a+h)3a3h         Since f(a+h)=(a+h)3,f(a)=a3=h3+3h2a+3ha2+a3a3h=h3+3h2a+3ha2h=h2+3ha+3a2

Thus, f(a)=a3,f(a+h)=h3+3h2a+3ha2+a3,f(a+h)f(a)h=h2+3ha+3a2

Have a homework question?

Subscribe to bartleby learn! Ask subject matter experts 30 homework questions each month. Plus, you’ll have access to millions of step-by-step textbook answers!