To evaluate f ( x ) = x 2 + 1 at a number, substitute the number for x in the definition of f ( x ) The given function of the values f ( a ) = a 2 + 1 , f ( a + h ) = a 2 + 2 a h + h 2 + 1 , f ( a + h ) − f ( a ) h = 2 a + h Given information: consider the function f ( x ) = x 2 + 1 Calculation: Start by assume x is the variable of a function f ( x ) . Evaluate the values of the function f ( x ) = x 2 + 1 for the given values of the variable x Plug the value x = a , in the function f ( x ) = x 2 + 1 then the value of f ( a ) . f ( a ) = a 2 + 1 Plug the value x = a + h , in the function f ( x ) = x 2 + 1 then the value of f ( a + h ) . f ( a + h ) = ( a + h ) 2 + 1 f ( a + h ) = a 2 + 2 a h + h 2 + 1 f ( a + h ) − f ( a ) h = a 2 + 2 a h + h 2 + 1 − ( a 2 + 1 ) h Since f ( a + h ) = a 2 + 2 a h + h 2 + 1 , f ( a ) = a 2 + 1 = a 2 + 2 a h + h 2 + 1 − a 2 − 1 h = 2 a h + h 2 h = h ( 2 a + h ) h = 2 a + h Thus, f ( a ) = a 2 + 1 , f ( a + h ) = a 2 + 2 a h + h 2 + 1 , f ( a + h ) − f ( a ) h = 2 a + h

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 36E
Solution

To evaluate f(x)=x2+1 at a number, substitute the number for x in the definition of f(x)

The given function of the values

  f(a)=a2+1,f(a+h)=a2+2ah+h2+1,f(a+h)f(a)h=2a+h

Given information:consider the function f(x)=x2+1

Calculation:

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

Evaluate the values of the function f(x)=x2+1 for the given values of the variable x

Plug the value x=a, in the function f(x)=x2+1 then the value of f(a).

  f(a)=a2+1

Plug the value x=a+h, in the function f(x)=x2+1 then the value of f(a+h).

  f(a+h)=(a+h)2+1f(a+h)=a2+2ah+h2+1

  f(a+h)f(a)h=a2+2ah+h2+1(a2+1)h        Since f(a+h)=a2+2ah+h2+1,f(a)=a2+1=a2+2ah+h2+1a21h=2ah+h2h=h(2a+h)h=2a+h

Thus, f(a)=a2+1,f(a+h)=a2+2ah+h2+1,f(a+h)f(a)h=2a+h

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