Evaluate the Difference Quotient f(x)=x^2-x+1 , (f(2+h)-f(2))/h , h!=0

$f (x) = x^{2} - x + 1$ , $\frac{f (2 + h) - f (2)}{h}$ , $h \neq 0$

Step 1

Consider the difference quotient formula.

$\frac{f (x + h) - f (x)}{h}$

Step 2

Find the components of the definition.

Evaluate the function at $x = x + h$ .

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = {(x + h)}^{2} - (x + h) + 1$

Simplify the result.

Simplify each term.

Rewrite ${(x + h)}^{2}$ as $(x + h) (x + h)$ .

$f (x + h) = (x + h) (x + h) - (x + h) + 1$

Expand $(x + h) (x + h)$ using the FOIL Method.

Apply the distributive property.

$f (x + h) = x (x + h) + h (x + h) - (x + h) + 1$

Apply the distributive property.

$f (x + h) = x \cdot x + x h + h (x + h) - (x + h) + 1$

Apply the distributive property.

$f (x + h) = x \cdot x + x h + h x + h \cdot h - (x + h) + 1$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x + h) = x^{2} + x h + h x + h \cdot h - (x + h) + 1$

Multiply $h$ by $h$ .

$f (x + h) = x^{2} + x h + h x + h^{2} - (x + h) + 1$

Add $x h$ and $h x$ .

Reorder $x$ and $h$ .

$f (x + h) = x^{2} + h x + h x + h^{2} - (x + h) + 1$

Add $h x$ and $h x$ .

$f (x + h) = x^{2} + 2 h x + h^{2} - (x + h) + 1$

Apply the distributive property.

$f (x + h) = x^{2} + 2 h x + h^{2} - x - h + 1$

The final answer is $x^{2} + 2 h x + h^{2} - x - h + 1$ .

$x^{2} + 2 h x + h^{2} - x - h + 1$

Reorder.

Move $- x$ .

$x^{2} + 2 h x + h^{2} - h - x + 1$

Move $x^{2}$ .

$2 h x + h^{2} + x^{2} - h - x + 1$

Reorder $2 h x$ and $h^{2}$ .

$h^{2} + 2 h x + x^{2} - h - x + 1$

Find the components of the definition.

$f (x + h) = h^{2} + 2 h x + x^{2} - h - x + 1$

$f (x) = x^{2} - x + 1$

$f (x + h) = h^{2} + 2 h x + x^{2} - h - x + 1$

$f (x) = x^{2} - x + 1$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{h^{2} + 2 h x + x^{2} - h - x + 1 - (x^{2} - x + 1)}{h}$

Step 4

Simplify.

Simplify the numerator.

Apply the distributive property.

$\frac{h^{2} + 2 h x + x^{2} - h - x + 1 - x^{2} - - x - 1 \cdot 1}{h}$

Simplify.

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

$\frac{h^{2} + 2 h x + x^{2} - h - x + 1 - x^{2} + 1 x - 1 \cdot 1}{h}$

Multiply $x$ by $1$ .

$\frac{h^{2} + 2 h x + x^{2} - h - x + 1 - x^{2} + x - 1 \cdot 1}{h}$

Multiply $- 1$ by $1$ .

$\frac{h^{2} + 2 h x + x^{2} - h - x + 1 - x^{2} + x - 1}{h}$

Subtract $x^{2}$ from $x^{2}$ .

$\frac{h^{2} + 2 h x - h - x + 1 + 0 + x - 1}{h}$

Add $h^{2}$ and $0$ .

$\frac{h^{2} + 2 h x - h - x + 1 + x - 1}{h}$

Add $- x$ and $x$ .

$\frac{h^{2} + 2 h x - h + 0 + 1 - 1}{h}$

Add $h^{2}$ and $0$ .

$\frac{h^{2} + 2 h x - h + 1 - 1}{h}$

Subtract $1$ from $1$ .

$\frac{h^{2} + 2 h x - h + 0}{h}$

Add $h^{2} + 2 h x - h$ and $0$ .

$\frac{h^{2} + 2 h x - h}{h}$

Factor $h$ out of $h^{2} + 2 h x - h$ .

Factor $h$ out of $h^{2}$ .

$\frac{h \cdot h + 2 h x - h}{h}$

Factor $h$ out of $2 h x$ .

$\frac{h (h) + h (2 x) - h}{h}$

Factor $h$ out of $- h$ .

$\frac{h (h) + h (2 x) + h \cdot - 1}{h}$

Factor $h$ out of $h (h) + h (2 x)$ .

$\frac{h (h + 2 x) + h \cdot - 1}{h}$

Factor $h$ out of $h (h + 2 x) + h \cdot - 1$ .

$\frac{h (h + 2 x - 1)}{h}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $h$ .

Cancel the common factor.

$\frac{h (h + 2 x - 1)}{h}$

Divide $h + 2 x - 1$ by $1$ .

$h + 2 x - 1$

Reorder $h$ and $2 x$ .

$2 x + h - 1$

Step 5

Replace the variable $x$ with $2$ in the expression.

$2 (2) + h - 1$

Step 6

Multiply $2$ by $2$ .

$4 + h - 1$

Step 7

Subtract $1$ from $4$ .

$h + 3$

Step 8

Evaluate the Difference Quotient f(x)=x^2-x+1 , (f(2+h)-f(2))/h , h!=0

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