Find the Difference Quotient f(x)=-2x^2-x

The problem asks you to calculate the difference quotient for the quadratic function f(x) = -2x^2 - x. The difference quotient is a way to express the slope of the secant line between two points on a graph of the function, which is a method used in calculus to approach the concept of derivatives. The difference quotient is typically represented by the formula:

(f(x + h) - f(x)) / h

where h is a small increment in x. For this specific problem, you are expected to substitute the given function into the difference quotient formula and simplify as much as possible.

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

Answer

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Solution:

Step 1:

Utilize the formula for the difference quotient: $\frac{f (x + h) - f (x)}{h}$ .

Step 2:

Determine the function values required by the formula.

Step 2.1:

Calculate $f (x + h)$ .

Step 2.1.1:

Substitute $x$ with $(x + h)$ in the function: $f (x + h) = - 2 (x + h)^{2} - (x + h)$ .

Step 2.1.2:

Expand and simplify the expression.

Step 2.1.2.1:

Expand each term individually.

Step 2.1.2.1.1:

Express $(x + h)^{2}$ as $(x + h) (x + h)$ : $f (x + h) = - 2 ((x + h) (x + h)) - (x + h)$ .

Step 2.1.2.1.2:

Use the FOIL method to expand $(x + h) (x + h)$ .

Step 2.1.2.1.2.1:

Distribute the terms: $f (x + h) = - 2 (x (x + h) + h (x + h)) - (x + h)$ .

Step 2.1.2.1.2.2:

Continue distribution: $f (x + h) = - 2 (x x + x h + h x + h h) - (x + h)$ .

Step 2.1.2.1.2.3:

Complete distribution: $f (x + h) = - 2 (x x + x h + h x + h h) - (x + h)$ .

Step 2.1.2.1.3:

Combine like terms and simplify.

Step 2.1.2.1.3.1:

Simplify each term.

Step 2.1.2.1.3.1.1:

Multiply $x$ by itself: $f (x + h) = - 2 (x^{2} + x h + h x + h h) - (x + h)$ .

Step 2.1.2.1.3.1.2:

Multiply $h$ by itself: $f (x + h) = - 2 (x^{2} + x h + h x + h^{2}) - (x + h)$ .

Step 2.1.2.1.3.2:

Combine $x h$ and $h x$ .

Step 2.1.2.1.3.2.1:

Rearrange terms: $f (x + h) = - 2 (x^{2} + h x + h x + h^{2}) - (x + h)$ .

Step 2.1.2.1.3.2.2:

Combine $h x$ and $h x$ : $f (x + h) = - 2 (x^{2} + 2 h x + h^{2}) - (x + h)$ .

Step 2.1.2.1.4:

Distribute the negative sign: $f (x + h) = - 2 x^{2} - 2 (2 h x) - 2 h^{2} - (x + h)$ .

Step 2.1.2.1.5:

Multiply $2$ by $- 2$ : $f (x + h) = - 2 x^{2} - 4 (h x) - 2 h^{2} - (x + h)$ .

Step 2.1.2.1.6:

Finish distribution: $f (x + h) = - 2 x^{2} - 4 h x - 2 h^{2} - x - h$ .

Step 2.1.2.2:

The simplified result is $- 2 x^{2} - 4 h x - 2 h^{2} - x - h$ .

Step 2.2:

Rearrange the terms.

Step 2.2.1:

Move $- x$ : $- 2 x^{2} - 4 h x - 2 h^{2} - h - x$ .

Step 2.2.2:

Move $- 2 x^{2}$ : $- 4 h x - 2 h^{2} - 2 x^{2} - h - x$ .

Step 2.2.3:

Rearrange $- 4 h x$ and $- 2 h^{2}$ : $- 2 h^{2} - 4 h x - 2 x^{2} - h - x$ .

Step 2.3:

Calculate $f (x)$ and combine with $f (x + h)$ .

Step 3:

Insert the function values into the difference quotient: $\frac{f (x + h) - f (x)}{h} = \frac{- 2 h^{2} - 4 h x - 2 x^{2} - h - x - (- 2 x^{2} - x)}{h}$ .

Step 4:

Simplify the expression.

Step 4.1:

Simplify the numerator.

Step 4.1.1:

Apply the distributive property: $\frac{- 2 h^{2} - 4 h x - 2 x^{2} - h - x + 2 x^{2} - (- x)}{h}$ .

Step 4.1.2:

Multiply $- 2$ by $- 1$ : $\frac{- 2 h^{2} - 4 h x - 2 x^{2} - h - x + 2 x^{2} + x}{h}$ .

Step 4.1.3:

Multiply $- (- x)$ .

Step 4.1.3.1:

Multiply $- 1$ by $- 1$ : $\frac{- 2 h^{2} - 4 h x - 2 x^{2} - h - x + 2 x^{2} + x}{h}$ .

Step 4.1.3.2:

Multiply $x$ by $1$ : $\frac{- 2 h^{2} - 4 h x - 2 x^{2} - h - x + 2 x^{2} + x}{h}$ .

Step 4.1.4:

Add $- 2 x^{2}$ and $2 x^{2}$ : $\frac{- 2 h^{2} - 4 h x - h - x + 0 + x}{h}$ .

Step 4.1.5:

Combine $- 2 h^{2}$ and $0$ : $\frac{- 2 h^{2} - 4 h x - h - x + x}{h}$ .

Step 4.1.6:

Combine $- x$ and $x$ : $\frac{- 2 h^{2} - 4 h x - h + 0}{h}$ .

Step 4.1.7:

Combine $- 2 h^{2} - 4 h x - h$ and $0$ : $\frac{- 2 h^{2} - 4 h x - h}{h}$ .

Step 4.1.8:

Factor out $h$ from $- 2 h^{2} - 4 h x - h$ .

Step 4.1.8.1:

Factor $h$ from $- 2 h^{2}$ : $\frac{h (- 2 h) - 4 h x - h}{h}$ .

Step 4.1.8.2:

Factor $h$ from $- 4 h x$ : $\frac{h (- 2 h) + h (- 4 x) - h}{h}$ .

Step 4.1.8.3:

Factor $h$ from $- h$ : $\frac{h (- 2 h) + h (- 4 x) + h (- 1)}{h}$ .

Step 4.1.8.4:

Factor $h$ from $h (- 2 h) + h (- 4 x)$ : $\frac{h (- 2 h - 4 x) + h (- 1)}{h}$ .

Step 4.1.8.5:

Factor $h$ from $h (- 2 h - 4 x) + h (- 1)$ : $\frac{h (- 2 h - 4 x - 1)}{h}$ .

Step 4.2:

Cancel out the common $h$ factors.

Step 4.2.1:

Eliminate the common $h$ factor.

Step 4.2.1.1:

Cancel $h$ : $\frac{h (- 2 h - 4 x - 1)}{h}$ .

Step 4.2.1.2:

Divide $- 2 h - 4 x - 1$ by $1$ : $- 2 h - 4 x - 1$ .

Step 4.2.2:

Rearrange $- 2 h$ and $- 4 x$ : $- 4 x - 2 h - 1$ .

Step 5:

The difference quotient is $- 4 x - 2 h - 1$ .

Knowledge Notes:

The problem involves finding the difference quotient for the function $f (x) = - 2 x^{2} - x$ . The difference quotient is a way to measure the average rate of change of a function over an interval and is defined as $\frac{f (x + h) - f (x)}{h}$ , where $h$ is the difference between two inputs of the function.

To solve this problem, one must understand several concepts:

Function evaluation: Substituting a specific value into a function.
Algebraic simplification: Combining like terms and simplifying expressions.
The FOIL method: A technique for multiplying two binomials, which stands for First, Outer, Inner, Last.
The distributive property: A property that allows one to distribute a multiplication over addition or subtraction within parentheses.
Factoring: The process of breaking down an expression into its component factors.
Cancelling common factors: Reducing fractions by eliminating common factors in the numerator and denominator.

The solution involves applying these concepts to evaluate $f (x + h)$ , simplify the resulting expression, and then calculate the difference quotient by subtracting $f (x)$ and dividing by $h$ . The final step is to simplify the expression by factoring and cancelling common factors, yielding the simplified difference quotient.