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

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: $f(x + h) = -2(xx + xh + hx + hh) - (x + h)$.

Step 2.1.2.1.2.3:

Complete distribution: $f(x + h) = -2(xx + xh + hx + hh) - (x + h)$.

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: $f(x + h) = -2(x^2 + xh + hx + hh) - (x + h)$.

Step 2.1.2.1.3.1.2:

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

Step 2.1.2.1.3.2:

Combine $xh$ and $hx$.

Step 2.1.2.1.3.2.1:

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

Step 2.1.2.1.3.2.2:

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

Step 2.1.2.1.4:

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

Step 2.1.2.1.5:

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

Step 2.1.2.1.6:

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

Step 2.1.2.2:

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

Step 2.2:

Rearrange the terms.

Step 2.2.1:

Move $-x$: $-2x^2 - 4hx - 2h^2 - h - x$.

Step 2.2.2:

Move $-2x^2$: $-4hx - 2h^2 - 2x^2 - h - x$.

Step 2.2.3:

Rearrange $-4hx$ and $-2h^2$: $-2h^2 - 4hx - 2x^2 - h - x$.

Step 2.3:

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

Step 3:

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

Step 4:

Simplify the expression.

Step 4.1:

Simplify the numerator.

Step 4.1.1:

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

Step 4.1.2:

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

Step 4.1.3:

Multiply $-(-x)$.

Step 4.1.3.1:

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

Step 4.1.3.2:

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

Step 4.1.4:

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

Step 4.1.5:

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

Step 4.1.6:

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

Step 4.1.7:

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

Step 4.1.8:

Factor out $h$ from $-2h^2 - 4hx - h$.

Step 4.1.8.1:

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

Step 4.1.8.2:

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

Step 4.1.8.3:

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

Step 4.1.8.4:

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

Step 4.1.8.5:

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

Step 4.2:

Cancel out the common $h$ factors.

Step 4.2.1:

Eliminate the common $h$ factor.

Step 4.2.1.1:

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

Step 4.2.1.2:

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

Step 4.2.2:

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

Step 5:

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

Knowledge Notes:

The problem involves finding the difference quotient for the function $f(x) = -2x^2 - x$. 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:

1. Function evaluation: Substituting a specific value into a function.

2. Algebraic simplification: Combining like terms and simplifying expressions.

3. The FOIL method: A technique for multiplying two binomials, which stands for First, Outer, Inner, Last.

4. The distributive property: A property that allows one to distribute a multiplication over addition or subtraction within parentheses.

5. Factoring: The process of breaking down an expression into its component factors.

6. 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.