Find the Difference Quotient f(x)=2x-3x^2

Solution:

Step:1

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

Step:2

Determine the function's values.

Step:2.1

Calculate $f(x+h)$.

Step:2.1.1

Substitute $x$ with $(x+h)$ in $f(x)$. $f(x+h)=2(x+h)-3(x+h{)}^2$.

Step:2.1.2

Expand and simplify the expression.

Step:2.1.2.1

Distribute and expand terms individually.

Step:2.1.2.1.1

Use distribution: $f(x+h)=2x+2h-3(x+h{)}^2$.

Step:2.1.2.1.2

Express $(x+h{)}^2$ as $(x+h)(x+h)$: $f(x+h)=2x+2h-3((x+h)(x+h))$.

Step:2.1.2.1.3

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

Step:2.1.2.1.3.1

Distribute: $f(x+h)=2x+2h-3(x(x+h)+h(x+h))$.

Step:2.1.2.1.3.2

Distribute again: $f(x+h)=2x+2h-3(xx+xh+hx+hh)$.

Step:2.1.2.1.3.3

Continue distribution: $f(x+h)=2x+2h-3(xx+xh+hx+hh)$.

Step:2.1.2.1.4

Combine like terms and simplify.

Step:2.1.2.1.4.1

Simplify each term: $f(x+h)=2x+2h-3(x^2+xh+hx+h^2)$.

Step:2.1.2.1.4.2

Combine $xh$ and $hx$: $f(x+h)=2x+2h-3(x^2+2hx+h^2)$.

Step:2.1.2.1.5

Apply distribution: $f(x+h)=2x+2h-3x^2-6hx-3h^2$.

Step:2.1.2.1.6

Simplify the expression: $f(x+h)=2x+2h-3x^2-6hx-3h^2$.

Step:2.1.2.2

The simplified form is $2x+2h-3x^2-6hx-3h^2$.

Step:2.2

Rearrange the terms.

Step:2.2.1

Move $2x$ to the end: $2h-3x^2-6hx-3h^2+2x$.

Step:2.2.2

Move $2h$ to the end: $-3x^2-6hx-3h^2+2h+2x$.

Step:2.2.3

Move $-3x^2$ to the end: $-6hx-3h^2-3x^2+2h+2x$.

Step:2.2.4

Reorder $-6hx$ and $-3h^2$: $-3h^2-6hx-3x^2+2h+2x$.

Step:2.3

Determine $f(x+h)$ and $f(x)$: $f(x+h)=-3h^2-6hx-3x^2+2h+2x$ and $f(x)=2x-3x^2$.

Step:3

Insert the values into the difference quotient: $\frac{f(x+h)-f(x)}{h}=\frac{-3h^2-6hx-3x^2+2h+2x-(2x-3x^2)}{h}$.

Step:4

Simplify the expression.

Step:4.1

Simplify the numerator.

Step:4.1.1

Distribute the negative sign: $\frac{-3h^2-6hx-3x^2+2h+2x-2x+3x^2}{h}$.

Step:4.1.2

Combine like terms: $\frac{-3h^2-6hx+2h}{h}$.

Step:4.1.3

Factor out $h$: $\frac{h(-3h-6x+2)}{h}$.

Step:4.2

Cancel the common $h$ factor.

Step:4.2.1

Cancel $h$: $\frac{h(-3h-6x+2)}{\cancel{h}}$.

Step:4.2.2

The simplified difference quotient is $-3h-6x+2$.

Knowledge Notes:

The difference quotient is a formula used in calculus to determine the slope of the secant line between two points on a graph of a function. It is given by $\frac{f(x+h)-f(x)}{h}$, where $f(x)$ is a function, $x$ is a point on the domain, and $h$ is the difference between $x$ and another point.

To solve for the difference quotient of a given function, you must:

1. Substitute $x+h$ into the function to find $f(x+h)$.

2. Simplify the expression for $f(x+h)$.

3. Subtract $f(x)$ from $f(x+h)$.

4. Divide the result by $h$.

5. Simplify the expression further, if possible, by factoring and canceling out common terms.

In this problem, we applied the distributive property, combined like terms, and factored out common factors to simplify the expression. The FOIL method (First, Outer, Inner, Last) was used to expand binomials. It is important to perform each step carefully to avoid errors in simplification.