Precalculus Examples

$R (x) = 56 x - 0.02 x^{2}$ , $0 \leq x \leq 2800$

Step 1

Write $R (x) = 56 x - 0.02 x^{2}$ as an equation.

$y = 56 x - 0.02 x^{2}$

Step 2

Substitute using the average rate of change formula.

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Step 2.1

The average rate of change of a function can be found by calculating the change in $y$ values of the two points divided by the change in $x$ values of the two points.

$\frac{f (2800) - f (0)}{(2800) - (0)}$

Step 2.2

Substitute the equation $y = 56 x - 0.02 x^{2}$ for $f (2800)$ and $f (0)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(56 (2800) - 0.02 {(2800)}^{2}) - (56 (0) - 0.02 {(0)}^{2})}{(2800) - (0)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $56$ by $2800$ .

$\frac{156800 - 0.02 \cdot 2800^{2} - (56 (0) - 0.02 \cdot 0^{2})}{2800 - (0)}$

Step 3.1.2

Raise $2800$ to the power of $2$ .

$\frac{156800 - 0.02 \cdot 7840000 - (56 (0) - 0.02 \cdot 0^{2})}{2800 - (0)}$

Step 3.1.3

Multiply $- 0.02$ by $7840000$ .

$\frac{156800 - 156800 - (56 (0) - 0.02 \cdot 0^{2})}{2800 - (0)}$

Step 3.1.4

Simplify each term.

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Step 3.1.4.1

Multiply $56$ by $0$ .

$\frac{156800 - 156800 - (0 - 0.02 \cdot 0^{2})}{2800 - (0)}$

Step 3.1.4.2

Raising $0$ to any positive power yields $0$ .

$\frac{156800 - 156800 - (0 - 0.02 \cdot 0)}{2800 - (0)}$

Step 3.1.4.3

Multiply $- 0.02$ by $0$ .

$\frac{156800 - 156800 - (0 + 0)}{2800 - (0)}$

Step 3.1.5

Add $0$ and $0$ .

$\frac{156800 - 156800 - 0}{2800 - (0)}$

Step 3.1.6

Multiply $- 1$ by $0$ .

$\frac{156800 - 156800 + 0}{2800 - (0)}$

Step 3.1.7

Add $156800 - 156800$ and $0$ .

$\frac{156800 - 156800}{2800 - (0)}$

Step 3.1.8

Subtract $156800$ from $156800$ .

$\frac{0}{2800 - (0)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $0$ .

$\frac{0}{2800 + 0}$

Step 3.2.2

Add $2800$ and $0$ .

$\frac{0}{2800}$

Step 3.3

Divide $0$ by $2800$ .

$0$