Finite Math Examples

$y = \frac{1}{2} x^{2}$ , $[1, 2]$

Step 1

Substitute using the average rate of change formula.

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Step 1.1

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.

$\frac{f (2) - f (1)}{(2) - (1)}$

Step 1.2

Substitute the equation $y = \frac{1}{2} x^{2}$ for $f (2)$ and $f (1)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\frac{1}{2} \cdot {(2)}^{2}) - (\frac{1}{2} \cdot {(1)}^{2})}{(2) - (1)}$

Step 2

Simplify the expression.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

Cancel the common factor of $2$ .

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Step 2.1.1.1

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

$\frac{\frac{1}{2} \cdot (2 \cdot 2) - \frac{1}{2} \cdot 1^{2}}{2 - (1)}$

Step 2.1.1.2

Cancel the common factor.

$\frac{\frac{1}{2} \cdot (2 \cdot 2) - \frac{1}{2} \cdot 1^{2}}{2 - (1)}$

Step 2.1.1.3

Rewrite the expression.

$\frac{2 - \frac{1}{2} \cdot 1^{2}}{2 - (1)}$

Step 2.1.2

One to any power is one.

$\frac{2 - \frac{1}{2} \cdot 1}{2 - (1)}$

Step 2.1.3

Multiply $- 1$ by $1$ .

$\frac{2 - \frac{1}{2}}{2 - (1)}$

Step 2.1.4

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{2 \cdot \frac{2}{2} - \frac{1}{2}}{2 - (1)}$

Step 2.1.5

Combine $2$ and $\frac{2}{2}$ .

$\frac{\frac{2 \cdot 2}{2} - \frac{1}{2}}{2 - (1)}$

Step 2.1.6

Combine the numerators over the common denominator.

$\frac{\frac{2 \cdot 2 - 1}{2}}{2 - (1)}$

Step 2.1.7

Simplify the numerator.

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Step 2.1.7.1

Multiply $2$ by $2$ .

$\frac{\frac{4 - 1}{2}}{2 - (1)}$

Step 2.1.7.2

Subtract $1$ from $4$ .

$\frac{\frac{3}{2}}{2 - (1)}$

Step 2.2

Simplify the denominator.

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Step 2.2.1

Multiply $- 1$ by $1$ .

$\frac{\frac{3}{2}}{2 - 1}$

Step 2.2.2

Subtract $1$ from $2$ .

$\frac{\frac{3}{2}}{1}$

Step 2.3

Divide $\frac{3}{2}$ by $1$ .

$\frac{3}{2}$