Precalculus Examples

$A (t) = 0.5 t + 78$

Step 1

Consider the difference quotient formula.

$\frac{f (t + h) - f t}{h}$

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = t + h$ .

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Step 2.1.1

Replace the variable $t$ with $t + h$ in the expression.

$A (t + h) = 0.5 (t + h) + 78$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

$A (t + h) = 0.5 t + 0.5 h + 78$

Step 2.1.2.2

The final answer is $0.5 t + 0.5 h + 78$ .

$0.5 t + 0.5 h + 78$

Step 2.2

Reorder $0.5 t$ and $0.5 h$ .

$0.5 h + 0.5 t + 78$

Step 2.3

Find the components of the definition.

$A (t + h) = 0.5 h + 0.5 t + 78$

$A (t) = 0.5 t + 78$

$A (t + h) = 0.5 h + 0.5 t + 78$

$A (t) = 0.5 t + 78$

Step 3

Plug in the components.

$\frac{A (t + h) - A t}{h} = \frac{0.5 h + 0.5 t + 78 - (0.5 t + 78)}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Factor $0.5$ out of $0.5 h + 0.5 t + 78 - (0.5 t + 78)$ .

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Step 4.1.1.1

Factor $0.5$ out of $0.5 h$ .

$\frac{0.5 h + 0.5 t + 78 - (0.5 t + 78)}{h}$

Step 4.1.1.2

Factor $0.5$ out of $0.5 t$ .

$\frac{0.5 h + 0.5 (t) + 78 - (0.5 t + 78)}{h}$

Step 4.1.1.3

Factor $0.5$ out of $78$ .

$\frac{0.5 h + 0.5 t + 0.5 \cdot 156 - (0.5 t + 78)}{h}$

Step 4.1.1.4

Factor $0.5$ out of $- (0.5 t + 78)$ .

$\frac{0.5 h + 0.5 t + 0.5 \cdot 156 + 0.5 (- 2 (0.5 t + 78))}{h}$

Step 4.1.1.5

Factor $0.5$ out of $0.5 h + 0.5 t$ .

$\frac{0.5 (h + t) + 0.5 \cdot 156 + 0.5 (- 2 (0.5 t + 78))}{h}$

Step 4.1.1.6

Factor $0.5$ out of $0.5 (h + t) + 0.5 \cdot 156$ .

$\frac{0.5 (h + t + 156) + 0.5 (- 2 (0.5 t + 78))}{h}$

Step 4.1.1.7

Factor $0.5$ out of $0.5 (h + t + 156) + 0.5 (- 2 (0.5 t + 78))$ .

$\frac{0.5 (h + t + 156 - 2 (0.5 t + 78))}{h}$

Step 4.1.2

Apply the distributive property.

$\frac{0.5 (h + t + 156 - 2 (0.5 t) - 2 \cdot 78)}{h}$

Step 4.1.3

Multiply $0.5$ by $- 2$ .

$\frac{0.5 (h + t + 156 - 1 t - 2 \cdot 78)}{h}$

Step 4.1.4

Multiply $- 2$ by $78$ .

$\frac{0.5 (h + t + 156 - 1 t - 156)}{h}$

Step 4.1.5

Rewrite $- 1 t$ as $- t$ .

$\frac{0.5 (h + t + 156 - t - 156)}{h}$

Step 4.1.6

Subtract $t$ from $t$ .

$\frac{0.5 (h + 0 + 156 - 156)}{h}$

Step 4.1.7

Add $h$ and $0$ .

$\frac{0.5 (h + 156 - 156)}{h}$

Step 4.1.8

Subtract $156$ from $156$ .

$\frac{0.5 (h + 0)}{h}$

Step 4.1.9

Add $h$ and $0$ .

$\frac{0.5 h}{h}$

Step 4.2

Cancel the common factor of $h$ .

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Step 4.2.1

Cancel the common factor.

$\frac{0.5 h}{h}$

Step 4.2.2

Divide $0.5$ by $1$ .

$0.5$

Step 5