Precalculus Examples

$y = (0.9) \frac{t}{12}$

Step 1

Write $y = (0.9) \frac{t}{12}$ as a function.

$f (t) = (0.9) \frac{t}{12}$

Step 2

Consider the difference quotient formula.

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

Step 3

Find the components of the definition.

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Step 3.1

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

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Step 3.1.1

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

$f (t + h) = (0.9) (\frac{t + h}{12})$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Cancel the common factor of $0.3$ .

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Step 3.1.2.1.1

Factor $0.3$ out of $0.9$ .

$f (t + h) = 0.3 (3) (\frac{t + h}{12})$

Step 3.1.2.1.2

Factor $0.3$ out of $12$ .

$f (t + h) = 0.3 \cdot (3 (\frac{t + h}{0.3 \cdot 40}))$

Step 3.1.2.1.3

Cancel the common factor.

$f (t + h) = 0.3 \cdot (3 (\frac{t + h}{0.3 \cdot 40}))$

Step 3.1.2.1.4

Rewrite the expression.

$f (t + h) = 3 (\frac{t + h}{40})$

Step 3.1.2.2

Combine $3$ and $\frac{t + h}{40}$ .

$f (t + h) = \frac{3 (t + h)}{40}$

Step 3.1.2.3

The final answer is $\frac{3 (t + h)}{40}$ .

$\frac{3 (t + h)}{40}$

Step 3.2

Find the components of the definition.

$f (t + h) = \frac{3 (t + h)}{40}$

$f (t) = \frac{3 t}{40}$

$f (t + h) = \frac{3 (t + h)}{40}$

$f (t) = \frac{3 t}{40}$

Step 4

Plug in the components.

$\frac{f (t + h) - f t}{h} = \frac{\frac{3 (t + h)}{40} - (\frac{3 t}{40})}{h}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Combine the numerators over the common denominator.

$\frac{\frac{3 (t + h) - 3 t}{40}}{h}$

Step 5.1.2

Rewrite $\frac{3 (t + h) - 3 t}{40}$ in a factored form.

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Step 5.1.2.1

Factor $3$ out of $3 (t + h) - 3 t$ .

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Step 5.1.2.1.1

Factor $3$ out of $- 3 t$ .

$\frac{\frac{3 (t + h) + 3 (- t)}{40}}{h}$

Step 5.1.2.1.2

Factor $3$ out of $3 (t + h) + 3 (- t)$ .

$\frac{\frac{3 (t + h - t)}{40}}{h}$

Step 5.1.2.2

Subtract $t$ from $t$ .

$\frac{\frac{3 (h + 0)}{40}}{h}$

Step 5.1.2.3

Add $h$ and $0$ .

$\frac{\frac{3 h}{40}}{h}$

Step 5.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{3 h}{40} \cdot \frac{1}{h}$

Step 5.3

Cancel the common factor of $h$ .

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Step 5.3.1

Factor $h$ out of $3 h$ .

$\frac{h \cdot 3}{40} \cdot \frac{1}{h}$

Step 5.3.2

Cancel the common factor.

$\frac{h \cdot 3}{40} \cdot \frac{1}{h}$

Step 5.3.3

Rewrite the expression.

$\frac{3}{40}$

Step 6