Precalculus Examples

$h (t) = (t + 3) \cdot 2 + 5$

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.

$h (t + h) = ((t + h) + 3) \cdot 2 + 5$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

Apply the distributive property.

$h (t + h) = t \cdot 2 + h \cdot 2 + 3 \cdot 2 + 5$

Step 2.1.2.1.2

Simplify.

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Step 2.1.2.1.2.1

Move $2$ to the left of $t$ .

$h (t + h) = 2 \cdot t + h \cdot 2 + 3 \cdot 2 + 5$

Step 2.1.2.1.2.2

Move $2$ to the left of $h$ .

$h (t + h) = 2 \cdot t + 2 \cdot h + 3 \cdot 2 + 5$

Step 2.1.2.1.2.3

Multiply $3$ by $2$ .

$h (t + h) = 2 \cdot t + 2 \cdot h + 6 + 5$

Step 2.1.2.1.3

Multiply $2$ by $h$ .

$h (t + h) = 2 t + 2 h + 6 + 5$

Step 2.1.2.2

Add $6$ and $5$ .

$h (t + h) = 2 t + 2 h + 11$

Step 2.1.2.3

The final answer is $2 t + 2 h + 11$ .

$2 t + 2 h + 11$

Step 2.2

Reorder $2 t$ and $2 h$ .

$2 h + 2 t + 11$

Step 2.3

Find the components of the definition.

$h (t + h) = 2 h + 2 t + 11$

$h (t) = 2 t + 11$

$h (t + h) = 2 h + 2 t + 11$

$h (t) = 2 t + 11$

Step 3

Plug in the components.

$\frac{h (t + h) - h t}{h} = \frac{2 h + 2 t + 11 - (2 t + 11)}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

$\frac{2 h + 2 t + 11 - (2 t) - 1 \cdot 11}{h}$

Step 4.1.2

Multiply $2$ by $- 1$ .

$\frac{2 h + 2 t + 11 - 2 t - 1 \cdot 11}{h}$

Step 4.1.3

Multiply $- 1$ by $11$ .

$\frac{2 h + 2 t + 11 - 2 t - 11}{h}$

Step 4.1.4

Subtract $2 t$ from $2 t$ .

$\frac{2 h + 0 + 11 - 11}{h}$

Step 4.1.5

Add $2 h$ and $0$ .

$\frac{2 h + 11 - 11}{h}$

Step 4.1.6

Subtract $11$ from $11$ .

$\frac{2 h + 0}{h}$

Step 4.1.7

Add $2 h$ and $0$ .

$\frac{2 h}{h}$

Step 4.2

Cancel the common factor of $h$ .

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Step 4.2.1

Cancel the common factor.

$\frac{2 h}{h}$

Step 4.2.2

Divide $2$ by $1$ .

$2$

Step 5