Precalculus Examples

$C (n) = 22 + 0.1 n$

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

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Step 2.1

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

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Step 2.1.1

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

$C (n + h) = 22 + 0.1 (n + h)$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

$C (n + h) = 22 + 0.1 n + 0.1 h$

Step 2.1.2.2

The final answer is $22 + 0.1 n + 0.1 h$ .

$22 + 0.1 n + 0.1 h$

Step 2.2

Reorder.

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Step 2.2.1

Move $22$ .

$0.1 n + 0.1 h + 22$

Step 2.2.2

Reorder $0.1 n$ and $0.1 h$ .

$0.1 h + 0.1 n + 22$

Step 2.3

Find the components of the definition.

$C (n + h) = 0.1 h + 0.1 n + 22$

$C (n) = 22 + 0.1 n$

$C (n + h) = 0.1 h + 0.1 n + 22$

$C (n) = 22 + 0.1 n$

Step 3

Plug in the components.

$\frac{C (n + h) - C n}{h} = \frac{0.1 h + 0.1 n + 22 - (22 + 0.1 n)}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Factor $0.1$ out of $0.1 h + 0.1 n + 22 - (22 + 0.1 n)$ .

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Step 4.1.1.1

Reorder $22$ and $0.1 n$ .

$\frac{0.1 h + 0.1 n + 22 - (0.1 n + 22)}{h}$

Step 4.1.1.2

Factor $0.1$ out of $0.1 h$ .

$\frac{0.1 h + 0.1 n + 22 - (0.1 n + 22)}{h}$

Step 4.1.1.3

Factor $0.1$ out of $0.1 n$ .

$\frac{0.1 h + 0.1 (n) + 22 - (0.1 n + 22)}{h}$

Step 4.1.1.4

Factor $0.1$ out of $22$ .

$\frac{0.1 h + 0.1 n + 0.1 \cdot 220 - (0.1 n + 22)}{h}$

Step 4.1.1.5

Factor $0.1$ out of $- (0.1 n + 22)$ .

$\frac{0.1 h + 0.1 n + 0.1 \cdot 220 + 0.1 (- 10 (0.1 n + 22))}{h}$

Step 4.1.1.6

Factor $0.1$ out of $0.1 h + 0.1 n$ .

$\frac{0.1 (h + n) + 0.1 \cdot 220 + 0.1 (- 10 (0.1 n + 22))}{h}$

Step 4.1.1.7

Factor $0.1$ out of $0.1 (h + n) + 0.1 \cdot 220$ .

$\frac{0.1 (h + n + 220) + 0.1 (- 10 (0.1 n + 22))}{h}$

Step 4.1.1.8

Factor $0.1$ out of $0.1 (h + n + 220) + 0.1 (- 10 (0.1 n + 22))$ .

$\frac{0.1 (h + n + 220 - 10 (0.1 n + 22))}{h}$

Step 4.1.2

Apply the distributive property.

$\frac{0.1 (h + n + 220 - 10 (0.1 n) - 10 \cdot 22)}{h}$

Step 4.1.3

Multiply $0.1$ by $- 10$ .

$\frac{0.1 (h + n + 220 - 1 n - 10 \cdot 22)}{h}$

Step 4.1.4

Multiply $- 10$ by $22$ .

$\frac{0.1 (h + n + 220 - 1 n - 220)}{h}$

Step 4.1.5

Rewrite $- 1 n$ as $- n$ .

$\frac{0.1 (h + n + 220 - n - 220)}{h}$

Step 4.1.6

Subtract $n$ from $n$ .

$\frac{0.1 (h + 0 + 220 - 220)}{h}$

Step 4.1.7

Add $h$ and $0$ .

$\frac{0.1 (h + 220 - 220)}{h}$

Step 4.1.8

Subtract $220$ from $220$ .

$\frac{0.1 (h + 0)}{h}$

Step 4.1.9

Add $h$ and $0$ .

$\frac{0.1 h}{h}$

Step 4.2

Cancel the common factor of $h$ .

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Step 4.2.1

Cancel the common factor.

$\frac{0.1 h}{h}$

Step 4.2.2

Divide $0.1$ by $1$ .

$0.1$

Step 5