Precalculus Examples

$b (t) = 4 {(2)}^{t}$

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.

$b (t + h) = 4 {(2)}^{t + h}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Multiply $4 \cdot 2^{t + h}$ .

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Step 2.1.2.1.1

Rewrite $4$ as $2^{2}$ .

$b (t + h) = 2^{2} \cdot 2^{t + h}$

Step 2.1.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$b (t + h) = 2^{2 + t + h}$

Step 2.1.2.2

The final answer is $2^{2 + t + h}$ .

$2^{2 + t + h}$

Step 2.2

Find the components of the definition.

$b (t + h) = 2^{2 + t + h}$

$b (t) = 2^{2 + t}$

$b (t + h) = 2^{2 + t + h}$

$b (t) = 2^{2 + t}$

Step 3

Plug in the components.

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

Step 4

Multiply $- 1$ by $2^{2 + t}$ .

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

Step 5