Precalculus Examples

$f (t) = 330 {(1.04)}^{30 t}$

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.

$f (t + h) = 330 {(1.04)}^{30 (t + h)}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

$f (t + h) = 330 \cdot {1.04}^{30 t + 30 h}$

Step 2.1.2.2

The final answer is $330 \cdot {1.04}^{30 t + 30 h}$ .

$330 \cdot {1.04}^{30 t + 30 h}$

Step 2.2

Find the components of the definition.

$f (t + h) = 330 \cdot {1.04}^{30 t + 30 h}$

$f (t) = 330 \cdot {1.04}^{30 t}$

$f (t + h) = 330 \cdot {1.04}^{30 t + 30 h}$

$f (t) = 330 \cdot {1.04}^{30 t}$

Step 3

Plug in the components.

$\frac{f (t + h) - f t}{h} = \frac{330 \cdot {1.04}^{30 t + 30 h} - (330 \cdot {1.04}^{30 t})}{h}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $- 1$ by $330$ .

$\frac{330 \cdot {1.04}^{30 t + 30 h} - 330 \cdot {1.04}^{30 t}}{h}$

Step 4.2

Rewrite $330 \cdot {1.04}^{30 t + 30 h} - 330 \cdot {1.04}^{30 t}$ in a factored form.

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Step 4.2.1

Factor $330$ out of $330 \cdot {1.04}^{30 t + 30 h} - 330 \cdot {1.04}^{30 t}$ .

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Step 4.2.1.1

Factor $330$ out of $330 \cdot {1.04}^{30 t + 30 h}$ .

$\frac{330 \cdot {1.04}^{30 t + 30 h} - 330 \cdot {1.04}^{30 t}}{h}$

Step 4.2.1.2

Factor $330$ out of $- 330 \cdot {1.04}^{30 t}$ .

$\frac{330 \cdot {1.04}^{30 t + 30 h} + 330 (- {1.04}^{30 t})}{h}$

Step 4.2.1.3

Factor $330$ out of $330 \cdot {1.04}^{30 t + 30 h} + 330 (- {1.04}^{30 t})$ .

$\frac{330 ({1.04}^{30 t + 30 h} - {1.04}^{30 t})}{h}$

Step 4.2.2

Rewrite ${1.04}^{30 t + 30 h}$ as ${({1.04}^{10 t + 10 h})}^{3}$ .

$\frac{330 ({({1.04}^{10 t + 10 h})}^{3} - {1.04}^{30 t})}{h}$

Step 4.2.3

Rewrite ${1.04}^{30 t}$ as ${({1.04}^{10 t})}^{3}$ .

$\frac{330 ({({1.04}^{10 t + 10 h})}^{3} - {({1.04}^{10 t})}^{3})}{h}$

Step 4.2.4

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = {1.04}^{10 t + 10 h}$ and $b = {1.04}^{10 t}$ .

$\frac{330 (({1.04}^{10 t + 10 h} - {1.04}^{10 t}) ({({1.04}^{10 t + 10 h})}^{2} + {1.04}^{10 t + 10 h} \cdot {1.04}^{10 t} + {({1.04}^{10 t})}^{2}))}{h}$

Step 4.2.5

Simplify.

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Step 4.2.5.1

Rewrite ${1.04}^{10 t + 10 h}$ as ${({1.04}^{5 t + 5 h})}^{2}$ .

$\frac{330 (({({1.04}^{5 t + 5 h})}^{2} - {1.04}^{10 t}) ({({1.04}^{10 t + 10 h})}^{2} + {1.04}^{10 t + 10 h} \cdot {1.04}^{10 t} + {({1.04}^{10 t})}^{2}))}{h}$

Step 4.2.5.2

Rewrite ${1.04}^{10 t}$ as ${({1.04}^{5 t})}^{2}$ .

$\frac{330 (({({1.04}^{5 t + 5 h})}^{2} - {({1.04}^{5 t})}^{2}) ({({1.04}^{10 t + 10 h})}^{2} + {1.04}^{10 t + 10 h} \cdot {1.04}^{10 t} + {({1.04}^{10 t})}^{2}))}{h}$

Step 4.2.5.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = {1.04}^{5 t + 5 h}$ and $b = {1.04}^{5 t}$ .

$\frac{330 (({1.04}^{5 t + 5 h} + {1.04}^{5 t}) ({1.04}^{5 t + 5 h} - {1.04}^{5 t}) ({({1.04}^{10 t + 10 h})}^{2} + {1.04}^{10 t + 10 h} \cdot {1.04}^{10 t} + {({1.04}^{10 t})}^{2}))}{h}$

Step 4.2.5.4

Multiply ${1.04}^{10 t + 10 h}$ by ${1.04}^{10 t}$ by adding the exponents.

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Step 4.2.5.4.1

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

$\frac{330 (({1.04}^{5 t + 5 h} + {1.04}^{5 t}) ({1.04}^{5 t + 5 h} - {1.04}^{5 t}) ({({1.04}^{10 t + 10 h})}^{2} + {1.04}^{10 t + 10 h + 10 t} + {({1.04}^{10 t})}^{2}))}{h}$

Step 4.2.5.4.2

Add $10 t$ and $10 t$ .

$\frac{330 (({1.04}^{5 t + 5 h} + {1.04}^{5 t}) ({1.04}^{5 t + 5 h} - {1.04}^{5 t}) ({({1.04}^{10 t + 10 h})}^{2} + {1.04}^{20 t + 10 h} + {({1.04}^{10 t})}^{2}))}{h}$

$\frac{330 ({1.04}^{5 t + 5 h} + {1.04}^{5 t}) ({1.04}^{5 t + 5 h} - {1.04}^{5 t}) ({({1.04}^{10 t + 10 h})}^{2} + {1.04}^{20 t + 10 h} + {({1.04}^{10 t})}^{2})}{h}$

Step 4.3

Simplify each term.

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Step 4.3.1

Multiply the exponents in ${({1.04}^{10 t + 10 h})}^{2}$ .

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Step 4.3.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{330 ({1.04}^{5 t + 5 h} + {1.04}^{5 t}) ({1.04}^{5 t + 5 h} - {1.04}^{5 t}) ({1.04}^{(10 t + 10 h) \cdot 2} + {1.04}^{20 t + 10 h} + {({1.04}^{10 t})}^{2})}{h}$

Step 4.3.1.2

Apply the distributive property.

$\frac{330 ({1.04}^{5 t + 5 h} + {1.04}^{5 t}) ({1.04}^{5 t + 5 h} - {1.04}^{5 t}) ({1.04}^{10 t \cdot 2 + 10 h \cdot 2} + {1.04}^{20 t + 10 h} + {({1.04}^{10 t})}^{2})}{h}$

Step 4.3.1.3

Multiply $2$ by $10$ .

$\frac{330 ({1.04}^{5 t + 5 h} + {1.04}^{5 t}) ({1.04}^{5 t + 5 h} - {1.04}^{5 t}) ({1.04}^{20 t + 10 h \cdot 2} + {1.04}^{20 t + 10 h} + {({1.04}^{10 t})}^{2})}{h}$

Step 4.3.1.4

Multiply $2$ by $10$ .

$\frac{330 ({1.04}^{5 t + 5 h} + {1.04}^{5 t}) ({1.04}^{5 t + 5 h} - {1.04}^{5 t}) ({1.04}^{20 t + 20 h} + {1.04}^{20 t + 10 h} + {({1.04}^{10 t})}^{2})}{h}$

Step 4.3.2

Multiply the exponents in ${({1.04}^{10 t})}^{2}$ .

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Step 4.3.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{330 ({1.04}^{5 t + 5 h} + {1.04}^{5 t}) ({1.04}^{5 t + 5 h} - {1.04}^{5 t}) ({1.04}^{20 t + 20 h} + {1.04}^{20 t + 10 h} + {1.04}^{10 t \cdot 2})}{h}$

Step 4.3.2.2

Multiply $2$ by $10$ .

$\frac{330 ({1.04}^{5 t + 5 h} + {1.04}^{5 t}) ({1.04}^{5 t + 5 h} - {1.04}^{5 t}) ({1.04}^{20 t + 20 h} + {1.04}^{20 t + 10 h} + {1.04}^{20 t})}{h}$

Step 5