Precalculus Examples

Find the Average Rate of Change f(x)=18+3e^(-3x)
Step 1
Consider the difference quotient formula.
Step 2
Find the components of the definition.
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Step 2.1
Evaluate the function at .
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Step 2.1.1
Replace the variable with in the expression.
Step 2.1.2
Simplify the result.
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Step 2.1.2.1
Apply the distributive property.
Step 2.1.2.2
The final answer is .
Step 2.2
Find the components of the definition.
Step 3
Plug in the components.
Step 4
Simplify the numerator.
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Step 4.1
Apply the distributive property.
Step 4.2
Multiply by .
Step 4.3
Multiply by .
Step 4.4
Subtract from .
Step 4.5
Add and .
Step 4.6
Rewrite in a factored form.
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Step 4.6.1
Factor out of .
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Step 4.6.1.1
Factor out of .
Step 4.6.1.2
Factor out of .
Step 4.6.1.3
Factor out of .
Step 4.6.2
Rewrite as .
Step 4.6.3
Rewrite as .
Step 4.6.4
Since both terms are perfect cubes, factor using the difference of cubes formula, where and .
Step 4.6.5
Simplify.
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Step 4.6.5.1
Multiply the exponents in .
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Step 4.6.5.1.1
Apply the power rule and multiply exponents, .
Step 4.6.5.1.2
Apply the distributive property.
Step 4.6.5.1.3
Multiply by .
Step 4.6.5.1.4
Multiply by .
Step 4.6.5.2
Multiply by by adding the exponents.
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Step 4.6.5.2.1
Use the power rule to combine exponents.
Step 4.6.5.2.2
Subtract from .
Step 4.6.5.3
Multiply the exponents in .
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Step 4.6.5.3.1
Apply the power rule and multiply exponents, .
Step 4.6.5.3.2
Multiply by .
Step 5