Precalculus Examples

Find the Average Rate of Change R(x)=45x^(9/10)
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
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
Multiply by .
Step 4.2
Rewrite in a factored form.
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Step 4.2.1
Factor out of .
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Step 4.2.1.1
Factor out of .
Step 4.2.1.2
Factor out of .
Step 4.2.1.3
Factor out of .
Step 4.2.2
Rewrite as .
Step 4.2.3
Rewrite as .
Step 4.2.4
Since both terms are perfect cubes, factor using the difference of cubes formula, where and .
Step 4.2.5
Simplify.
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Step 4.2.5.1
Rewrite as .
Step 4.2.5.2
Rewrite as .
Step 4.2.5.3
Since both terms are perfect cubes, factor using the difference of cubes formula, where and .
Step 4.2.5.4
Simplify.
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Step 4.2.5.4.1
Multiply the exponents in .
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Step 4.2.5.4.1.1
Apply the power rule and multiply exponents, .
Step 4.2.5.4.1.2
Cancel the common factor of .
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Step 4.2.5.4.1.2.1
Factor out of .
Step 4.2.5.4.1.2.2
Cancel the common factor.
Step 4.2.5.4.1.2.3
Rewrite the expression.
Step 4.2.5.4.2
Multiply the exponents in .
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Step 4.2.5.4.2.1
Apply the power rule and multiply exponents, .
Step 4.2.5.4.2.2
Cancel the common factor of .
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Step 4.2.5.4.2.2.1
Factor out of .
Step 4.2.5.4.2.2.2
Cancel the common factor.
Step 4.2.5.4.2.2.3
Rewrite the expression.
Step 4.2.5.5
Multiply the exponents in .
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Step 4.2.5.5.1
Apply the power rule and multiply exponents, .
Step 4.2.5.5.2
Cancel the common factor of .
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Step 4.2.5.5.2.1
Factor out of .
Step 4.2.5.5.2.2
Cancel the common factor.
Step 4.2.5.5.2.3
Rewrite the expression.
Step 4.2.5.6
Multiply the exponents in .
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Step 4.2.5.6.1
Apply the power rule and multiply exponents, .
Step 4.2.5.6.2
Cancel the common factor of .
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Step 4.2.5.6.2.1
Factor out of .
Step 4.2.5.6.2.2
Cancel the common factor.
Step 4.2.5.6.2.3
Rewrite the expression.
Step 5