Precalculus Examples

Find the Average Rate of Change f(x)=1/8x^3-x^2
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
Simplify each term.
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Step 2.1.2.1.1
Use the Binomial Theorem.
Step 2.1.2.1.2
Apply the distributive property.
Step 2.1.2.1.3
Simplify.
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Step 2.1.2.1.3.1
Combine and .
Step 2.1.2.1.3.2
Multiply .
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Step 2.1.2.1.3.2.1
Combine and .
Step 2.1.2.1.3.2.2
Combine and .
Step 2.1.2.1.3.2.3
Combine and .
Step 2.1.2.1.3.3
Multiply .
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Step 2.1.2.1.3.3.1
Combine and .
Step 2.1.2.1.3.3.2
Combine and .
Step 2.1.2.1.3.3.3
Combine and .
Step 2.1.2.1.3.4
Combine and .
Step 2.1.2.1.4
Simplify each term.
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Step 2.1.2.1.4.1
Move to the left of .
Step 2.1.2.1.4.2
Move to the left of .
Step 2.1.2.1.5
Rewrite as .
Step 2.1.2.1.6
Expand using the FOIL Method.
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Step 2.1.2.1.6.1
Apply the distributive property.
Step 2.1.2.1.6.2
Apply the distributive property.
Step 2.1.2.1.6.3
Apply the distributive property.
Step 2.1.2.1.7
Simplify and combine like terms.
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Step 2.1.2.1.7.1
Simplify each term.
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Step 2.1.2.1.7.1.1
Multiply by .
Step 2.1.2.1.7.1.2
Multiply by .
Step 2.1.2.1.7.2
Add and .
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Step 2.1.2.1.7.2.1
Reorder and .
Step 2.1.2.1.7.2.2
Add and .
Step 2.1.2.1.8
Apply the distributive property.
Step 2.1.2.1.9
Multiply by .
Step 2.1.2.2
The final answer is .
Step 2.2
Reorder.
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Step 2.2.1
Move .
Step 2.2.2
Move .
Step 2.2.3
Move .
Step 2.2.4
Move .
Step 2.2.5
Move .
Step 2.2.6
Move .
Step 2.2.7
Reorder and .
Step 2.3
Find the components of the definition.
Step 3
Plug in the components.
Step 4
Simplify.
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Step 4.1
Simplify the numerator.
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Step 4.1.1
Apply the distributive property.
Step 4.1.2
Multiply .
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Step 4.1.2.1
Multiply by .
Step 4.1.2.2
Multiply by .
Step 4.1.3
Subtract from .
Step 4.1.4
Add and .
Step 4.1.5
Add and .
Step 4.1.6
Add and .
Step 4.1.7
Combine the numerators over the common denominator.
Step 4.1.8
Factor out of .
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Step 4.1.8.1
Factor out of .
Step 4.1.8.2
Factor out of .
Step 4.1.8.3
Factor out of .
Step 4.1.9
Combine the numerators over the common denominator.
Step 4.1.10
Simplify the numerator.
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Step 4.1.10.1
Factor out of .
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Step 4.1.10.1.1
Factor out of .
Step 4.1.10.1.2
Factor out of .
Step 4.1.10.1.3
Factor out of .
Step 4.1.10.2
Apply the distributive property.
Step 4.1.10.3
Multiply by .
Step 4.1.10.4
Rewrite using the commutative property of multiplication.
Step 4.1.11
To write as a fraction with a common denominator, multiply by .
Step 4.1.12
Combine and .
Step 4.1.13
Combine the numerators over the common denominator.
Step 4.1.14
Simplify the numerator.
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Step 4.1.14.1
Factor out of .
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Step 4.1.14.1.1
Factor out of .
Step 4.1.14.1.2
Factor out of .
Step 4.1.14.2
Multiply by .
Step 4.1.15
To write as a fraction with a common denominator, multiply by .
Step 4.1.16
Combine and .
Step 4.1.17
Combine the numerators over the common denominator.
Step 4.1.18
Simplify the numerator.
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Step 4.1.18.1
Factor out of .
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Step 4.1.18.1.1
Factor out of .
Step 4.1.18.1.2
Factor out of .
Step 4.1.18.2
Multiply by .
Step 4.2
Multiply the numerator by the reciprocal of the denominator.
Step 4.3
Combine.
Step 4.4
Cancel the common factor of .
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Step 4.4.1
Cancel the common factor.
Step 4.4.2
Rewrite the expression.
Step 4.5
Multiply by .
Step 5