Precalculus Examples

Find the Average Rate of Change y=1/2x^2
Step 1
Write as a function.
Step 2
Consider the difference quotient formula.
Step 3
Find the components of the definition.
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Step 3.1
Evaluate the function at .
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Step 3.1.1
Replace the variable with in the expression.
Step 3.1.2
Simplify the result.
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Step 3.1.2.1
Rewrite as .
Step 3.1.2.2
Expand using the FOIL Method.
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Step 3.1.2.2.1
Apply the distributive property.
Step 3.1.2.2.2
Apply the distributive property.
Step 3.1.2.2.3
Apply the distributive property.
Step 3.1.2.3
Simplify and combine like terms.
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Step 3.1.2.3.1
Simplify each term.
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Step 3.1.2.3.1.1
Multiply by .
Step 3.1.2.3.1.2
Multiply by .
Step 3.1.2.3.2
Add and .
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Step 3.1.2.3.2.1
Reorder and .
Step 3.1.2.3.2.2
Add and .
Step 3.1.2.4
Apply the distributive property.
Step 3.1.2.5
Simplify.
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Step 3.1.2.5.1
Combine and .
Step 3.1.2.5.2
Cancel the common factor of .
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Step 3.1.2.5.2.1
Factor out of .
Step 3.1.2.5.2.2
Cancel the common factor.
Step 3.1.2.5.2.3
Rewrite the expression.
Step 3.1.2.5.3
Combine and .
Step 3.1.2.6
The final answer is .
Step 3.2
Reorder.
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Step 3.2.1
Move .
Step 3.2.2
Reorder and .
Step 3.3
Find the components of the definition.
Step 4
Plug in the components.
Step 5
Simplify.
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Step 5.1
Simplify the numerator.
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Step 5.1.1
Subtract from .
Step 5.1.2
Add and .
Step 5.1.3
Factor out of .
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Step 5.1.3.1
Factor out of .
Step 5.1.3.2
Factor out of .
Step 5.1.3.3
Factor out of .
Step 5.1.4
To write as a fraction with a common denominator, multiply by .
Step 5.1.5
Combine and .
Step 5.1.6
Combine the numerators over the common denominator.
Step 5.1.7
Move to the left of .
Step 5.2
Combine and .
Step 5.3
Multiply the numerator by the reciprocal of the denominator.
Step 5.4
Cancel the common factor of .
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Step 5.4.1
Cancel the common factor.
Step 5.4.2
Rewrite the expression.
Step 6