Precalculus Examples

Find the Average Rate of Change g(t)=t^4-t^3+t^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
Use the Binomial Theorem.
Step 2.1.2.1.3
Apply the distributive property.
Step 2.1.2.1.4
Simplify.
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Step 2.1.2.1.4.1
Multiply by .
Step 2.1.2.1.4.2
Multiply by .
Step 2.1.2.1.5
Remove parentheses.
Step 2.1.2.1.6
Rewrite as .
Step 2.1.2.1.7
Expand using the FOIL Method.
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Step 2.1.2.1.7.1
Apply the distributive property.
Step 2.1.2.1.7.2
Apply the distributive property.
Step 2.1.2.1.7.3
Apply the distributive property.
Step 2.1.2.1.8
Simplify and combine like terms.
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Step 2.1.2.1.8.1
Simplify each term.
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Step 2.1.2.1.8.1.1
Multiply by .
Step 2.1.2.1.8.1.2
Multiply by .
Step 2.1.2.1.8.2
Add and .
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Step 2.1.2.1.8.2.1
Reorder and .
Step 2.1.2.1.8.2.2
Add and .
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
Move .
Step 2.2.8
Move .
Step 2.2.9
Move .
Step 2.2.10
Move .
Step 2.2.11
Move .
Step 2.2.12
Move .
Step 2.2.13
Move .
Step 2.2.14
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
Subtract from .
Step 4.1.8
Add and .
Step 4.1.9
Factor out of .
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Step 4.1.9.1
Factor out of .
Step 4.1.9.2
Factor out of .
Step 4.1.9.3
Factor out of .
Step 4.1.9.4
Factor out of .
Step 4.1.9.5
Factor out of .
Step 4.1.9.6
Factor out of .
Step 4.1.9.7
Factor out of .
Step 4.1.9.8
Factor out of .
Step 4.1.9.9
Factor out of .
Step 4.1.9.10
Factor out of .
Step 4.1.9.11
Factor out of .
Step 4.1.9.12
Factor out of .
Step 4.1.9.13
Factor out of .
Step 4.1.9.14
Factor out of .
Step 4.1.9.15
Factor out of .
Step 4.1.9.16
Factor out of .
Step 4.1.9.17
Factor out of .
Step 4.2
Reduce the expression by cancelling the common factors.
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Step 4.2.1
Cancel the common factor of .
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Step 4.2.1.1
Cancel the common factor.
Step 4.2.1.2
Divide by .
Step 4.2.2
Simplify the expression.
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Step 4.2.2.1
Move .
Step 4.2.2.2
Move .
Step 4.2.2.3
Move .
Step 4.2.2.4
Move .
Step 4.2.2.5
Move .
Step 4.2.2.6
Move .
Step 4.2.2.7
Move .
Step 4.2.2.8
Move .
Step 4.2.2.9
Reorder and .
Step 5