Precalculus Examples

Find the Average Rate of Change V(r)=4/3pir^3
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
Use the Binomial Theorem.
Step 2.1.2.2
Apply the distributive property.
Step 2.1.2.3
Remove parentheses.
Step 2.1.2.4
Simplify each term.
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Step 2.1.2.4.1
Move to the left of .
Step 2.1.2.4.2
Move to the left of .
Step 2.1.2.5
Apply the distributive property.
Step 2.1.2.6
Simplify.
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Step 2.1.2.6.1
Multiply .
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Step 2.1.2.6.1.1
Combine and .
Step 2.1.2.6.1.2
Combine and .
Step 2.1.2.6.2
Cancel the common factor of .
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Step 2.1.2.6.2.1
Factor out of .
Step 2.1.2.6.2.2
Cancel the common factor.
Step 2.1.2.6.2.3
Rewrite the expression.
Step 2.1.2.6.3
Cancel the common factor of .
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Step 2.1.2.6.3.1
Factor out of .
Step 2.1.2.6.3.2
Cancel the common factor.
Step 2.1.2.6.3.3
Rewrite the expression.
Step 2.1.2.6.4
Multiply .
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Step 2.1.2.6.4.1
Combine and .
Step 2.1.2.6.4.2
Combine and .
Step 2.1.2.7
Simplify each term.
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Step 2.1.2.7.1
Move to the left of .
Step 2.1.2.7.2
Move to the left of .
Step 2.1.2.8
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
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
Subtract from .
Step 4.1.2
Add and .
Step 4.1.3
Factor out of .
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Step 4.1.3.1
Factor out of .
Step 4.1.3.2
Factor out of .
Step 4.1.3.3
Factor out of .
Step 4.1.3.4
Factor out of .
Step 4.1.3.5
Factor out of .
Step 4.1.4
To write as a fraction with a common denominator, multiply by .
Step 4.1.5
Combine and .
Step 4.1.6
Combine the numerators over the common denominator.
Step 4.1.7
Simplify the numerator.
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Step 4.1.7.1
Factor out of .
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Step 4.1.7.1.1
Factor out of .
Step 4.1.7.1.2
Factor out of .
Step 4.1.7.1.3
Factor out of .
Step 4.1.7.2
Move to the left of .
Step 4.1.8
To write as a fraction with a common denominator, multiply by .
Step 4.1.9
Combine and .
Step 4.1.10
Combine the numerators over the common denominator.
Step 4.1.11
Simplify the numerator.
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Step 4.1.11.1
Apply the distributive property.
Step 4.1.11.2
Multiply by .
Step 4.1.11.3
Rewrite using the commutative property of multiplication.
Step 4.1.11.4
Move to the left of .
Step 4.1.12
Combine exponents.
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Step 4.1.12.1
Combine and .
Step 4.1.12.2
Combine and .
Step 4.1.12.3
Combine and .
Step 4.1.13
Remove unnecessary parentheses.
Step 4.1.14
Move to the left of .
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