Precalculus Examples

Find the Average Rate of Change P(t)=(600t)/(2t^2+9)
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 the denominator.
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Step 2.1.2.1.1
Rewrite as .
Step 2.1.2.1.2
Expand using the FOIL Method.
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Step 2.1.2.1.2.1
Apply the distributive property.
Step 2.1.2.1.2.2
Apply the distributive property.
Step 2.1.2.1.2.3
Apply the distributive property.
Step 2.1.2.1.3
Simplify and combine like terms.
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Step 2.1.2.1.3.1
Simplify each term.
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Step 2.1.2.1.3.1.1
Multiply by .
Step 2.1.2.1.3.1.2
Multiply by .
Step 2.1.2.1.3.2
Add and .
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Step 2.1.2.1.3.2.1
Reorder and .
Step 2.1.2.1.3.2.2
Add and .
Step 2.1.2.1.4
Apply the distributive property.
Step 2.1.2.1.5
Multiply by .
Step 2.1.2.2
The final answer is .
Step 2.2
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
To write as a fraction with a common denominator, multiply by .
Step 4.1.2
To write as a fraction with a common denominator, multiply by .
Step 4.1.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 4.1.3.1
Multiply by .
Step 4.1.3.2
Multiply by .
Step 4.1.3.3
Reorder the factors of .
Step 4.1.4
Combine the numerators over the common denominator.
Step 4.1.5
Rewrite in a factored form.
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Step 4.1.5.1
Factor out of .
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Step 4.1.5.1.1
Factor out of .
Step 4.1.5.1.2
Factor out of .
Step 4.1.5.1.3
Factor out of .
Step 4.1.5.2
Expand using the FOIL Method.
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Step 4.1.5.2.1
Apply the distributive property.
Step 4.1.5.2.2
Apply the distributive property.
Step 4.1.5.2.3
Apply the distributive property.
Step 4.1.5.3
Simplify each term.
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Step 4.1.5.3.1
Rewrite using the commutative property of multiplication.
Step 4.1.5.3.2
Multiply by by adding the exponents.
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Step 4.1.5.3.2.1
Move .
Step 4.1.5.3.2.2
Multiply by .
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Step 4.1.5.3.2.2.1
Raise to the power of .
Step 4.1.5.3.2.2.2
Use the power rule to combine exponents.
Step 4.1.5.3.2.3
Add and .
Step 4.1.5.3.3
Move to the left of .
Step 4.1.5.3.4
Rewrite using the commutative property of multiplication.
Step 4.1.5.3.5
Move to the left of .
Step 4.1.5.4
Apply the distributive property.
Step 4.1.5.5
Simplify.
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Step 4.1.5.5.1
Rewrite using the commutative property of multiplication.
Step 4.1.5.5.2
Multiply by by adding the exponents.
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Step 4.1.5.5.2.1
Move .
Step 4.1.5.5.2.2
Multiply by .
Step 4.1.5.5.3
Rewrite using the commutative property of multiplication.
Step 4.1.5.5.4
Multiply by .
Step 4.1.5.6
Simplify each term.
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Step 4.1.5.6.1
Multiply by by adding the exponents.
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Step 4.1.5.6.1.1
Move .
Step 4.1.5.6.1.2
Multiply by .
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Step 4.1.5.6.1.2.1
Raise to the power of .
Step 4.1.5.6.1.2.2
Use the power rule to combine exponents.
Step 4.1.5.6.1.3
Add and .
Step 4.1.5.6.2
Multiply by .
Step 4.1.5.6.3
Rewrite using the commutative property of multiplication.
Step 4.1.5.6.4
Multiply by .
Step 4.1.5.6.5
Multiply by .
Step 4.1.5.7
Subtract from .
Step 4.1.5.8
Add and .
Step 4.1.5.9
Subtract from .
Step 4.1.5.10
Add and .
Step 4.1.5.11
Subtract from .
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Step 4.1.5.11.1
Move .
Step 4.1.5.11.2
Subtract from .
Step 4.1.5.12
Factor out of .
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Step 4.1.5.12.1
Factor out of .
Step 4.1.5.12.2
Factor out of .
Step 4.1.5.12.3
Factor out of .
Step 4.1.5.12.4
Factor out of .
Step 4.1.5.12.5
Factor out 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 4.6
Factor out of .
Step 4.7
Rewrite as .
Step 4.8
Factor out of .
Step 4.9
Factor out of .
Step 4.10
Factor out of .
Step 4.11
Simplify the expression.
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Step 4.11.1
Rewrite as .
Step 4.11.2
Move the negative in front of the fraction.
Step 5