Precalculus Examples

Find the Average Rate of Change f(x)=3x natural log of x ; [1,e^2]
;
Step 1
Write as an equation.
Step 2
Substitute using the average rate of change formula.
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Step 2.1
The average rate of change of a function can be found by calculating the change in values of the two points divided by the change in values of the two points.
Step 2.2
Substitute the equation for and , replacing in the function with the corresponding value.
Step 3
Simplify the expression.
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Step 3.1
Simplify the numerator.
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Step 3.1.1
Use logarithm rules to move out of the exponent.
Step 3.1.2
The natural logarithm of is .
Step 3.1.3
Multiply by .
Step 3.1.4
Multiply by .
Step 3.1.5
Multiply .
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Step 3.1.5.1
Multiply by .
Step 3.1.5.2
Multiply by .
Step 3.1.6
The natural logarithm of is .
Step 3.1.7
Multiply by .
Step 3.1.8
Add and .
Step 3.2
Simplify the denominator.
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Step 3.2.1
Rewrite as .
Step 3.2.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .