Precalculus Examples

Find the Average Rate of Change f(x)=cos(x) , [-pi/3,pi/3]
,
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
Multiply the numerator and denominator of the fraction by .
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Step 3.1.1
Multiply by .
Step 3.1.2
Combine.
Step 3.2
Apply the distributive property.
Step 3.3
Cancel the common factor of .
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Step 3.3.1
Cancel the common factor.
Step 3.3.2
Rewrite the expression.
Step 3.4
Simplify the numerator.
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Step 3.4.1
Factor out of .
Step 3.4.2
The exact value of is .
Step 3.4.3
Add full rotations of until the angle is greater than or equal to and less than .
Step 3.4.4
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 3.4.5
The exact value of is .
Step 3.4.6
Combine the numerators over the common denominator.
Step 3.4.7
Subtract from .
Step 3.4.8
Divide by .
Step 3.5
Simplify the denominator.
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Step 3.5.1
Multiply .
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Step 3.5.1.1
Multiply by .
Step 3.5.1.2
Multiply by .
Step 3.5.2
Cancel the common factor of .
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Step 3.5.2.1
Cancel the common factor.
Step 3.5.2.2
Rewrite the expression.
Step 3.5.3
Add and .
Step 3.6
Reduce the expression by cancelling the common factors.
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Step 3.6.1
Multiply by .
Step 3.6.2
Cancel the common factor of and .
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Step 3.6.2.1
Factor out of .
Step 3.6.2.2
Cancel the common factors.
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Step 3.6.2.2.1
Factor out of .
Step 3.6.2.2.2
Cancel the common factor.
Step 3.6.2.2.3
Rewrite the expression.
Step 3.6.3
Divide by .