Precalculus Examples

Find the Average Rate of Change f(x)=cot(x) , [pi/6,(3pi)/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
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
Simplify by cancelling.
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Step 3.3.1
Cancel the common factor of .
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Step 3.3.1.1
Factor out of .
Step 3.3.1.2
Cancel the common factor.
Step 3.3.1.3
Rewrite the expression.
Step 3.3.2
Multiply by .
Step 3.3.3
Cancel the common factor of .
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Step 3.3.3.1
Move the leading negative in into the numerator.
Step 3.3.3.2
Cancel the common factor.
Step 3.3.3.3
Rewrite the expression.
Step 3.4
Simplify the numerator.
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Step 3.4.1
Factor out of .
Step 3.4.2
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cotangent is negative in the fourth quadrant.
Step 3.4.3
The exact value of is .
Step 3.4.4
Multiply by .
Step 3.4.5
The exact value of is .
Step 3.4.6
Subtract from .
Step 3.4.7
Combine exponents.
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Step 3.4.7.1
Factor out negative.
Step 3.4.7.2
Multiply by .
Step 3.5
Simplify terms.
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Step 3.5.1
Subtract from .
Step 3.5.2
Cancel the common factor of and .
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Step 3.5.2.1
Factor out of .
Step 3.5.2.2
Cancel the common factors.
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Step 3.5.2.2.1
Factor out of .
Step 3.5.2.2.2
Cancel the common factor.
Step 3.5.2.2.3
Rewrite the expression.
Step 3.5.3
Move the negative in front of the fraction.
Step 4
The result can be shown in multiple forms.
Exact Form:
Decimal Form: