Precalculus Examples

Find the Average Rate of Change f(x)=cot(x) , [pi/3,pi/2]
,
Step 1
Write as an equation.
Step 2
Substitute using the average rate of change formula.
Tap for more steps...
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.
Tap for more steps...
Step 3.1
Multiply the numerator and denominator of the fraction by .
Tap for more steps...
Step 3.1.1
Multiply by .
Step 3.1.2
Combine.
Step 3.2
Apply the distributive property.
Step 3.3
Simplify by cancelling.
Tap for more steps...
Step 3.3.1
Cancel the common factor of .
Tap for more steps...
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
Cancel the common factor of .
Tap for more steps...
Step 3.3.2.1
Move the leading negative in into the numerator.
Step 3.3.2.2
Factor out of .
Step 3.3.2.3
Cancel the common factor.
Step 3.3.2.4
Rewrite the expression.
Step 3.3.3
Multiply by .
Step 3.4
Simplify the numerator.
Tap for more steps...
Step 3.4.1
Factor out of .
Step 3.4.2
The exact value of is .
Step 3.4.3
The exact value of is .
Step 3.4.4
Multiply by .
Step 3.4.5
Combine and simplify the denominator.
Tap for more steps...
Step 3.4.5.1
Multiply by .
Step 3.4.5.2
Raise to the power of .
Step 3.4.5.3
Raise to the power of .
Step 3.4.5.4
Use the power rule to combine exponents.
Step 3.4.5.5
Add and .
Step 3.4.5.6
Rewrite as .
Tap for more steps...
Step 3.4.5.6.1
Use to rewrite as .
Step 3.4.5.6.2
Apply the power rule and multiply exponents, .
Step 3.4.5.6.3
Combine and .
Step 3.4.5.6.4
Cancel the common factor of .
Tap for more steps...
Step 3.4.5.6.4.1
Cancel the common factor.
Step 3.4.5.6.4.2
Rewrite the expression.
Step 3.4.5.6.5
Evaluate the exponent.
Step 3.4.6
Subtract from .
Step 3.4.7
Combine exponents.
Tap for more steps...
Step 3.4.7.1
Factor out negative.
Step 3.4.7.2
Combine and .
Step 3.4.8
Cancel the common factor of and .
Tap for more steps...
Step 3.4.8.1
Factor out of .
Step 3.4.8.2
Cancel the common factors.
Tap for more steps...
Step 3.4.8.2.1
Factor out of .
Step 3.4.8.2.2
Cancel the common factor.
Step 3.4.8.2.3
Rewrite the expression.
Step 3.4.8.2.4
Divide by .
Step 3.4.9
Multiply by .
Step 3.5
Simplify by adding terms.
Tap for more steps...
Step 3.5.1
Subtract from .
Step 3.5.2
Move the negative in front of the fraction.
Step 4
The result can be shown in multiple forms.
Exact Form:
Decimal Form: