Precalculus Examples

Find the Average Rate of Change f(x)=3x^2-x^3 , (-1,4)
f(x)=3x2-x3 , (-1,4)
Step 1
Write f(x)=3x2-x3 as an equation.
y=3x2-x3
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 y values of the two points divided by the change in x values of the two points.
f(4)-f(-1)(4)-(-1)
Step 2.2
Substitute the equation y=3x2-x3 for f(4) and f(-1), replacing x in the function with the corresponding x value.
(3(4)2-(4)3)-(3(-1)2-(-1)3)(4)-(-1)
(3(4)2-(4)3)-(3(-1)2-(-1)3)(4)-(-1)
Step 3
Simplify the expression.
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Step 3.1
Simplify the numerator.
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Step 3.1.1
Raise 4 to the power of 2.
316-43-(3(-1)2-(-1)3)4-(-1)
Step 3.1.2
Multiply 3 by 16.
48-43-(3(-1)2-(-1)3)4-(-1)
Step 3.1.3
Raise 4 to the power of 3.
48-164-(3(-1)2-(-1)3)4-(-1)
Step 3.1.4
Multiply -1 by 64.
48-64-(3(-1)2-(-1)3)4-(-1)
Step 3.1.5
Simplify each term.
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Step 3.1.5.1
Raise -1 to the power of 2.
48-64-(31-(-1)3)4-(-1)
Step 3.1.5.2
Multiply 3 by 1.
48-64-(3-(-1)3)4-(-1)
Step 3.1.5.3
Multiply -1 by (-1)3 by adding the exponents.
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Step 3.1.5.3.1
Multiply -1 by (-1)3.
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Step 3.1.5.3.1.1
Raise -1 to the power of 1.
48-64-(3+(-1)1(-1)3)4-(-1)
Step 3.1.5.3.1.2
Use the power rule aman=am+n to combine exponents.
48-64-(3+(-1)1+3)4-(-1)
48-64-(3+(-1)1+3)4-(-1)
Step 3.1.5.3.2
Add 1 and 3.
48-64-(3+(-1)4)4-(-1)
48-64-(3+(-1)4)4-(-1)
Step 3.1.5.4
Raise -1 to the power of 4.
48-64-(3+1)4-(-1)
48-64-(3+1)4-(-1)
Step 3.1.6
Add 3 and 1.
48-64-144-(-1)
Step 3.1.7
Multiply -1 by 4.
48-64-44-(-1)
Step 3.1.8
Subtract 64 from 48.
-16-44-(-1)
Step 3.1.9
Subtract 4 from -16.
-204-(-1)
-204-(-1)
Step 3.2
Simplify the denominator.
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Step 3.2.1
Multiply -1 by -1.
-204+1
Step 3.2.2
Add 4 and 1.
-205
-205
Step 3.3
Divide -20 by 5.
-4
-4
f(x)=3x2-x3,(-1,4)
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