Precalculus Examples

Find the Average Rate of Change p(t)=((t^2-4)(t+1))/(t^2+3) on [-3,1]
p(t)=(t2-4)(t+1)t2+3 on [-3,1]
Step 1
Write p(t)=(t2-4)(t+1)t2+3 as an equation.
y=(t2-4)(t+1)t2+3
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 t values of the two points.
f(1)-f(-3)(1)-(-3)
Step 2.2
Substitute the equation y=(t2-4)(t+1)t2+3 for f(1) and f(-3), replacing t in the function with the corresponding t value.
(((1)2-4)((1)+1)(1)2+3)-(((-3)2-4)((-3)+1)(-3)2+3)(1)-(-3)
(((1)2-4)((1)+1)(1)2+3)-(((-3)2-4)((-3)+1)(-3)2+3)(1)-(-3)
Step 3
Simplify the expression.
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Step 3.1
Multiply the numerator and denominator of the fraction by (12+3)((-3)2+3).
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Step 3.1.1
Multiply (12-4)(1+1)12+3-((-3)2-4)(-3+1)(-3)2+31-(-3) by (12+3)((-3)2+3)(12+3)((-3)2+3).
(12+3)((-3)2+3)(12+3)((-3)2+3)(12-4)(1+1)12+3-((-3)2-4)(-3+1)(-3)2+31-(-3)
Step 3.1.2
Combine.
(12+3)((-3)2+3)((12-4)(1+1)12+3-((-3)2-4)(-3+1)(-3)2+3)(12+3)((-3)2+3)(1-(-3))
(12+3)((-3)2+3)((12-4)(1+1)12+3-((-3)2-4)(-3+1)(-3)2+3)(12+3)((-3)2+3)(1-(-3))
Step 3.2
Apply the distributive property.
(12+3)((-3)2+3)(12-4)(1+1)12+3+(12+3)((-3)2+3)(-((-3)2-4)(-3+1)(-3)2+3)(12+3)((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
Step 3.3
Simplify by cancelling.
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Step 3.3.1
Cancel the common factor of 12+3.
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Step 3.3.1.1
Cancel the common factor.
(12+3)((-3)2+3)(12-4)(1+1)12+3+(12+3)((-3)2+3)(-((-3)2-4)(-3+1)(-3)2+3)(12+3)((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
Step 3.3.1.2
Rewrite the expression.
((-3)2+3)((12-4)(1+1))+(12+3)((-3)2+3)(-((-3)2-4)(-3+1)(-3)2+3)(12+3)((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
((-3)2+3)((12-4)(1+1))+(12+3)((-3)2+3)(-((-3)2-4)(-3+1)(-3)2+3)(12+3)((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
Step 3.3.2
Cancel the common factor of (-3)2+3.
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Step 3.3.2.1
Move the leading negative in -((-3)2-4)(-3+1)(-3)2+3 into the numerator.
((-3)2+3)((12-4)(1+1))+(12+3)((-3)2+3)-((-3)2-4)(-3+1)(-3)2+3(12+3)((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
Step 3.3.2.2
Factor (-3)2+3 out of (12+3)((-3)2+3).
((-3)2+3)((12-4)(1+1))+((-3)2+3)(12+3)-((-3)2-4)(-3+1)(-3)2+3(12+3)((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
Step 3.3.2.3
Cancel the common factor.
((-3)2+3)((12-4)(1+1))+((-3)2+3)(12+3)-((-3)2-4)(-3+1)(-3)2+3(12+3)((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
Step 3.3.2.4
Rewrite the expression.
((-3)2+3)((12-4)(1+1))+(12+3)(-((-3)2-4)(-3+1))(12+3)((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
((-3)2+3)((12-4)(1+1))+(12+3)(-((-3)2-4)(-3+1))(12+3)((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
((-3)2+3)((12-4)(1+1))+(12+3)(-((-3)2-4)(-3+1))(12+3)((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
Step 3.4
Simplify the numerator.
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Step 3.4.1
Raise -3 to the power of 2.
(9+3)(12-4)(1+1)+(12+3)(-((-3)2-4)(-3+1))(12+3)((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
Step 3.4.2
Add 9 and 3.
12(12-4)(1+1)+(12+3)(-((-3)2-4)(-3+1))(12+3)((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
Step 3.4.3
One to any power is one.
12(1-4)(1+1)+(12+3)(-((-3)2-4)(-3+1))(12+3)((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
Step 3.4.4
Subtract 4 from 1.
12-3(1+1)+(12+3)(-((-3)2-4)(-3+1))(12+3)((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
Step 3.4.5
Multiply 12 by -3.
-36(1+1)+(12+3)(-((-3)2-4)(-3+1))(12+3)((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
Step 3.4.6
Add 1 and 1.
-362+(12+3)(-((-3)2-4)(-3+1))(12+3)((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
Step 3.4.7
Multiply -36 by 2.
-72+(12+3)(-((-3)2-4)(-3+1))(12+3)((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
Step 3.4.8
One to any power is one.
-72+(1+3)(-((-3)2-4)(-3+1))(12+3)((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
Step 3.4.9
Add 1 and 3.
-72+4(-((-3)2-4)(-3+1))(12+3)((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
Step 3.4.10
Raise -3 to the power of 2.
-72+4(-(9-4)(-3+1))(12+3)((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
Step 3.4.11
Subtract 4 from 9.
-72+4(-15(-3+1))(12+3)((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
Step 3.4.12
Multiply -1 by 5.
-72+4(-5(-3+1))(12+3)((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
Step 3.4.13
Add -3 and 1.
-72+4(-5-2)(12+3)((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
Step 3.4.14
Multiply 4(-5-2).
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Step 3.4.14.1
Multiply -5 by -2.
-72+410(12+3)((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
Step 3.4.14.2
Multiply 4 by 10.
-72+40(12+3)((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
-72+40(12+3)((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
Step 3.4.15
Add -72 and 40.
-32(12+3)((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
-32(12+3)((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
Step 3.5
Simplify the denominator.
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Step 3.5.1
One to any power is one.
-32(1+3)((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
Step 3.5.2
Add 1 and 3.
-324((-3)2+3)1+(12+3)((-3)2+3)(-(-3))
Step 3.5.3
Raise -3 to the power of 2.
-324(9+3)1+(12+3)((-3)2+3)(-(-3))
Step 3.5.4
Add 9 and 3.
-324121+(12+3)((-3)2+3)(-(-3))
Step 3.5.5
Multiply 4121.
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Step 3.5.5.1
Multiply 4 by 12.
-32481+(12+3)((-3)2+3)(-(-3))
Step 3.5.5.2
Multiply 48 by 1.
-3248+(12+3)((-3)2+3)(-(-3))
-3248+(12+3)((-3)2+3)(-(-3))
Step 3.5.6
One to any power is one.
-3248+(1+3)((-3)2+3)(-(-3))
Step 3.5.7
Add 1 and 3.
-3248+4((-3)2+3)(-(-3))
Step 3.5.8
Raise -3 to the power of 2.
-3248+4(9+3)(-(-3))
Step 3.5.9
Add 9 and 3.
-3248+412(-(-3))
Step 3.5.10
Multiply 412(-(-3)).
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Step 3.5.10.1
Multiply 4 by 12.
-3248+48(-(-3))
Step 3.5.10.2
Multiply -1 by -3.
-3248+483
Step 3.5.10.3
Multiply 48 by 3.
-3248+144
-3248+144
Step 3.5.11
Add 48 and 144.
-32192
-32192
Step 3.6
Reduce the expression by cancelling the common factors.
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Step 3.6.1
Cancel the common factor of -32 and 192.
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Step 3.6.1.1
Factor 32 out of -32.
32(-1)192
Step 3.6.1.2
Cancel the common factors.
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Step 3.6.1.2.1
Factor 32 out of 192.
32-1326
Step 3.6.1.2.2
Cancel the common factor.
32-1326
Step 3.6.1.2.3
Rewrite the expression.
-16
-16
-16
Step 3.6.2
Move the negative in front of the fraction.
-16
-16
-16
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