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QuestionFind the average rate of change of f(x)=x3−6x+4f(x)=x^{3}-6 x+4 for the intervals: (a) -7 to -4, (b) -2 to 2, (c) 2 to 7.

Studdy Solution

STEP 1

Assumptions1. The function is f(x)=x3−6x+4f(x)=x^{3}-6x+4 . We need to find the average rate of change over the intervals -7 to -4, - to, and to7

STEP 2

The average rate of change of a function over an interval [a, b] is given by the formulaAverage rate of change=f(b)−f(a)b−aAverage\, rate\, of\, change = \frac{f(b) - f(a)}{b - a}

STEP 3

We first calculate the average rate of change over the interval -7 to -. Substitute a = -7 and b = - into the formula.
Average rate of change=f(−)−f(−7)−−(−7)Average\, rate\, of\, change = \frac{f(-) - f(-7)}{- - (-7)}

STEP 4

Evaluate the function at -4 and -7.
f(−4)=(−4)3−6(−4)+4f(-4) = (-4)^3 -6(-4) +4f(−7)=(−7)3−6(−7)+4f(-7) = (-7)^3 -6(-7) +4

STEP 5

Calculate the values of f(-4) and f(-7).
f(−4)=−64+24+4=−36f(-4) = -64 +24 +4 = -36f(−7)=−343+42+4=−297f(-7) = -343 +42 +4 = -297

STEP 6

Substitute these values back into the formula.
Average rate of change=−36−(−297)−4−(−)Average\, rate\, of\, change = \frac{-36 - (-297)}{-4 - (-)}

STEP 7

implify the expression to get the average rate of change from -7 to -4.
Average rate of change=2613=87Average\, rate\, of\, change = \frac{261}{3} =87

STEP 8

Repeat the same process for the interval -2 to2. Substitute a = -2 and b =2 into the formula.
Average rate of change=f(2)−f(−2)2−(−2)Average\, rate\, of\, change = \frac{f(2) - f(-2)}{2 - (-2)}

STEP 9

Evaluate the function at2 and -2.
f(2)=(2)3−6(2)+4f(2) = (2)^3 -6(2) +4f(−2)=(−2)3−6(−2)+4f(-2) = (-2)^3 -6(-2) +4

STEP 10

Calculate the values of f(2) and f(-2).
f(2)=8−12+4=0f(2) =8 -12 +4 =0f(−2)=−8+12+4=8f(-2) = -8 +12 +4 =8

STEP 11

Substitute these values back into the formula.
Average rate of change=0−8−(−)Average\, rate\, of\, change = \frac{0 -8}{ - (-)}

STEP 12

implify the expression to get the average rate of change from -2 to2.
Average rate of change=−84=−2Average\, rate\, of\, change = \frac{-8}{4} = -2

STEP 13

Repeat the same process for the interval2 to7. Substitute a =2 and b =7 into the formula.
Average rate of change=f(7)−f(2)7−2Average\, rate\, of\, change = \frac{f(7) - f(2)}{7 -2}

STEP 14

Evaluate the function at7 and2.
f(7)=(7)3−6(7)+4f(7) = (7)^3 -6(7) +4f(2)=(2)3−6(2)+4f(2) = (2)^3 -6(2) +4

STEP 15

Calculate the values of f(7) and f(2).
f(7)=343−42+4=305f(7) =343 -42 +4 =305f(2)=8−12+4=0f(2) =8 -12 +4 =0

STEP 16

Substitute these values back into the formula.
Average rate of change=305−0−2Average\, rate\, of\, change = \frac{305 -0}{ -2}

STEP 17

implify the expression to get the average rate of change from2 to7.
Average rate of change=3055=61Average\, rate\, of\, change = \frac{305}{5} =61The average rate of change of f(x)=x3−6x+4f(x)=x^{3}-6x+4 over the intervals are(a) From -7 to -4 is87(b) From -2 to2 is -2(c) From2 to7 is61

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