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QuestionFind the average rate of change of f(x)=−2x3+4x+7f(x)=-2 x^{3}+4 x+7 on [−1,3][-1,3]. Is it even, odd, or neither?

Studdy Solution

STEP 1

Assumptions1. The function is f(x)=−x3+4x+7f(x)=- x^{3}+4 x+7 . The interval is [−1,3][-1,3]
3. The average rate of change of a function on an interval [a,b][a,b] is given by the formula f(b)−f(a)b−a\frac{f(b) - f(a)}{b - a}
4. A function is even if f(−x)=f(x)f(-x) = f(x) for all xx in the function's domain5. A function is odd if f(−x)=−f(x)f(-x) = -f(x) for all xx in the function's domain6. If a function is neither even nor odd, then it does not satisfy either of the above conditions

STEP 2

First, we need to find the average rate of change for the function on the interval [−1,][-1,]. We can do this by plugging the values of −1-1 and $$ into the function and then applying the formula for the average rate of change.
Average rate of change=f()−f(−1)−(−1)Average\, rate\, of\, change = \frac{f() - f(-1)}{ - (-1)}

STEP 3

Now, plug in the values of −1-1 and 33 into the function f(x)f(x).
f(3)=−2(3)3+(3)+7f(3) = -2(3)^{3}+(3)+7f(−1)=−2(−1)3+(−1)+7f(-1) = -2(-1)^{3}+(-1)+7

STEP 4

Calculate the values of f(3)f(3) and f(−1)f(-1).
f(3)=−2(27)+12+7=−54+12+7=−35f(3) = -2(27)+12+7 = -54+12+7 = -35f(−1)=−2(−1)+4(−1)+7=2−4+7=f(-1) = -2(-1)+4(-1)+7 =2-4+7 =

STEP 5

Plug in the values of f(3)f(3) and f(−1)f(-1) into the formula for the average rate of change.
Average rate of change=−35−53−(−1)Average\, rate\, of\, change = \frac{-35 -5}{3 - (-1)}

STEP 6

Calculate the average rate of change.
Average rate of change=−404=−10Average\, rate\, of\, change = \frac{-40}{4} = -10The average rate of change for the function on the interval [−1,3][-1,3] is −10-10.

STEP 7

Next, we need to determine whether the function is even, odd or neither. We can do this by plugging −x-x into the function and comparing the result with f(x)f(x) and −f(x)-f(x).
f(−x)=−2(−x)3+4(−x)+7f(-x) = -2(-x)^{3}+4(-x)+7

STEP 8

implify the expression for f(−x)f(-x).
f(−x)=−2x3−4x+7f(-x) = -2x^{3}-4x+7

STEP 9

Compare f(−x)f(-x) with f(x)f(x) and −f(x)-f(x).
f(−x)f(-x) is not equal to f(x)f(x), so the function is not even. f(−x)f(-x) is not equal to −f(x)-f(x), so the function is not odd.
Therefore, the function is neither even nor odd.

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