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Math  /  Calculus

QuestionFind the average rate of change of k(x)=−8x3k(x) = -8x^3 over the interval [−1,1][-1, 1].
Write your answer as an integer, fraction, or decimal rounded to the nearest tenth. Simplify any fractions.

Studdy Solution

STEP 1

1. We are given the function k(x)=−8x3 k(x) = -8x^3 .
2. We need to find the average rate of change over the interval [−1,1][-1, 1].

STEP 2

1. Calculate the value of k(x) k(x) at the endpoints of the interval.
2. Use the formula for the average rate of change.
3. Simplify the result to obtain the final answer.

STEP 3

Calculate k(−1) k(-1) :
k(−1)=−8(−1)3 k(-1) = -8(-1)^3 k(−1)=−8(−1) k(-1) = -8(-1) k(−1)=8 k(-1) = 8

STEP 4

Calculate k(1) k(1) :
k(1)=−8(1)3 k(1) = -8(1)^3 k(1)=−8(1) k(1) = -8(1) k(1)=−8 k(1) = -8

STEP 5

Use the formula for the average rate of change, which is:
Average rate of change=k(b)−k(a)b−a \text{Average rate of change} = \frac{k(b) - k(a)}{b - a}
where a=−1 a = -1 and b=1 b = 1 .

STEP 6

Substitute the values into the formula:
Average rate of change=k(1)−k(−1)1−(−1) \text{Average rate of change} = \frac{k(1) - k(-1)}{1 - (-1)} Average rate of change=−8−81+1 \text{Average rate of change} = \frac{-8 - 8}{1 + 1} Average rate of change=−162 \text{Average rate of change} = \frac{-16}{2}

STEP 7

Simplify the fraction:
Average rate of change=−8 \text{Average rate of change} = -8
The average rate of change of k(x)=−8x3 k(x) = -8x^3 over the interval [−1,1][-1, 1] is:
−8 \boxed{-8}

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