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Math  /  Calculus

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Given the function f(x)=x2−5x+0f(x)=x^{2}-5 x+0, determine the average rate of change of the function over the interval −3≤x≤9-3 \leq x \leq 9.
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Studdy Solution

STEP 1

1. The function given is f(x)=x2−5x f(x) = x^2 - 5x .
2. The average rate of change of a function over an interval [a,b][a, b] is given by the formula f(b)−f(a)b−a\frac{f(b) - f(a)}{b - a}.
3. We are asked to find this average rate of change over the interval [−3,9][-3, 9].

STEP 2

1. Evaluate the function at the endpoints of the interval.
2. Calculate the difference in function values.
3. Calculate the difference in the interval endpoints.
4. Use the average rate of change formula.

STEP 3

Evaluate the function at the endpoints of the interval. First, calculate f(−3) f(-3) :
f(−3)=(−3)2−5(−3) f(-3) = (-3)^2 - 5(-3) f(−3)=9+15 f(-3) = 9 + 15 f(−3)=24 f(-3) = 24
Next, calculate f(9) f(9) :
f(9)=(9)2−5(9) f(9) = (9)^2 - 5(9) f(9)=81−45 f(9) = 81 - 45 f(9)=36 f(9) = 36

STEP 4

Calculate the difference in function values:
f(9)−f(−3)=36−24=12 f(9) - f(-3) = 36 - 24 = 12

STEP 5

Calculate the difference in the interval endpoints:
9−(−3)=9+3=12 9 - (-3) = 9 + 3 = 12

STEP 6

Use the average rate of change formula:
Average Rate of Change=f(9)−f(−3)9−(−3)=1212=1\text{Average Rate of Change} = \frac{f(9) - f(-3)}{9 - (-3)} = \frac{12}{12} = 1
The average rate of change of the function over the interval [−3,9][-3, 9] is 1 \boxed{1} .

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