Find the Average Value of the Derivative y=25-x^2 , [-5,5]

Problem

Average Value of d(25−x2)/d(x)* on *[−5,5]

Solution

1. Find the derivative of the function y=25−x2 with respect to x

d(y)/d(x)=d(25−x2)/d(x)

d(y)/d(x)=−2*x

2. Identify the formula for the average value of a function ƒ(x) on the interval [a,b] which is 1/(b−a)*(∫_a^b)(ƒ(x)*d(x))

Average Value=1/(5−(−5))*(∫_−5^5)(−2*x*d(x))

3. Simplify the constant outside the integral.

1/(5−(−5))=1/10

4. Evaluate the definite integral of the derivative function.

(∫_−5^5)(−2*x*d(x))=[−x2]5(−5)

(∫_−5^5)(−2*x*d(x))=−(5)2−(−(−5)2)

(∫_−5^5)(−2*x*d(x))=−25−(−25)

(∫_−5^5)(−2*x*d(x))=0

5. Calculate the final average value by multiplying the integral result by the constant.

Average Value=1/10⋅0

Average Value=0

Final Answer

Average Value of d(25−x2)/d(x)* on *[−5,5]=0

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