Calculus: Average Value of a Function

The video tutorial explains how to find the average value of a function over a given interval. It begins by introducing the concept of average value and the integral formula used to calculate it. The function is piecewise-defined, so the integral is split into two parts. The tutorial then demonstrates how to calculate the antiderivatives for each part and evaluate the definite integrals. Finally, the results are simplified to find the average value of the function over the interval.

What is the formula for finding the average value of a function over an interval?

• The product of the function values at the endpoints of the interval

• The integral of the function over the interval divided by the length of the interval

• The sum of the function values at the endpoints of the interval

• The derivative of the function at the midpoint of the interval

Why is the integral broken into two parts in this problem?

• To make the calculation easier

• Because the function is piecewise-defined

• To avoid complex numbers

• To simplify the derivative

What is the antiderivative of 1 with respect to x?

• x^2

• 1

• x

• 0

What is the antiderivative of e to the negative 4x?

• Negative 4e to the negative 4x

• e to the negative 4x

• 4e to the negative 4x

• Negative e to the negative 4x divided by 4

What rule is applied to find the antiderivative of e to the negative 4x?

• Power rule

• Chain rule

• Product rule

• Quotient rule

Tags

CCSS.HSF.TF.A.2

What is the value of cosine of 0?

• 1

• 0

• -1

• 2

What is the final expression for the average value of the function?

• 3 plus 2 cosine of negative 1 plus e to the negative 4 over 4

• 13/4 minus 2 cosine of negative 1 minus e to the negative 4 over 4

• 1/2 times 2 plus cosine of negative 1 plus e to the negative 4 over 4

• 2 minus cosine of negative 1 minus e to the negative 4 over 4

What is the result of evaluating the antiderivative at 0 for the function 1 minus 2 sine of x?

• 2

• 0

• 1

• -2

What is the value of e to the 0?

• 0

• 1

• -1

• e

Tags

CCSS.5.NF.A.1

CCSS.5.NF.A.2

What is the simplified form of 3 plus 1/4?

• 13/4

• 3.25

• 4

• 7/4

Tags

CCSS.5.NF.A.1

CCSS.5.NF.A.2