Understanding the Mean Value Theorem

The video tutorial explains how to determine the mean slope of the function f(x) = 1/x on the interval from 2 to 5 using the mean value theorem. It begins by introducing the function and the concept of mean slope, followed by a graphical representation. The tutorial then details the calculation of the mean slope and the derivation of the function to find the slope of the tangent line. Finally, it solves for the value of C that satisfies the mean value theorem, ensuring the solution lies within the specified interval.

What is the function f(x) that is being considered in the closed interval [2, 5]?

• f(x) = 1/x

• f(x) = x + 1

• f(x) = x^2

• f(x) = x - 1

According to the Mean Value Theorem, what must exist in the open interval (2, 5)?

• A point where the function is undefined

• A point where the function is zero

• A point where the derivative is zero

• A point where the derivative equals the average slope

How is the average slope between two points on a function calculated?

• By multiplying the function values at the two points

• By dividing the change in y by the change in x

• By dividing the change in x by the change in y

• By adding the function values at the two points

What is the average slope of the function f(x) = 1/x on the interval [2, 5]?

• -1/10

• -1/5

• 1/10

• 1/5

What is the derivative of the function f(x) = 1/x?

• 1/x^2

• -1/x^2

• x^2

• -x^2

What equation must be solved to find the value of c where the derivative equals the average slope?

• x^2 = 10

• x^2 = -10

• x = -10

• x = 10

What is the approximate value of c that satisfies the Mean Value Theorem for this function?

• 4.5

• 2.5

• 3.162

• 5

Why is the negative solution for c not considered in this problem?

• Because it is outside the interval [2, 5]

• Because it does not satisfy the function

• Because it is not a valid solution

• Because it is not a real number

What does the graph verify about the value of c?

• That the slope of the tangent line at c equals the average slope

• That c is outside the interval

• That c is not a solution

• That c is exactly 5

What is the significance of the Mean Value Theorem in this context?

• It indicates the function has a maximum

• It proves the function is differentiable

• It guarantees a point where the derivative equals the average slope

• It shows that the function is continuous