Analyzing Derivatives and Concavity

In session six, Virj Cornelius and Mark Koralik tackle a non-calculator free response problem. They explore the application of the chain rule to find the derivative of a composite function, analyze concavity using the second derivative, and apply the fundamental theorem of calculus to solve an integration problem. The session concludes with a discussion on determining if a function is increasing or decreasing using the first derivative.

What is the primary rule used to find the derivative of a composite function like h(x) = f(g(x))?

• Product Rule

• Quotient Rule

• Chain Rule

• Power Rule

When calculating h'(7), what is the value of g(7) used in the process?

• 7

• 0

• 4

• 2

Which rule is applied in conjunction with the chain rule to find the second derivative of k?

• Quotient Rule

• Product Rule

• Power Rule

• Sum Rule

What is the concavity of the graph of k at x=4?

• Neither

• Concave down

• Concave up

• Cannot be determined

What is the significance of the second derivative being less than zero for the function k?

• The function is increasing

• The function is concave down

• The function is decreasing

• The function is concave up

What theorem is used to calculate the integral part of m(x) when finding m(2)?

• Mean Value Theorem

• Fundamental Theorem of Calculus

• Intermediate Value Theorem

• Rolle's Theorem

What is the value of m(2) after performing the necessary calculations?

• 45

• 42

• 40

• 37

Which derivative is used to determine if m is increasing or decreasing at x=2?

• Third Derivative

• Second Derivative

• Fourth Derivative

• First Derivative

What is the value of m'(2) that indicates whether m is increasing or decreasing?

• 68

• 60

• 52

• 75

At x=2, is the function m increasing, decreasing, or neither?

• Cannot be determined

• Decreasing

• Increasing

• Neither