Analyzing Functions and Derivatives

This video is an AP review focusing on graph analysis, covering key vocabulary, first and second derivatives, critical points, concavity, and extrema. It explains the fundamental theorem of calculus and provides strategies for tackling typical AP questions.

What is a critical point on a graph?

• A point where the function is undefined

• A point where the first derivative is zero or does not exist

• A point where the function has a maximum value

• A point where the second derivative is zero

How do you justify that a function is increasing on an interval?

• By showing that the first derivative is greater than or equal to zero

• By showing that the second derivative is greater than zero

• By showing that the first derivative is less than zero

• By showing that the function is concave up

What does the second derivative tell us about a graph?

• It tells us where the function is increasing or decreasing

• It tells us the absolute maximum and minimum values

• It tells us the critical points of the function

• It tells us the concavity of the function

What is a point of inflection?

• A point where the function is undefined

• A point where the function has a maximum value

• A point where the second derivative changes signs

• A point where the first derivative is zero

How do you determine if a function is concave up?

• By showing that the function has a maximum value

• By showing that the first derivative is positive

• By showing that the second derivative is greater than zero

• By showing that the first derivative is zero

What is required for a relative maximum to occur?

• The function must be concave up

• The first derivative must be zero

• The second derivative must be zero

• The first derivative must change from positive to negative

How do you find the absolute maximum of a function?

• By finding the points where the second derivative is zero

• By considering both critical points and endpoints

• By finding the points where the function is concave up

• By finding the points where the first derivative is zero

What does the Fundamental Theorem of Calculus state?

• The second derivative is the slope of the first derivative

• Derivatives and integrals are opposites

• The first derivative is the slope of the second derivative

• Derivatives and integrals are the same

How do you find the second derivative of an accumulation function?

• By taking the derivative of the first derivative

• By taking the integral of the accumulation function twice

• By taking the integral of the first derivative

• By taking the derivative of the accumulation function twice

When given the graph of the derivative function, what is the first step in analyzing it?

• Identify the critical points

• Find the absolute maximum and minimum

• Label the graph as the derivative function

• Determine the concavity