Understanding Functions Using Calculus

This video tutorial explains how to analyze a function using calculus. It covers determining intervals where the function is increasing or decreasing, identifying concavity, and finding relative extrema and points of inflection. The process involves calculating the first and second derivatives, using test values to determine signs, and verifying results graphically.

What is the first step in analyzing a function using calculus?

• Identifying critical numbers

• Determining the domain

• Finding the second derivative

• Graphing the function

How do you find critical numbers of a function?

• By setting the second derivative to zero

• By finding where the first derivative is zero or undefined

• By solving the original function for zero

• By evaluating the function at endpoints

What does a positive first derivative indicate about a function on an interval?

• The function is constant

• The function is increasing

• The function is concave down

• The function is decreasing

What is a relative maximum?

• A point where the function changes from decreasing to increasing

• A point where the function changes from increasing to decreasing

• A point where the function is constant

• A point where the function is undefined

How do you determine concavity of a function?

• By evaluating the function at critical points

• By finding the first derivative

• By finding the second derivative

• By graphing the function

What does a negative second derivative indicate about a function's concavity?

• The function is decreasing

• The function is concave up

• The function is increasing

• The function is concave down

What method is used to find points of inflection when the second derivative is not factorable?

• Completing the square

• Using the quadratic formula

• Finding the first derivative

• Graphing the function

Why is it important to use exact values when finding points of inflection?

• To find the domain of the function

• To simplify the function

• To avoid rounding errors in calculations

• To ensure the function is continuous

What is verified by graphing the original function?

• The points of inflection and relative extrema

• The critical numbers

• The domain of the function

• The intervals of increase and decrease

What does a change in concavity at a point indicate?

• A relative minimum

• A relative maximum

• A critical number

• A point of inflection