Understanding the First Derivative Test

What does a change in the sign of the first derivative indicate?

• A discontinuity in the function

• A change in the function's domain

• A local maximum or minimum

• A point of inflection

What is a critical point in the context of the first derivative test?

• A point where the second derivative is zero

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

• A point where the function has a vertical tangent

• A point where the function is undefined

What happens at a left endpoint if the first derivative is negative?

• There is no local extrema

• The function is undefined

• There is a local minimum

• There is a local maximum

How do you determine the sign of the derivative in a given interval?

• By evaluating the second derivative

• By graphing the function

• By checking the sign of the derivative at a test point within the interval

• By finding the limit of the function

What is the relative maximum value found in the example with critical points -3 and 3?

• 4 at x = -2

• 57 at x = -3

• 51 at x = 3

• 0 at x = 0

In the second example, what is the relative minimum value?

• 4 * e^-2 at x = -2

• 0 at x = 0

• 51 at x = 3

• 57 at x = -3

What does the limit as x approaches negative infinity tell us about the function?

• The function has a vertical asymptote

• The function is undefined

• The function approaches zero but never equals it

• The function has an absolute maximum

What is the velocity of a particle in terms of derivatives?

• The second derivative of position

• The integral of acceleration

• The first derivative of position

• The derivative of acceleration

At what time does the particle change direction?

• At t = 0

• At t = 4/3

• At t = 1

• At t = 2

Where does the particle start its motion?

• At position 4

• At position 3

• At position 0

• At position 1