Understanding the First Derivative Test

The video tutorial explains how to use the first derivative test to find relative extrema of a function. It begins by identifying critical numbers where the derivative is zero or undefined. The tutorial applies the quotient rule to find the derivative of a given function, then divides the domain into intervals to test the sign of the derivative. By analyzing these intervals, the video determines where the function is increasing or decreasing, leading to the identification of relative minimum and maximum points. The tutorial concludes by verifying these points graphically.

What is the first step in using the First Derivative Test to find relative extrema?

• Determine the second derivative of the function.

• Graph the function.

• Solve the function for x.

• Find the possible locations of relative extrema.

When applying the Quotient Rule, what do you need to identify first?

• The x-intercepts of the function.

• The numerator and denominator functions.

• The critical points of the function.

• The second derivative of the function.

How do you determine where a rational function's derivative is undefined?

• By finding where the function is increasing.

• By finding where the denominator is zero.

• By finding where the numerator is zero.

• By finding where the function is decreasing.

What is the significance of a critical number in the context of the First Derivative Test?

• It is where the function has a horizontal tangent.

• It is where the function is undefined.

• It is where the derivative is zero or undefined.

• It is where the function has a vertical tangent.

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

• By substituting a test value from the interval into the derivative.

• By graphing the function.

• By evaluating the second derivative.

• By solving the function for y.

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

• The function is constant.

• The function is increasing.

• The function is undefined.

• The function is decreasing.

What does a negative derivative indicate about the function on an interval?

• The function is increasing.

• The function is undefined.

• The function is decreasing.

• The function is constant.

What is the relative minimum point of the function in the example?

• (-1, -5/2)

• (2, 5)

• (1, 5/2)

• (0, 0)

What is the relative maximum point of the function in the example?

• (2, 5)

• (1, 5/2)

• (-1, -5/2)

• (0, 0)

How can you verify the relative extrema graphically?

• By plotting the original function.

• By plotting the derivative function.

• By plotting the second derivative.

• By plotting the inverse function.