Understanding Horizontal Tangent Lines

The video tutorial explains how to find points on a function where the tangent line is horizontal. It begins by discussing the concept of horizontal tangents and their slopes, which are zero. The tutorial then demonstrates how to find the derivative of a function to determine the slopes of tangent lines. By setting the derivative equal to zero, the x-coordinates of horizontal tangents are found. The tutorial continues by substituting these x-values back into the original function to find the corresponding y-coordinates, resulting in the points of tangency. Finally, the video verifies these points graphically, showing horizontal tangent lines at the origin and another point.

What is the condition for a tangent line to be horizontal?

• The slope of the tangent line is positive.

• The slope of the tangent line is negative.

• The slope of the tangent line is zero.

• The slope of the tangent line is undefined.

What does the derivative of a function represent?

• The maximum value of the function.

• The minimum value of the function.

• The slope of the tangent line at any point on the function.

• The y-intercept of the function.

How do you find the x-coordinates where the tangent line is horizontal?

• Set the second derivative of the function equal to zero.

• Set the derivative of the function equal to zero.

• Set the function's y-intercept equal to zero.

• Set the original function equal to zero.

What is the derivative of -3x^3?

• -6x^2

• -9x^2

• -3x^2

• -3x

What is the greatest common factor used to factor the equation -3x^2 + 6x = 0?

• 3

• -3

• 3x

• -3x

What are the x-coordinates where the tangent line is horizontal?

• x = -1 and x = 1

• x = 2 and x = 4

• x = 1 and x = 3

• x = 0 and x = 2

How do you find the y-coordinates of the points where the tangent line is horizontal?

• By solving the derivative equation.

• By substituting the x-values into the derivative function.

• By substituting the x-values into the original function.

• By setting the y-values to zero.

What is the y-coordinate of the point when x = 0?

• 4

• 2

• -4

• 0

What is the y-coordinate of the point when x = 2?

• 0

• 4

• 2

• -4

What is the equation of the horizontal tangent line at the point (2,4)?

• y = 0

• y = 2

• y = 4

• y = -4