Understanding Horizontal Tangent Lines

The video tutorial explains how to determine the points at which a graph has horizontal tangent lines. It begins by defining horizontal tangents and their significance in calculus. The tutorial then demonstrates how to find these points by calculating the first derivative of a function and setting it to zero. Two example problems are solved to illustrate the process, including factoring and solving for x to find the points where the graph has horizontal tangents.

What is the characteristic of a tangent line compared to a secant line?

• A tangent line touches the curve at two points.

• A tangent line touches the curve at only one point.

• A tangent line is always vertical.

• A tangent line is parallel to the x-axis.

At what points do horizontal tangent lines occur on a graph?

• At points where the graph crosses the y-axis

• At points where the graph is undefined

• At maximum or minimum points

• At points of inflection

What is the first step in finding a horizontal tangent line for a function?

• Find the second derivative

• Graph the function

• Set the function equal to zero

• Find the first derivative and set it equal to zero

What is the derivative of the function x^3 - 6x^2 + 15?

• 3x^2 + 12x

• 3x^2 - 6x

• x^2 - 12x + 15

• 3x^2 - 12x

What are the x-values where the function x^3 - 6x^2 + 15 has horizontal tangent lines?

• x = 1 and x = 5

• x = 0 and x = 4

• x = -1 and x = 3

• x = 2 and x = 6

What is the y-value when x = 0 for the function x^3 - 6x^2 + 15?

• 0

• 15

• 30

• -15

What is the first derivative of the function 4x^3 - 7x^2 - 6x + 4?

• 12x^2 - 7x - 6

• 12x^2 - 14x + 4

• 12x^2 - 14x - 6

• 4x^2 - 14x + 4

What is the greatest common factor (GCF) taken out from the derivative 12x^2 - 14x - 6?

• 2

• 6

• 3

• 4

What are the x-values where the function 4x^3 - 7x^2 - 6x + 4 has horizontal tangent lines?

• x = 0 and x = 4

• x = 3/2 and x = -1/3

• x = 1 and x = -2

• x = 2 and x = -1

What is the process called when you replace a term in a trinomial to factor it?

• Expansion

• Elimination

• Grouping

• Substitution