Tangent Lines and Derivatives

This video tutorial explains how to find the equation of a tangent line at a specific point on a function. It begins by identifying the point of tangency, which involves evaluating the function at the given x-coordinate. Next, the slope of the tangent line is determined by calculating the derivative of the function and evaluating it at the same x-coordinate. With the point and slope known, the point-slope form is used to derive the equation of the tangent line. The tutorial concludes with a graphical verification of the solution.

How do we find the y-coordinate of the point of tangency?

• By evaluating the function at the given x-coordinate

• By using the y-intercept

• By guessing

• By using the slope

What is the y-coordinate of the point of tangency when x = -1?

• 3

• 1

• -3

• 0

What do we need to determine the equation of a tangent line?

• Neither the slope nor a point on the line

• Both the slope and a point on the line

• Only a point on the line

• Only the slope of the line

What is the x-coordinate of the point of tangency in this problem?

• 0

• 1

• -1

• 2

What is the derivative of the function 3x^3 - 3x?

• 3x^2 - 3

• 9x^2 - 3

• 9x^3 - 3

• 3x^3 - 3

What is the slope of the tangent line at x = -1?

• 9

• 3

• 6

• 0

Which form of a line equation is used to determine the equation of the tangent line?

• Slope-intercept form

• Standard form

• Point-slope form

• General form

What is the equation of the tangent line in slope-intercept form?

• y = 6x - 1

• y = 6x + 6

• y = 6x - 6

• y = 6x + 1

What is the y-intercept of the tangent line?

• 6

• 1

• -1

• 0

What is the slope of the tangent line as verified graphically?

• 3

• 6

• 9

• 0