Tangent Lines and Derivatives

This video tutorial covers differentiation techniques, including the constant rule, power rule, constant multiple rule, and the sum of difference rule. It aims to teach how to find the equation of a tangent line and identify points on a function with a horizontal tangent line. The video provides examples, such as finding the tangent line equation at a specific point and determining points where the tangent line is horizontal. The tutorial also emphasizes verifying results using graphs and highlights the relationship between derivatives and relative extrema.

Which of the following rules is NOT covered in the introduction to differentiation techniques?

• Sum of Difference Rule

• Power Rule

• Product Rule

• Constant Rule

What is the first step in finding the equation of the tangent line for the function y = x^2 - √x?

• Graph the function

• Find the derivative directly

• Rewrite the function with rational exponents

• Use the constant rule

What is the slope of the tangent line at the point (4,14) for the function y = x^2 - √x?

• 7.75

• 8.25

• 6.5

• 9.0

Which form is used to find the equation of the tangent line after determining the slope?

• Slope-intercept form

• Standard form

• Point-slope form

• Quadratic form

What is the equation of the tangent line at the point (4,14) for the function y = x^2 - √x?

• y = 7.75x + 14

• y = 7.75x - 17

• y = 7.5x - 17

• y = 8x - 14

What is the slope of a horizontal tangent line?

• 0

• 1

• -1

• Undefined

What must be true for the derivative of a function at points where the tangent line is horizontal?

• Derivative is positive

• Derivative is negative

• Derivative is zero

• Derivative is undefined

What are the x-coordinates of the points where the tangent line is horizontal for the given function?

• x = 11 and x = 1

• x = 3 and x = 7

• x = 0 and x = 5

• x = 2 and x = 8

What is the y-coordinate of the point where x = 11 for the function with a horizontal tangent line?

• 111.3

• 120.7

• 55.3

• 100.5

Where do relative extrema occur in relation to the derivative of a function?

• Where the derivative is zero

• Where the derivative is positive

• Where the derivative is undefined

• Where the derivative is negative