Understanding Tangent Lines and Derivatives

The video tutorial explains how to find the slope and equation of a tangent line at a given point for the function f(x) = 2x - x^3. It begins by introducing the problem and using the limit definition of the derivative to find the derivative function. The tutorial then simplifies the derivative and calculates the slope at the point (-2, 4). Finally, it derives the equation of the tangent line using point-slope form and visualizes the result graphically.

What is the initial function given in the problem?

• f(x) = 2x - x^3

• f(x) = 2x + x^3

• f(x) = x^2 - 2x

• f(x) = x^3 - 2x

What is the purpose of using the limit definition of the derivative?

• To find the maximum value of the function

• To determine the slope of the tangent line

• To solve the function for x

• To calculate the area under the curve

What is the result of cubing the binomial (x + h) using Pascal's Triangle?

• x^3 + 3x^2h + 3xh^2 + h^3

• x^3 + 3x^2h + 2xh^2 + h^3

• x^3 + 2x^2h + 3xh^2 + h^3

• x^3 + 2x^2h + 2xh^2 + h^3

Why do we factor out h from the numerator in the simplification process?

• To calculate the derivative directly

• To eliminate the constant term

• To simplify the expression and avoid division by zero

• To find the maximum value of the function

What is the derivative function f'(x) obtained after simplification?

• f'(x) = 3x^2 + 2

• f'(x) = 2 - 3x^2

• f'(x) = 2 + 3x^2

• f'(x) = 3x^2 - 2

How do you calculate the slope of the tangent line at x = -2?

• By using the point-slope form

• By finding the average rate of change

• By substituting x = -2 into the derivative function

• By substituting x = -2 into the original function

What is the slope of the tangent line at the point (-2, 4)?

• -10

• -4

• 4

• 10

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

• Slope-intercept form

• Point-slope form

• Standard form

• Quadratic form

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

• y = 10x - 16

• y = -10x - 16

• y = -10x + 16

• y = 10x + 16

What is the y-intercept of the tangent line?

• -10

• 16

• 10

• -16