Understanding Tangent Lines and Derivatives

The video tutorial explains how to find the slope and equation of a tangent line to a parabola at a given point. It begins by introducing the problem and graphing the parabola. The derivative of the function is found using the limit definition, and the expression is simplified. The slope of the tangent line is calculated by evaluating the derivative at the given point. Using the slope and point, the equation of the tangent line is determined in both point-slope and slope-intercept forms. The solution is verified graphically, confirming the slope and y-intercept.

What is the initial function given in the problem?

• f(x) = x^2 + 2x

• f(x) = 2x + x^2

• f(x) = 2x - x^2

• f(x) = x^2 - 2x

What is the purpose of finding the derivative function in this problem?

• To find the y-intercept of the function

• To find the maximum value of the function

• To calculate the area under the curve

• To determine the slope of the tangent line

What is the expression for the difference quotient used in the limit definition of the derivative?

• (f(x+h) + f(x))/h

• (f(x) - f(x+h))/h

• (f(x) + f(x+h))/h

• (f(x+h) - f(x))/h

What is the derivative function f'(x) derived from the given function?

• f'(x) = 2 - 2x

• f'(x) = 2x + 2

• f'(x) = 2 + 2x

• f'(x) = 2x - 2

How do you find the slope of the tangent line at a specific point using the derivative function?

• Evaluate the derivative at the given x-coordinate

• Integrate the derivative function

• Evaluate the derivative at x = 0

• Find the second derivative

What is the slope of the tangent line at the point (-2, -8)?

• 4

• 5

• 6

• 7

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

• Standard form

• Slope-intercept form

• Point-slope form

• Vertex form

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

• y = 4x - 6

• y = 6x - 4

• y = 4x + 6

• y = 6x + 4

What is the y-intercept of the tangent line?

• 2

• 4

• 3

• 5

How can the slope of the tangent line be verified graphically?

• By moving up 6 units and right 1 unit from the point of tangency

• By observing the color of the line

• By checking the x-intercept

• By measuring the angle with the x-axis