Understanding Derivatives and Tangent Lines

The video tutorial explains how to estimate the derivative of a function at a specific point using a graph. It begins by identifying the tangent line at x=4 and calculating its slope using two methods: the slope formula and coordinate plane analysis. The tutorial concludes by interpreting the result, explaining that the derivative represents the rate of change of the function at that point.

What is the derivative of a function at a specific point also known as?

• The area under the curve

• The maximum value of the function

• The slope of the tangent line

• The average rate of change

What is the first step in estimating the derivative at x = 4?

• Determine the y-intercept

• Calculate the area under the curve

• Locate the point on the function

• Find the maximum value of the function

Which two points are used to find the slope of the tangent line?

• (4,3) and (5,1)

• (3,4) and (1,5)

• (2,2) and (3,3)

• (5,5) and (6,6)

How is the slope of the tangent line calculated using coordinates?

• By multiplying the x and y values

• By adding the x and y values

• By dividing the change in y by the change in x

• By dividing the change in x by the change in y

What is the change in y when moving from the point (4,3) to (5,1)?

• -2

• -1

• 2

• 1

What is the change in x when moving from the point (4,3) to (5,1)?

• 0

• -1

• 2

• 1

What is another method to find the slope of the tangent line besides using coordinates?

• Using the area under the curve

• Using the maximum value of the function

• Using the y-intercept

• Using the coordinate plane

What is the result of the slope calculation using the slope formula?

• 2

• 1

• -1

• -2

What does a derivative value of -2 at x = 4 indicate about the function?

• The function is increasing at a rate of 1 unit per 2 units change in x

• The function is decreasing at a rate of 2 units per 1 unit change in x

• The function is increasing at a rate of 2 units per 1 unit change in x

• The function is constant

What does the slope of the tangent line represent in terms of the function's behavior?

• The function's maximum value

• The function's minimum value

• The instantaneous rate of change

• The average rate of change