Understanding Derivatives Using Limits

The video tutorial explains how to calculate the derivative of the function f(x) = 4/x at x = 2 using the limit definition of a derivative. It involves substituting values into the limit definition, simplifying the expression, and determining the derivative value. The tutorial concludes with a graphical verification of the derivative by examining the slope of the tangent line at the point x = 2 on the graph of the function.

What is the function f(x) that we are analyzing in this tutorial?

• f(x) = 4x

• f(x) = 4/x

• f(x) = x/4

• f(x) = x^4

What is the purpose of finding the derivative at x = 2?

• To calculate the area under the curve

• To find the maximum value of the function

• To determine the slope of the tangent line at x = 2

• To find the y-intercept of the function

In the limit definition of the derivative, what does the expression 'H approaches zero' signify?

• H is increasing indefinitely

• H is decreasing indefinitely

• H is getting closer to zero

• H is equal to zero

What is the common denominator used to simplify the difference quotient?

• H

• 2

• 2 + H

• 4

How is the complex fraction simplified before finding the limit?

• By adding H to both sides

• By multiplying by the reciprocal of H

• By subtracting H from both sides

• By dividing by 2

What is the final value of the limit as H approaches zero?

• 2

• 1

• 0

• -2

What does the derivative value of 1 represent in the context of the function?

• The slope of the tangent line at x = 2

• The minimum value of the function

• The y-intercept of the function

• The maximum value of the function

How is the slope of the tangent line verified graphically?

• By calculating the area under the curve

• By finding the y-intercept

• By comparing with another function

• By selecting two points on the tangent line

What is the coordinate of the point on the function when x = 2?

• (2, 4)

• (4, 2)

• (2, 2)

• (1, 2)

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

• 1

• 0

• -1

• 2