Understanding Derivatives Using Limits

The video tutorial explains how to calculate the derivative of the function f(x) = 2x^2 at x = 1 using the limit definition of a derivative. It walks through the process of setting up the problem, simplifying the difference quotient, and finding the limit as h approaches zero. The tutorial concludes with a graphical verification of the derivative, showing that the slope of the tangent line at x = 1 is 4.

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

• f(x) = x^2

• f(x) = 2x^2

• f(x) = 3x^2

• f(x) = 4x^2

What is the purpose of finding the derivative at x = 1?

• To calculate the area under the curve

• To determine the slope of the tangent line at x = 1

• To find the maximum value of the function

• To find the y-intercept of the function

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

• (f(1 + h) - f(1)) / h

• (f(x + h) - f(x)) / h

• (f(1 + h) + f(1)) / h

• (f(x + h) + f(x)) / h

What is the result of squaring the expression (1 + h)?

• 1 + h^2

• 1 + 2h + h^2

• 1 + h

• 1 + 3h + h^2

What happens to the numerator after distributing and simplifying?

• It becomes 2h + 2h^2

• It becomes 4h + h^2

• It becomes 2h + h^2

• It becomes 4h + 2h^2

How can the fraction be simplified further?

• By factoring out h

• By multiplying by h

• By subtracting a constant

• By adding a constant

What is the final value of the derivative at x = 1?

• 2

• 3

• 5

• 4

What is the graphical representation used to verify the derivative?

• The area under the curve

• The slope of the tangent line

• The y-intercept of the function

• The maximum point of the function

What are the coordinates of the point of tangency on the graph?

• (1, 2)

• (2, 6)

• (1, 1)

• (2, 4)

How many units do you move up and right to verify the slope of the tangent line?

• Up 5 units, right 1 unit

• Up 2 units, right 1 unit

• Up 4 units, right 1 unit

• Up 3 units, right 1 unit