Understanding Derivatives Using the Limit Definition

The video tutorial explains how to determine the derivative of a function using the limit definition. It covers the process of simplifying the difference quotient by multiplying with the conjugate, leading to the derivative of the function. The tutorial then evaluates the derivative at x=3 to find the slope of the tangent line and verifies the result graphically.

What is the purpose of finding the derivative of a function using the limit definition?

• To find the maximum value of the function

• To solve the function for x

• To determine the slope of the tangent line at a specific point

• To calculate the area under the curve

What is the initial expression used in the limit definition of the derivative?

• f(x) - f(x+h)

• f(x+h) - f(x)

• f(x) + f(x+h)

• f(x) * f(x+h)

Why do we multiply the numerator and denominator by the conjugate in the difference quotient?

• To change the function into a polynomial

• To find the maximum value of the function

• To simplify the expression and avoid division by zero

• To eliminate the variable x

What happens to the common factor of h in the numerator and denominator as h approaches zero?

• It becomes undefined

• It simplifies to zero

• It cancels out

• It doubles in value

What is the derivative of the function f(x) = √(x + 1) at x = 3?

• 1/5

• 1/3

• 1/4

• 1/2

What is the point of tangency for the function f(x) = √(x + 1) at x = 3?

• (3, 1)

• (3, 4)

• (3, 2)

• (3, 3)

What is the slope of the tangent line to the function at x = 3?

• 1/5

• 1/3

• 1/4

• 1/2

What is the graphical representation of the function f(x) = √(x + 1)?

• A cubic curve

• A square root curve

• A parabola

• A straight line

What does the derivative tell us about the function at a specific point?

• The maximum value of the function

• The slope of the tangent line

• The y-intercept of the function

• The x-intercept of the function

How can we verify the point of tangency graphically?

• By finding the x-intercept

• By calculating the area under the curve

• By checking the slope of the tangent line

• By plotting the derivative