Linear Approximations and Derivatives

The video tutorial explains how to find the linear approximation of the function f(x) = sin(3x) at x = 0. It involves determining the point of tangency and the slope of the tangent line using derivatives. The linear approximation is derived using the point-slope form, resulting in the equation y = 3x. This approximation is then used to estimate the function value at x = 0.1, which is approximately 0.3. The tutorial concludes with a graphical representation of the function and its linear approximation.

What is the main goal of finding a linear approximation for a function?

• To estimate the function's value near a point

• To find the exact value of the function at a point

• To calculate the integral of the function

• To determine the maximum value of the function

What is the y-coordinate of the point of tangency for f(x) = sin(3x) at x = 0?

• 1

• sin(3)

• 3

• 0

Which mathematical rule is used to find the derivative of f(x) = sin(3x)?

• Chain Rule

• Quotient Rule

• Product Rule

• Power Rule

What is the slope of the tangent line to f(x) = sin(3x) at x = 0?

• 1

• 0

• 3

• sin(3)

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

• Slope-intercept form

• Standard form

• Point-slope form

• Vertex form

What is the linear approximation L(x) for f(x) = sin(3x) at x = 0?

• L(x) = x

• L(x) = sin(x)

• L(x) = 0

• L(x) = 3x

How is the linear approximation used to estimate f(0.1)?

• By substituting 0.1 into the linear approximation

• By finding the integral from 0 to 0.1

• By calculating the exact value of f(0.1)

• By using the derivative at x = 0.1

What is the estimated value of f(0.1) using the linear approximation?

• 0.3

• 0.1

• 1.0

• 0.5

What is the actual function value of f(0.1) compared to the approximation?

• Completely different

• Slightly more

• Slightly less

• Exactly equal

What does the graphical representation show about the linear approximation?

• It is irrelevant to the actual function

• It is identical to the actual function

• It is close to the actual function

• It is far from the actual function