Tangent Plane and Linear Approximation

The video tutorial explains how to use a tangent line to approximate the value of a function at a specific point. It begins by introducing the concept of tangent line approximation, also known as linear approximation. The tutorial then discusses the selection of a convenient point near the given point to simplify calculations. It proceeds to calculate the first-order partial derivatives of the function and evaluates these derivatives at the convenient point. Finally, the video demonstrates how to use these evaluations to calculate the tangent line approximation, concluding with the approximate value of the function at the desired point.

What is the function that needs to be approximated using a tangent line?

• f(x, y) = sqrt(14 - x^2 - y^2)

• f(x, y) = x^2 + y^2

• f(x, y) = x^2 - y^2

• f(x, y) = 14 - x^2 - y^2

Why is a convenient point used for tangent line approximation?

• To change the function

• To avoid using derivatives

• To increase accuracy

• To simplify calculations

What is the formula for the linear approximation L(x, y)?

• L(x, y) = f(a, b) * (df/dx)(x-a) * (df/dy)(y-b)

• L(x, y) = f(a, b) / (df/dx)(x-a) / (df/dy)(y-b)

• L(x, y) = f(a, b) + (df/dx)(x-a) + (df/dy)(y-b)

• L(x, y) = f(a, b) - (df/dx)(x-a) - (df/dy)(y-b)

How is the partial derivative of f with respect to x calculated?

• By differentiating with respect to x while treating y as a constant

• By differentiating with respect to y while treating x as a constant

• By integrating with respect to x

• By integrating with respect to y

What is the value of the partial derivative of f with respect to x at the point (1, 2)?

• -2/3

• -1/3

• 1/3

• 2/3

What is the value of the partial derivative of f with respect to y at the point (1, 2)?

• -2/3

• 2/3

• -1/3

• 1/3

What is the value of the original function f at the point (1, 2)?

• 3

• 2

• 5

• 4

What is the approximate value of f(0.9, 1.9) using the tangent line?

• 3.2

• 3.1

• 3.0

• 2.9

What is the purpose of determining the equation of the tangent plane?

• To find the Z value on the tangent plane

• To find the X value on the tangent plane

• To find the Y value on the tangent plane

• To find the slope of the tangent plane

What is the relationship between linear approximation and tangent plane?

• They are the same concept

• They are unrelated

• Linear approximation is a subset of tangent plane

• Tangent plane is a subset of linear approximation