Linear Approximation and Partial Derivatives

The video tutorial explains how to find the linear approximation of a function of two variables at a given point and use it to estimate new function values. It covers the graphical representation using tangent planes, the calculation and evaluation of partial derivatives, and the formulation of the linear approximation equation. Finally, it demonstrates how to apply this approximation to estimate new values close to the given point.

What is the purpose of finding a linear approximation for a function of two variables?

• To solve the function for all values of x and y

• To find the exact value of the function

• To approximate the function value near a given point

• To determine the maximum value of the function

In the context of linear approximation, what does the tangent plane represent?

• A plane that intersects the function at multiple points

• The exact surface of the function

• A plane that approximates the function near a specific point

• A plane that is parallel to the x-y plane

What is the role of the point of tangency in linear approximation?

• It is irrelevant to the approximation

• It is the point where the function is maximized

• It is the point where the tangent plane is constructed

• It is where the function is minimized

What is the significance of partial derivatives in linear approximation?

• They are used to find the maximum value of the function

• They help in constructing the tangent plane

• They are not used in linear approximation

• They determine the curvature of the function

How do you treat the other variable when finding the partial derivative with respect to one variable?

• As a variable

• As zero

• As infinity

• As a constant

What is the first step in finding the linear approximation of a function?

• Graph the function

• Find the partial derivatives

• Evaluate the function at the new point

• Solve for x and y

What is the next step after finding the partial derivatives in linear approximation?

• Graph them

• Evaluate them at the point of tangency

• Ignore them

• Use them to find the maximum value

What is the final step in using linear approximation to find a new function value?

• Find the maximum value of the function

• Solve for z

• Substitute the new x and y values into the linear approximation

• Graph the function

What is the linear approximation of f(x, y) at the given point in the video?

• 54x + 5y

• 10x + 5y

• 5x + 54y

• 5x + 10y

What is the approximate value of f(2.17, 4.29) using the linear approximation?

• 10.7875

• 11.0000

• 10.0000

• 9.5000