Find the Linearization L(x, y) of f(x, y) = sqrt(x)y at (1, 4)

TL;DR

This video explains how to find the linearization (or tangent line approximation) of a function at a given point.

Transcript

hi in this video we're going to find the linearization of this function so it's l of X Y of this function here at 1 comma four let's go ahead and work through it so first I'm going to give you the formula so the linearization or the tangent line approximation same thing of this function at a point is going to be F of a b where a b is your point in ... Read More

Key Insights

• 😥 The linearization of a function at a point provides an approximation of the function when the ordered pair is close to the given point.
• ❣️ Differentiating x and treating y as a constant gives the partial derivative with respect to x.
• ❣️ Differentiating y and treating x as a constant gives the partial derivative with respect to y.
• 😥 The linearization formula involves adding the function value at the point to the partial derivatives multiplied by the differences between the point's coordinates and the given point.
• 😀 The linearization is written as L(x, y) = 2x + 1/4y - 1 in this particular example.
• ✊ Using the power rule makes it easier to differentiate functions.
• ❓ The linearization is often used in calculus for approximating functions.

Questions & Answers

Q: What is the formula for finding the linearization of a function at a point?

The formula is F(a, b) + f(x)b + f(y)(y - b), where (a, b) is the point coordinates.

Q: How do you find the partial derivative with respect to x?

To find the partial derivative with respect to x, treat y as a constant and differentiate x.

Q: How do you find the partial derivative with respect to y?

To find the partial derivative with respect to y, treat x as a constant and differentiate y.

Q: How do you find the linearization of a function at a given point?

Plug in the values for the partial derivatives into the linearization formula: F(a, b) + f(x)b + f(y)(y - b).

Summary & Key Takeaways

• The linearization of a function at a point can be found using the formula F(a, b) + f(x)b + f(y)(y - b), where (a, b) is the point.

• To find the partial derivative with respect to x, treat y as a constant and differentiate x.

• To find the partial derivative with respect to y, treat x as a constant and differentiate y.

• Plugging in the values for the partial derivatives into the linearization formula gives the linearization of the function at the given point.