Chapter 12: Q. 13 (page 953)

What does it mean for a function of two variables, $f (x, y)$ , to be differentiable at a point $(a, b)$ ?

Short Answer

Expert verified

$f (x, y)$ to be differentiated at some extent $(a, b)$ when it on an open set containing the purpose and if $\nabla f (x, y) = f (a + Δ x, b + Δ y) - f (a, b)$ be a function of $(x, y)$ .

Step by step solution

Differentiability.

A differentiable function of one real variable is $1$ whose derivative occurs at each point in its domain, per mathematics.

In other words, each interior point within the domain of a differentiable function includes a non-vertical tangent line on its graph.

Differentiability for two functions of variables.

Let $f (x, y)$ be a function of two variables defined on an open set containing the purpose $(a, b)$ , and and let $\nabla f (x, y) = f (a + Δ x, b + Δ y) - f (a, b)$ be a function $(x, y)$ .

If the partial derivatives $f_{x} (a, b)$ and $f_{y} (a, b)$ both exist, the function f is claimed to be differentiable at $(a, b)$ .

$\nabla f (x, y) = f_{x} (a, b) Δ x + f_{y} (a, b) Δ y + ϵ_{1} Δ x + ϵ_{2} Δ y$

where $ϵ_{1}$ and $ϵ_{2}$ are $∆ x$ and $∆ y$ functions, and both move to zero when $(∆ x, ∆ y)$ $\to (0, 0)$

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What does it mean for a function of two variables, $f (x, y)$ , to be differentiable at a point $(a, b)$ ?

Short Answer

Step by step solution

Differentiability.

Differentiability for two functions of variables.

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What does it mean for a function of two variables, f(x,y), to be differentiable at a point (a,b)?

Short Answer

Step by step solution

Differentiability.

Differentiability for two functions of variables.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

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Study anywhere. Anytime. Across all devices.

What does it mean for a function of two variables, $f (x, y)$ , to be differentiable at a point $(a, b)$ ?