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Find the local linear approximation, $L(x, y),$ to $F(x, y)$ at the point $(\mathrm{a}, \mathrm{b})$. For each case, compute$$\frac{|F(x, y)-L(x, y)|}{\sqrt{(x-a)^{2}+(y-b)^{2}}} \quad \text { for } \quad(x, y)=(a+0.1, b), \quad \text { and } \quad(x, y)=(a+0.01, b+0.01) .$$a. $\quad F(x, y)=4 x+7 y-16 \quad(a, b)=$b. $\quad F(x, y)=x y \quad(a, b)=$c. $\quad F(x, y)=\frac{x}{y+1} \quad(a, b)=\quad(1,0)$d. $\quad F(x, y)=x e^{-y} \quad(a, b)=\quad(1,0)$e. $F(x, y)=\sin \pi(x+y) \quad(a, b)=(1 / 2,1 / 4)$

Calculus 3

Chapter 13

Two Variable Calculus and Diffusion

Section 1

Partial derivatives of functions of two variables

Partial Derivatives

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Lectures

12:15

In calculus, partial deriv…

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04:38

Compute the linear approxi…

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Find the linear approximat…

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Linear approximationa.…

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02:03

For the following exercise…

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In each part, find the loc…

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(a) Find the local linear …

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(a) Find the linear approx…

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A function $f$ is given al…

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Linear approximation a. Fi…

05:50

in this problem, really finding the name, your approximation of the given function at the given point. First, we are going to find a partial derivative with respect to X and Y. Little specially relative with respect to X is equal to this is cool. Son White, by the way, would be equaled soon. Co sign it Time score saying Why we're and a derivative with respect to why is equal to negative Fine time Ninoy. Now we're going to evaluate the function and the 0.0 pie. So what we have here is kind of sterile time school, sign of price. And those are both mirror. So that would give us a value of zero. Well, actually noticed. Zero times negative one gives us still gives us here. Okay, now we'll evaluate the partial derivatives and our given point. We have course Island's Zero Times Co sign a pry King. That's one time Were you born that is equal to nearly born and the partial derivative of why evaluated at that point would be equal to negative sign of zero time's flying high that is equal to zero. They're both equal to zero. So here we find that Alina approximation is equal to negative horn times. Ex miners Nero. I'm good bye. Still, ***. At our second point, we're going to evaluate our function. And the partial derivatives. No. Prior over to pry frying of Priore too. Terms school sign Well, pie when we get hair one times negative born nickel to negative one No, we value would do Partial derivative and X. We have cool sign. Oh, pilot too. Times cool sign of pie That IHS zero times negative one which is zero. And when would rather wait? Special do really worth respect A why you have negative nine of Priore too. Times sign a buy. Is this negative one times zero which is also equal to zero So are linear approximation is equal to negative one These are zero So everything else become zero.

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In calculus, partial derivatives are derivatives of a function with respect …