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3 years ago

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Express f (x) = |x-5| as a piecewise defined function

1 answer:

mrs_skeptik [129]3 years ago

3 0

If x>5......f(x)= x-5
if x <5......f (x)= -x+5

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Can you define f(0, 0) = c for some c that extends f(x, y) to be continuous at (0, 0)? If so, for what value of c? If not, expla

Ahat [919]

(i) Yes. Simplify $f(x,y)$ .

$\displaystyle \frac{x^2 - x^2y^2 + y^2}{x^2 + y^2} = 1 - \frac{x^2y^2}{x^2 + y^2}$

Now compute the limit by converting to polar coordinates.

$\displaystyle \lim_{(x,y)\to(0,0)} \frac{x^2y^2}{x^2+y^2} = \lim_{r\to0} \frac{r^4 \cos^2(\theta) \sin^2(\theta)}{r^2} = 0$

This tells us

$\displaystyle \lim_{(x,y)\to(0,0)} f(x,y) = 1$

so we can define $f(0,0)=1$ to make the function continuous at the origin.

Alternatively, we have

$\dfrac{x^2y^2}{x^2+y^2} \le \dfrac{x^4 + 2x^2y^2 + y^4}{x^2 + y^2} = \dfrac{(x^2+y^2)^2}{x^2+y^2} = x^2 + y^2$

and

$\dfrac{x^2y^2}{x^2+y^2} \ge 0 \ge -x^2 - y^2$

Now,

$\displaystyle \lim_{(x,y)\to(0,0)} -(x^2+y^2) = 0$

$\displaystyle \lim_{(x,y)\to(0,0)} (x^2+y^2) = 0$

so by the squeeze theorem,

$\displaystyle 0 \le \lim_{(x,y)\to(0,0)} \frac{x^2y^2}{x^2+y^2} \le 0 \implies \lim_{(x,y)\to(0,0)} \frac{x^2y^2}{x^2+y^2} = 0$

and $f(x,y)$ approaches 1 as we approach the origin.

(ii) No. Expand the fraction.

$\displaystyle \frac{x^2 + y^3}{xy} = \frac xy + \frac{y^2}x$

$f(0,y)$ and $f(x,0)$ are undefined, so there is no way to make $f(x,y)$ continuous at (0, 0).

(iii) No. Similarly,

$\dfrac{x^2 + y}y = \dfrac{x^2}y + 1$

is undefined when $y=0$ .

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2 years ago

Which of the following equations is equivalent to 2/3 x - 1/2 y = 6? 2x - y = 6 2x - y = 36 4x - 3y = 36

boyakko [2]

Answer: THE LAST OPTION

Step-by-step explanation:

1. You have the following equation given in the problem:

$\frac{2}{3}x-\frac{1}{2}y=6$

2. If you multiply both sides of the equation shown above by 6, you obtain the following:

$\frac{2*6}{3}x-\frac{1*6}{2}y=6*6$

$\frac{12}{3}x-\frac{6}{2}y=36$

3. When you simplify the equation, you obtain the following equivalent equation:

$4x-3y=36$

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ASHA 777 [7]

What type of diagram?

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There are three elements in that set.

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