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3 years ago

Please help me with this :)

Mathematics

1 answer:

jolli1 [7]3 years ago

5 0

Answer:

x ≤ 7/2 or x ≥ 13/2

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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

The quotient Of triple a number and 9 is decreased by the sum of the number and 5? Expression please

Grace [21]

X/3x -x+5
Hope it helps!!!!!!!!!

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3 years ago

What does 24 divided by 1.5 equal.

choli [55]

16 would be the answer

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3 years ago

Read 2 more answers

David wants to make a box plot showing histeam's points for the year. The median score was 7, first quartile was 4, and third qu

vichka [17]

A dot above 7, 4, 10, 2, and 20. Draw a box around the median, first quartile, and and the third quartile. Then connect the box to the minimum and maximum by drawing a line. Should look like this:

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3 years ago

Simplify to create an equivalent expression. -k-(-8k+7)

const2013 [10]

Answer:

7k -7

Step-by-step explanation:

-k-(-8k+7)

Distribute the minus sign to all terms in the parentheses

-k--8k-7

Ad negative negative is a positive

-k + 8k -7

Combine like terms

7k -7

8 0

3 years ago