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4vir4ik [10]
3 years ago
7

Is this correct? Yes or nah

Mathematics
1 answer:
lara31 [8.8K]3 years ago
5 0
There are 20 squares total you shaded 5
\frac{5}{20} = \frac{1}{4} = 25\%
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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.

5 0
2 years ago
Which equations are related equations to 14 + x = 26?
sertanlavr [38]

Answer: x=26-14

Step-by-step explanation :you need to subtract on both sides so that why it is x=26-14.

5 0
3 years ago
A salesperson earned a salary of $12000 per year plus 5% commission on sales of $100000. What is his total income
Papessa [141]

If im doing right....

100,000*.5=50,000

Then 12,000+50,000=62,000 per year.

-Seth


3 0
4 years ago
Read 2 more answers
What is the intersection of plane STXW and plane TUYX?
levacccp [35]
Explain a little more

4 0
3 years ago
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If g(x) = 2(x − 4), find the value of x if g(x) = 20
Natalka [10]
Substitute for x.
g(20)= 2(20-4)
g(20)=32.
x=32.
~god bless you!
4 0
3 years ago
Read 2 more answers
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