STRUGGLING with functions!!!

[OrangeOne]

Hello,

Im an engineering student and have encountered on problems solving some of the questions in my calculus book. I hope you can help me, if you are not able to answer all questions, either one of them will do.

1. I want to learn how to draw level curves and actual to the following function:
f(x,y)=x^2+4y^2

2. Describe with words and figures the following:
(x,y,z)ER^3 : x^2 + y^2+z^2 < 5
Is it open, closed, limited or compact?

3. draw f(x,y)= k for f(x,y)= x^2+4y^2 when k=1,2,3 (level curves)

4. The temperature in a point (x,y,z) in a body is described by the function T(x,y,z)=x^2+y^2+z^2+2x-2y (degrees celsius). Describe the areas of the body where the temperature is :
a) larger than 2 degrees celsius
b) smaller than 3 degrees celsius

5. Describe the area x^2+y^2 (is equal to or less than 4), x < 0, in polar coordinates. It has to do with sin and cos, right?

Any kind of help is highly appreciated!!

 

[BigGlenntheHeavy]

\(\displaystyle 1) \ f(x,y) \ = \ x^2+4y^2, \ z \ = \ f(x,y ), \ hence \ z \ = \ x^2+4y^2.\)

\(\displaystyle x^2+4y^2 \ge \ 0, \ ergo \ z \ \ge \ 0.\)

\(\displaystyle Let \ y \ = \ 0, \ then \ z \ = \ x^2, \ parabola.\)

\(\displaystyle Let \ x \ = \ 0, \ then \ z \ = \ 4y^2, \ parabola\)

\(\displaystyle Let \ z \ = \ 4, \ then \ \frac{x^2}{2^2}+\frac{y^2}{1^2} \ = \ 1, \ parallel \ to \ xy-plane.\)

\(\displaystyle Hence, \ we \ have \ an \ elliptic \ paraboloid, \ see \ graph.\)

[attachment=0:b2k756ip]aaa.jpg[/attachment:b2k756ip]

 

[galactus]

2. Describe with words and figures the following:
(x,y,z)ER^3 : x^2 + y^2+z^2 < 5
Is it open, closed, limited or compact?

\(\displaystyle (x,y,z)\in R^{3}: \;\ x^{2}+y^{2}+z^{2}<5\)

This describes a sphere with radius less than sqrt(5).

Is a sphere open, closed, or compact?. If you do not know the defintions of these terms they can be found in a calc book or online.

4. Describe the area x^2+y^2 (is equal to or less than 4), x < 0, in polar coordinates. It has to do with sin and cos, right?

Any kind of help is highly appreciated!!

In polar, \(\displaystyle x=rcos{\theta}, \;\ y=rsin{\theta}\)

Making the subs gives us \(\displaystyle r^{2}\leq 4\)

\(\displaystyle r\leq \sqrt{4}\Rightarrow r\leq 2\)

It describes circles with radius less than or equal to 2.

But, they also say that x<0. What does this tell us?.

 

[OrangeOne]

Thanks so much to both of you.

Galactus: the x<0 it tells us that the circle lies on the negative x-axis, ie on the "left" side of origo, am i correct?

 

[galactus]

Well, it's more like they are asking for the portion to the left of the x-axis. Not the whole circle lies to the left.

Take the point (-1,0) or (-3/4, sqrt(7)/4). These lie in the upper left quadrant of the circle.

It represents the left half of a circle of radius less than or equal to 2 centered at the origin.


Hey, this is post 6000.

The balloons come down, a free steak dinner, and presented with one of those big checks.