Answer to Question #124119 in Calculus for Muzamil

Question

.(a) Consider a function h(u,v)=4u^2+4v^2+1. Find the level curves for k=n+1,n+2,n+3,n+4,n+5,n+6,n+7.Where n is your arid number (for example if your arid number is 19-arid-12345 then choose n=12345). Perform all steps clearly in detail. Also Sketch the neat Graph.
(b) Let u=f(v),where f(v)=b(v^2-2v) draw the graph for different values of b∈[-5,5]. Just imagine that the values of b lies on the axis which is perpendicular to the laptop screen. Explain the behavior of graph when we change the values of b by assuming that b denotes the graph distance in back or in front of laptop screen? Also explain about the shape of graph (how it look like)? What are the necessary conditions for different types of functions to be defined in two variables explain with examples? How the conditions affect the input and output variables? )

Answer 1

(a) Hint : For plotting, I will use the specified value $n=4$ .

We transform the equation of the level curve so that it is geometrically clear what it is

$$h\left(u,v\right)=4u^2+4v^2+1=k\longrightarrow\\[0.3cm] \left.4u^2+4v^2=k-1\right|\div\left(4\right)\\[0.3cm] \boxed{u^2+v^2=\frac{k-1}{4}}\\[0.3cm] \text{It is a circle centered at}\,\,p.O\left(0,0\right)\,\,\text{and}\,\,R=\sqrt{\frac{k-1}{4}}$$

Hint : For plotting, I used the site https://www.geogebra.org/3d

The intersection of two surfaces is a level curve

k=5

k=6

k=7

k=8

k=9

k=10

k=11

(b)

Hint : Performing paragraph (a), I realized that I can’t draw spatial curves, so at this point I will draw all the curves in the xOy plane, that is, I will make a projection so that they are all in the same plane.

For $b<0$ , the graph is a parabola with branches down. When approaching $b=0$ , the parabola becomes more and more “open”. When $b=0$ it turns into a horizontal line $y=0$ . Further, the parameter is $b>0$ and this means that the graph is a parabola with branches up. As parameter $b$ increases, the parabola "narrows". It looks like this (I will take only the integer values of the paramete $b$ from the specified segment $[-5,5]$ ):