QUESTION

Pick any exercise in Chapter 12 of the calculus textbook. Post the solution or your attempt. Discuss any issues you might have had.

What were the most valuable elements you took away from this course?

Apex CALCULUS v4.0

Chapter 12. FUNCTIONS OF SEVERAL VARIABLES

12.1 Introduction to Multivariable Functions

Exercises 12.1

Page 689

In Exercises 15 - 22, describe in words and sketch the level curves for the function and given c values.

\(f(x,y) = x - y^2\); c = -2,0,2

ANSWER

Solution

Consider \(c=0\). Then \(f(x,y) = x-y^2 = c = 0\), so \(y = \pm\sqrt{x}\).

The horizontal plane at \(c=0\) will intersect the surface following a parabola that is facing right.

With \(c=-2\), \(y=\pm\sqrt{x+2}\) and with \(c=2\), \(y=\pm\sqrt{x-2}\), so the intersection will have the same shape (parabola) just shifted by 2.

Let us plot the 3 level curves.

x <- seq(-5,20,0.05)
y <- seq(-5, 5,0.05)
xy <- data.frame(expand.grid(x=x, y=y))
z <- xy$x-xy$y^2
f <- data.frame(xy, z)

ggplot(f, aes(x, y, z=z))+
  geom_raster(aes(fill=z))+
  geom_contour(breaks=c(-2,0,2), colour="white")+
  xlab("x")+ylab("y")

Even with contour coloring, it is somewhat difficult to imagine the surface. Luckily, rgl makes it incredible easy. And you can use your mouse to adjust graph perspective. Embedding the graph into an RMarkdown file proved to be more difficult.

myColorRamp <- function(colors, values) {
    v <- (values - min(values))/diff(range(values))
    x <- colorRamp(colors)(v)
    rgb(x[,1], x[,2], x[,3], maxColorValue = 255)
}
cols <- myColorRamp(c("red", "blue"), f$y) 
plot3d(f$x, f$y, f$z, xlab="x", ylab="y", zlab="", col = cols)

Most valuable elements you took away from this course?

After a long time I came in touch with my beloved higher mathematics again especially calculus and revisit old engineering mathematics concepts such as LU decomposition and many new R concepts which I was not aware, especially use of Latex to solve any mathematical problem without hand-written functions. This will really help me in my data science journey especially in machine learning and deep learning both in understanding and solving them.

DATA 605 : Week 16 Discussion

Debabrata Kabiraj

May 14, 2020