21. Solution:

\(f(x,y) = \sqrt {x^2 + 4y^2}\)

Let the output of the equation be \(c = 1\), \(f(x,y) = c\)

= \(\sqrt {x^2 + 4y^2} = c\)

= \(\sqrt {x^2 + 4y^2} = 1\)

= \(x^2 + 4y^2 = 1\)

Above equation can be true only when \(\{x|-1\le x \le 1\}\) and \(y=0\).

Hence Domain is \([-1,1]\), \(\{x|-1\le x \le 1\}\)

Using algebra,

\(y = \sqrt \frac{1-x^2}{4}\)

For values where \(x = 1~ or~ x=-1\), \(y = 0\), however, when \(x = 0,~ y = \frac{1}{2}\).

Therefore, range is \(\left[0,\frac{1}{2}\right]\), \(\bigg\{y|0\le y \le \frac{1}{2}\bigg\}\)

When \(c=2\)

= \(\sqrt {x^2 + 4y^2} = 2\)

= \(x^2 + 4y^2 = 4\)

Above equation can be true only when \(\{x|-2\le x \le 2\}\) and \(y=0\).

Hence Domain is \([-2,2]\), \(\{x|-2\le x \le 2\}\)

Using algebra,

\(y = \sqrt \frac{4-x^2}{4}\)

For values where \(x = 2~ or~ x=-2\), \(y = 0\), however, when \(x = 0,~ y = 1\).

Therefore, range is \([0,1]\), \(\{y|0\le y \le 1\}\)

When \(c=3\)

= \(\sqrt {x^2 + 4y^2} = 3\)

= \(x^2 + 4y^2 = 9\)

Above equation can be true only when \(\{x|-3\le x \le 3\}\) and \(y=0\).

Hence Domain is \([-3,3]\), \(\{x|-3\le x \le 3\}\)

Using algebra,

\(y = \sqrt \frac{9-x^2}{4}\)

For values where \(x = 3~ or~ x=-3\), \(y = 0\), however, when \(x = 0,~ y = \frac{3}{2}\).

Therefore, range is \([0,\frac{3}{2}]\), \(\{y|0\le y \le \frac{3}{2}\}\)

When \(c=4\)

= \(\sqrt {x^2 + 4y^2} = 4\)

= \(x^2 + 4y^2 = 16\)

Above equation can be true only when \(\{x|-4\le x \le 4\}\) and \(y=0\).

Hence Domain is \([-4,4]\), \(\{x|-4\le x \le 4\}\)

Using algebra,

\(y = \sqrt \frac{16-x^2}{4}\)

For values where \(x = 4~ or~ x=-4\), \(y = 0\), however, when \(x = 0,~ y = 2\).

Therefore, range is \([0,2]\), \(\{y|0\le y \le 2\}\)

Level Curves

#When c = 1
yValue1 <- function(x){
  return(sqrt((1-x^2)/4))
}

#When c = 2
yValue2 <- function(x){
  return(sqrt((4-x^2)/4))
}

#When c = 3
yValue3 <- function(x){
  return(sqrt((9-x^2)/4))
}

#When c = 4
yValue4 <- function(x){
  return(sqrt((16-x^2)/4))
}


#Plot the graphs
plot(yValue1,from=-5, to=5,
     main="Level Curves",
     ylab="Y",
     ylim=c(0,2.5),
     type="l",
     col="blue"
     )
curve(  yValue2, col="red", add=TRUE )
curve(  yValue3, col="green", add=TRUE )
curve(  yValue4, col="yellow", add=TRUE )

legend("topright", c("c=1", "c=2","c=3", "c=4" ),col=c("blue","red","green","yellow"), lwd=c(2.5,2.5,2.5,2.5), lty=c(1,1,1,1))

References