Exercise 5: Evaluate fx (x, y) and fy (x, y) at the indicated point.

f(x,y)=x2y−x+2y+3 at (1,2)

First, compute the partial derivatives with respect to x and y.

\(\frac { \partial }{ \partial x } ({ x }^{ 2 }y−x+2y+3)\quad =\quad \frac { \partial }{ \partial x } { x }^{ 2 }y-\frac { \partial }{ \partial x } x+\frac { \partial }{ \partial x } 2y+\frac { \partial }{ \partial x }\)

\(=\quad 2xy-1\)

\(\frac { \partial }{ \partial y } ({ x }^{ 2 }y−x+2y+3)\quad =\quad { x }^{ 2 }\frac { \partial }{ \partial y } y-\frac { \partial }{ \partial y } x+2\frac { \partial }{ \partial y } y+\quad 0\)

\(=\quad { x }^{ 2 }+2\)

Next, solve for z=(1,2):

\({ f }_{ x }(1,2)=4-1=3,\quad { f }_{ y }(1,2)=1+2=3\)

Plot our function f(x,y):

cone <- function(x, y){
sqrt(x^2*y-x+2*y+3)
}

x <- y <- seq(-4, 4, length= 30)
z <- outer(x, y, cone)
## Warning in sqrt(x^2 * y - x + 2 * y + 3): NaNs produced
persp(x, y, z, col='lightblue',
      box=TRUE, main=expression(x^2*y-x+2*y+3),
      axes=TRUE)