Chapter 13.6, Problem 52E

(a)

To determine

To calculate: $f\left(x,y\right)=x-y^2$, $P\left(4,-1\right)$

Is given, Find the gradient of the given function at P where

(b)

To determine

To calculate: $f\left(x,y\right)=x-y^2=3$, $\nabla f\left(4,-1\right)=\text{i}+2\text{j}$, $c=3$.

Is given, Find unit normal vector to the level curve

(c)

To determine

To calculate: $f_x\left(x,y\right)=1$, $f_y\left(x,y\right)=-2y$, $P\left(4,-1\right)$. is given, find tangent line to the level curve $f\left(x,y\right)=x-y^2$ at $P\left(4,-1\right)$

(d)

To determine

To graph: The curve, $f\left(x,y\right)=x-y^2-3$, and the point $P\left(4,-1\right)$.

Is given, find the level curve, the unit normal vector, and the tangent line in the x y-plane where