Find slope tangent line to the graph of $f$ at $(\pi/2,\pi/2,0)$ in direction parallel to xy-plane.

Question

Let $f(x,y)=x^2\sin(x+y)$ be any surface. Find the slope of the tangent line to the graph of $f$ at $(\pi/2,\pi/2,0)$ in the direction parallel to $xy$-plane. I am new to multivariable calculus and have no idea how to approach this problem.

Answer 1

$f(x,y)=x^2\sin(x+y)$

Tangent plane at $P(\pi/2,\pi/2,0)$

Let's write partial derivatives $$f'_x=x^2 \cos (x+y)+2 x \sin (x+y);\;f'_y=x^2 \cos (x+y)$$ Tangent plane has equation $$z=f(x_P,y_P)+f'_x(x_P,y_P)(x-x_P)+f'_y(x_P,y_P)(y-y_P)$$ That is $$z=-\frac{\pi ^2 x}{4}-\frac{\pi ^2 y}{4}+\frac{\pi ^3}{4}$$ If we want the equation of the tangent line parallel to $xy$ plane we consider the intersection of the tangent plane with the plane parallel to the plane $xy$ passing through $P$ which is $z=0$

In the plane $z=0$ the equation of the line is

$$-\frac{\pi ^2 x}{4}-\frac{\pi ^2 y}{4}+\frac{\pi ^3}{4}=0$$ which simplified gives

$\color{red}{y=-x+\pi}$

So the answer is that the slope is $m=-1$

Hope this is useful