# Is my thought proces right? Equation of tangent plane at level surface at given point.

So the question is: Give the equation of the tangent plane at the surface level of the function $$f(x, y, z)=\cos (x+2 y+3 z)$$ at the point $$(x, y, z)=(\pi / 2, \pi, \pi)$$

I don't actually want you to calculate it (but if you like, the bettter :) ), I really want to know if my thought process is correct (If I didn't make a mistake e.g. my equation for a tangent plane).

So first I calculated the partial derivatives (at the given point)

$$f_x=1$$ $$f_x=2$$ $$f_x=3$$

and then I used the equation for a tangent plane (don't really know if this is correct) $$z=f(a,b,c)+f_x(a,b)(x-a)+f_y(a,b)(y-a)+f_z(a,b)(z-a)$$ and got $$x+2y+3z-\frac{11}{2}\pi$$

If you can help me, it would be very much appreciated.

• $$x+2y+3z=\frac{11}{2}\pi$$ Dec 29, 2020 at 18:46
• If you're going to find the tangent plane of the graph $w=f(x,y,z)$ (which is the formula you're trying to use here), you need another variable. Try not to confuse the graph of a function with a level set of the function. Dec 29, 2020 at 19:40

This is asking you to first find the level surface of the function $$f(x,y,z)$$ that contains the point $$(\pi/2,\pi,\pi)$$. This would just be the surface implicitly defined by $$0=\cos(x+2y+3z)$$. The correct equation for the tangent plane of an implicitly defined surface is $$0=f_x(a,b,c)(x-a)+f_y(a,b,c)(y-b)+f_z(a,b,c)(z-c).$$
The equation you have in the question is a combination of this equation and the tangent plane of an explicitly defined surface $$z=f(x,y)$$, which is $$z=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)$$
Notice that $$f(x,y,z) = k_{1} \implies x+2y+3z=k_{2}$$ i.e. the tangent plane equation is simply $$x+2y+3z=\frac{\pi}{2}+2\pi+3\pi$$