How do you go about finding the appropriate partial derivatives to define a tangent plane given a function that is not explicitly defined as f(x,y,z)= like in this case?
I already tried defining the equation as f(y,z)=x=(z+17)/e^ycosz and finding the partial derivatives of f with respect to y and z and obtained x=17+z. I also tried defining the equation as f(x,z)=y=ln(z+17)-ln(xcosz) and obtained y=-1/17x+1/17z. What am I missing here?
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Well, as we mentioned in class today, the gradient of a two variable function is perpendicular to the tangent line of a level curve. Similarly, the gradient of a three variable function <∂f/∂x, ∂f/∂y, ∂f/∂z> is perpendicular to the tangent plane of the level surface. specifically the level surface L_17(f) for the function xe^y cos(z) -z . This gives you a normal vector and the problem gives you a point in the surface, which is all you need for the equation of the tangent plane. I think this is covered on page 654 of the textbook FYI, unless this version of the text has moved it around.
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