partial derivatives

MATH 3090 Problem Set #6 Directions: Work each problem on clean paper. Your solutions should be clear, errorfree, cogent, and didactic. You should assume that you are writing so that a C calculus student can understand how to solve the problem by reading your solution. To that end, writing short explanations and notes are a good. 𝜕𝑧 𝜕𝑧 1. Suppose that 𝑧 = 𝑥 cos(𝑥𝑦) , 𝑥(𝑠, 𝑡) = 𝑒 𝑠+𝑡 , 𝑦(𝑠, 𝑡) = 𝑒 𝑠−𝑡 . Find 𝜕𝑠 and 𝜕𝑡 . Evaluate both partial derivatives when 𝑠 = 1, 𝑡 = 2. 2. Find the directional derivative of 𝜙(𝑥, 𝑦, 𝑧) = 𝑥𝑦 2 𝑧 3 at the point (2, −1, 3) in the direction of 𝑣⃗ = √2 𝑖̂ + √3 𝑗̂ − 3 𝑘̂ . 3. In the previous problem, in what direction is the derivative maximal? What is the value of the derivative in that direction? Express your direction as a unit vector. 4. Recall from the second week of class that the vector ⟨ 𝑎, 𝑏, 𝑐 ⟩ is normal to the plane given by 𝑎(𝑥 − 𝑥0 ) + 𝑏(𝑦 − 𝑦0 ) + 𝑐(𝑧 − 𝑧0 ) = 0 (This was one way that we could identify planes, through their normal vectors). Suppose that 𝑃 = (𝑥0 , 𝑦0 , 𝑧0 ) is located on a surface in 3-space. Let T represent the tangent plane to the surface at the point, P. We know from Monday’s class that the gradient, ∇𝑓(𝑥0, , 𝑦0 , 𝑧0 ), is a vector normal to plane P. Therefore, substituting ∇𝑓(𝑥0, , 𝑦0 , 𝑧0 ) for a, b, and c, we have an alternative means of finding the tangent plane to a surface at a point. 𝑓𝑥 (𝑥0 , 𝑦0 , 𝑧0 )(𝑥 − 𝑥0 ) + 𝑓𝑦 (𝑥0 , 𝑦0 , 𝑧0 )(𝑦 − 𝑦0 )+𝑓𝑧 (𝑥0 , 𝑦0 , 𝑧0 )(𝑧 − 𝑧0 ) = 0. Find the equation of the tangent plane to the ellipsoid given by point (−2, −1, −3). 𝑥2 4 + 𝑦2 + 𝑧2 9 = 3 at the 5. Find the symmetric equations of the normal line to the surface in problem #7 at the point (−2, −1, −3). 6. Find the general form of the parametric representation of the normal line to the surface from the previous problem at some general point (𝑥0 , 𝑦0 , 𝑧0 ). 7. Show that every normal line to the surface of the sphere given by 𝑥 2 +𝑦 2 +𝑧 2 = 𝑟 2 passes through the origin. 2 2 8. Find all critical points of the function 𝑧 = 𝑥𝑒 −2𝑥 −2𝑦 . You do not have to classify the problems as minima, maxima, or saddle points.