Homework 10 questions

MATH 3090 Problem Set #5 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. 𝜕𝑓 1. Suppose that 𝑓(𝑥, 𝑦) = 𝑥 3 + 𝑥𝑦 2, 𝑥(𝑡) = 𝑒 2𝑡 , 𝑦(𝑡) = 𝑒 −2𝑡 . Find 𝜕𝑡 . 2. 𝑓(𝑥, 𝑦) = 𝑥 𝑥 2 +𝑦 2 𝜕𝑓 , 𝑥(𝑡) = sin(2𝑡) , 𝑦(𝑡) = cos (3𝑡), Find 𝜕𝑡 . 3. Suppose that 𝑧(𝑥, 𝑦) = 𝑥𝑦 2 , 𝑥(𝑡) = sin(𝑘𝑡) , 𝑦(𝑡) = cos (𝑘𝑡), where k is a constant. 𝜕𝑧 Find 𝜕𝑡 . 𝜕𝑧 𝜕𝑧 4. 𝑧 = 𝑥 2 ln (1 + 𝑦), 𝑥(𝑠, 𝑡) = sin(𝑠𝑡) , 𝑦(𝑠, 𝑡) = 𝑒 𝑠𝑡 , Find 𝜕𝑠 and 𝜕𝑡 . 𝜕𝑧 𝜕𝑧 5. 𝑧(𝑥, 𝑦) = (𝑥 2 − 𝑦 2 )2, 𝑥(𝑠, 𝑡) = 𝑡𝑒 𝑠𝑡 , 𝑦(𝑠, 𝑡) = 𝑠𝑡. Find 𝜕𝑠 and 𝜕𝑡 . 6. Find the equation of the tangent plane to the surface given by 𝑓(𝑥, 𝑦) = 1 + 𝑥 cos(𝑦 2 ) at the point (1, 0, 2). 7. Find the equation of the tangent plane to the surface given by 𝑥 3 + 𝑦 2 + 𝑧 = 1 at the point (−1, −1, 1). 8. Find the directional derivative of the function 𝑓(𝑥, 𝑦) = 𝑥 𝑦 at the point (3, 4) in the direction of 𝑢 ⃗ = < 1, 2 >. 9. Find the directional derivative of the function 𝜙(𝑥, 𝑦, 𝑧) = 𝑥𝑦 + 𝑧 2 at the point (5, 4, 3) in the direction of the vector 𝑣 = < 2, 3, 6 >.