SPRING 2013 – Calculus 101C – Test 2 Show ALL work for full credit 1 A) Graph the surface 𝑧 = 1 − 𝑥 ! − 𝑦 ! (and the path on the surface from part B) B) Find the vector function and its domain that represents the path of steepest ascent ON THE surface if you are starting at the point ! ! , ! ! , 0 and stopping at the extrema. Intuitively, this path should be easy to see on the surface. Verify your intuition using the math theory and techniques from this chapter. (your vector function should have 3 components since it’s on the surface) 2 Create a list of all possible candidates for the absolute maximum and minimum values of the following function that are in the closed set bounded by the triangle with vertices ( 0 , 0 ) , ( 0 , 3 ) , and ( 2 , 3 ). 𝐹 𝑥, 𝑦 = 3𝑥 ! + 4𝑦 ! − 9𝑥 − 16𝑦 + 2 3 Find the equation of the plane through the points (2 , 0 , 3) , (1 , 1 , 0) , and (3 , 2 , -‐1). 4 The temperature at any point in a homogeneous body is given by 𝑇 𝑥, 𝑦, 𝑧 = 𝑒 ! !! + sin 𝑥𝑧 What is the direction of the greatest drop in temperature at the point 2 , 0 , 𝜋 ? 5 𝑓 𝑥, 𝑦 = 2𝑥 ! + 𝑥𝑦 Compute the total differential as ( x , y ) changes from ( 2 , 1 ) to ( 2 , 2 ). 6 A) Identify the following surface by circling the correct name. B) Also draw a rough sketch of the surface. (Hyperboloid of 2 sheets, Hyperboloid of 1 sheet, Cone, Hyperbolic Paraboloid, Elliptic Paraboloid, Ellipsoid, or Plane) 𝑥 ! = 𝑧 ! + 𝑦 ! + 1 7 Find the equation of the plane containing the lines 𝑟! 𝑡 = 6 and 𝑟! 𝑡 = 4 5 3 + 𝑡 1 −2 4 1 + 𝑡 2 −4 0 0 . 8 Let ℎ 𝑡 = 𝑚 𝑥 , 𝑦 where 𝑥 𝑢, 𝑣 = 𝑓 𝑢 𝑡 , 𝑣 𝑡 and 𝑦 𝑢, 𝑣 = 𝑔 𝑢 𝑡 , 𝑣 𝑡 Find the formula for !! !" (hint: draw a tree and use the chain rule) 9 What is the greatest area that a rectangle can have if the length of its diagonal is 2 ? Let the rectangle be in the first quadrant with two of its sides along the coordinate axes; then the vertex opposite the origin has coordinates ( x , y ), with x and y positive. The length of its diagonal is 𝑥 ! + 𝑦 ! = 2 , and its area is xy. Thus, we formulate the problem to be that of maximizing 𝑓 𝑥, 𝑦 = 𝑥𝑦 subject to the constraint 𝑥 ! + 𝑦 ! − 4 = 0 . A) Use Lagrange’s Method to find the maximum. B) Substitute the constraint into the function to find the maximum. You should get exactly the same answer as in part A. (You can parameterize the constraint first if you want, whatever you find the easiest) 10 A) Sketch the surface A: 𝑥 ! + 𝑦 ! − 𝑧 ! = 1 B) The plane 2𝑥 − 3𝑦 + 4𝑧 = 4 and the surface A intersect creating a space curve C. Find a Vector Function that represents the tangent line to the space curve C at the point ( 1 , 2 , 2 ). (your vector function should have 3 components) 11 Find the directional derivative of the function 𝑓 𝑥, 𝑦, 𝑧 = 𝑥𝑦 sin 𝑧 at the point 1 , 2 , ! ! in the direction of the vector 𝑎 = 𝚤 + 2𝚥 + 2𝑘 . 12 Find the linear approximation of the function 𝑓 (𝑥, 𝑦) = 𝑥 ! + 𝑦 ! at the point 3 , −4 , 5 .