Calculus 3 (Math-UA-123) Fall 2016 Homework 5 Due: Thursday, October 20 at the start of class Please give complete, well-written solutions to the following exercises. 1. The figure on the left shows a contour diagram for the temperature T (in Celcius) along a wall in a heated room as a function of distance x in meters along the wall and time t in minutes. Estimate T /x and T /t at the given points. Give the units and interpret your answers. (a) x = 15, t = 20 (b) x = 5, t = 12 2. A function f (x, y, z) is called a harmonic function if its second-order partial derivatives exist and if it satisfies Laplace's equation: 2f 2f 2f + + = 0. x2 y 2 z 2 (a) Is f (x, y, z) = x2 + y 2 2z 2 harmonic? What about f (x, y, z) = x2 y 2 + z 2 ? (b) We may generalize Laplace's equation to functions of n variables as: 2f 2f 2f + + . . . + = 0. x21 x22 x2n Create an example of a harmonic function with 7 variables, and verify that your example is correct. 3. A surface that has the least surface area among all surfaces with a given boundary is called a minimal surface. It is a fact that a surface z = f (x, y) (where the second-order partial derivatives of f exist) is a minimal surface if it satisfies the partial differential equation: (1 + zy2 )zxx + (1 + zx2 )zyy = 2zx zy zxy . (a) Show that a plane is a minimal surface. (Hint: Write an equation for an arbitrary plane in the form z = ..., and show that it satisfies the above partial differential equation. (b) Use a computer algebra system (e.g., WolframAlpha) to sketch the graph of the helicoid: x = y tan z. (c) Show that the helicoid is a minimal surface. 1 Calculus 3 (Math-UA-123) Fall 2016 4. A friend was asked to find the equation of the tangent plane to the surface z = x3 y 2 at the point (x, y) = (2, 3). The friend's answer was z = 3x2 (x 2) 2y(y 3) 1. (a) At a glance, without doing any computation, how do you know that this is incorrect? What mistake did the friend make? (b) Answer the question correctly. 5. Wind chill, a measure of the apparent temperature felt on exposed skin, is a function of air temperature T and wind speed v. The following table contains the values of the wind chill W (v, T ) for some values of v and T . T = 10 T = 5 T = 0 T = 10 v=5 1 5 11 22 v = 20 9 15 22 35 v = 25 11 17 24 37 v = 30 12 19 26 39 (a) Find a linearization of the function W (v, T ) at the point (v, T ) = (25, 5). (b) Use the above linearization to approximate W (24, 6). (c) Use the above linearization to approximate W (5, 10), and explain why this value is very different from the actual value in the table above. 6. Find the values of z x and z y at the given point: xey + yez + 2 ln(x) = 2 + 3 ln(2), (1, ln(2), ln(3)). 7. If f (u, v, w) is differentiable and u = x y, v = y z, and w = z x, show that f f f + + = 0. x y z 8. Let f (x, y) = direction of: x+y . 1+x2 Find the directional derivative of f at the point (1, 2) in the (a) u = 3i 2j (b) v = i + 4j 9. (a) Two surfaces are called orthogonal at a point of intersection if their normal lines are perpendicular at that point. Show that surfaces with equations F (x, y, z) = 0 and G(x, y, z) = 0 are orthogonal at a point P where F 6= 0 and G 6= 0 if and only if at the point P , Fx Gx + Fy Gy + Fz Gz = 0. (b) Use part (a) to show that the surfaces z 2 = x2 + y 2 (a cone) and x2 + y 2 + z 2 = r2 (a sphere) are orthogonal at every point of intersection. 2