Test 3 Math 273 July 12, 2012 Name: Problem Value 1 10 2 10 3 10 4 10 5 10 Total 50 Score Show all appropriate work. z z and where x y sh is ar stu ed d vi y re aC s ou ou rc rs e eH w er as o. co m 1. Assume z = f (x, y), and nd 3x2 z x2 y 2 + 2z 3 + 3yz 5 = 0. w w 2. Use the chain rule to nd and where w = x cos(yz), x = s2 , y = t2 , and s t z = s 2t. 3. Use the Second Derivative Test to nd and classify the critical points of the function f (x, y) = 3xy x2 y xy 2 . 4. Find the directional derivative of f (x, y, z) = x2 y + x 1 + z at the point (1, 2, 3) in the direction of v = 2i + j 2k. Th 5. Find the equation of the plane tangent to the hyperboloid given by z 2 2x2 2y 2 = 12 at the point (1, 1, 4). https://www.coursehero.com/file/7758811/Summer-Test-3-Example/ Powered by TCPDF (www.tcpdf.org) Test 2 Math 273 March 1, 2013 Name: Problem Value 1 5 2 5 3 15 4 10 5 5 6 15 7 15 8 10 Total 80 Score Instructions. sh is ar stu ed d vi y re aC s ou ou rc rs e eH w er as o. co m 1. Show all work to receive full credit. 2. Leave space at the top of each page for your staple. If I can't read your work I will not grade it. 3. Simplify all answers. Perform all arithmetic, dot products, cross products, etc. Evaluate all trig. functions. 4. Write your answers only on one side of the paper provided. Any answers written on the cover sheet will not be graded. 5. Start a new page every two problems unless you need to start a new page earlier. Problems 7 and 8 should be on their own page. Failure to do this will result in a ve point deduction from you nal score. Th 6. When you hand in the test staple the cover sheet on top of the test. https://www.coursehero.com/file/7758806/Test-2-Example/ 2 1. Answer the following True or False. Write out the whole word. Illegible answers will receive zero credit. (a) If fxy and fyx are continuous then fxy = fyx . (b) The gradient, f , is normal to the surface created by f . (c) If f is continuous on a closed domain then f attains both its maximum and minimum values. (d) Two dierent level curves can intersect each other. sh is ar stu ed d vi y re aC s ou ou rc rs e eH w er as o. co m (e) The gradient points in the direction of steepest ascent. 2. Use polar coordinates to evaluate lim (x,y)(0,0) xy . + y2 x2 3. Find the following partial derivatives. (a) Find fx for the function f (x, y) = x cos(xy) + exy . (b) Find fxy for the function f (x, y) = x cos(xy) + exy . (c) Find hxxyzw x2 + y 3 for the function h(x, y, z, w) = x y sin(w) + cos . yz 3 3 4 4. Find an equation of the plane tangent to the hyperboloid given by z 2 2x2 2y 2 = 12 at the point (1, 1, 4). f for the function f (x, y) = cos(x2 + y). Th 5. Calculate 6. Given that (1, 1) is a critical point for the function h(x, y) = x4 4xy + 2y 2 classify it as a local max, local min, or saddle point. Use the partial derivatives hxx = 12x2 , hyy = 4, hxy = 4. 7. Use the chain rule to calculate x = s2 , y = s 2t, z = t2 . f f and for the functions f (x, y, z) = xz sin(y) and s t 8. Find the global extrema for the function on the given domain g(x, y) = x3 +y 3 3x3y, 0 x 1, 0 y 1. Note: The only critical point of this function in the domain is P = (1, 1). https://www.coursehero.com/file/7758806/Test-2-Example/ Powered by TCPDF (www.tcpdf.org)