1 Suppose f x y has values and partial derivatives as in the table at right a Find all the critical points you can from the given data and classify them into local mins local maxes and saddles 3 points x y f fx fy f xx fx y fyy 1 0 0 1 2 0 1 2 2 3 0 5 0 0 1 0 1 0 1 0 1 3 1 1 0 1 0 3 2 2 2 2 Local mins if any Local maxes if any Saddles if any b Let L x y denote the linear approximation to f x y at the point 0 1 Is L 0 1 1 1 likely to be larger than equal to or smaller than f 0 1 1 1 Circle your answer below 1 point L 0 1 1 1 f 0 1 1 1 L 0 1 1 1 f 0 1 1 1 L 0 1 1 1 f 0 1 1 1 2 Find an equation of the tangent plane to the surface defined by the equation point 2 1 1 3 points Equation x 1 2 x x y 3y z 2 7 at the 2 y z 3 A tiny spaceship is orbiting on the path given by x 2 y 2 4 The solar radiation at a point x y in the plane is f x y x 2 y x y 2y a Use the method of Lagrange multipliers to find the maximum value M and minimum value m of solar radiation experienced by the tiny spaceship in its orbit 5 points M m b The orbit of the spaceship is shown below On the same axes graph the level set f x y m 2 points 3 2 1 4 3 2 1 1 1 2 3 Tiny rocket from xkcd com 2 3 4 4 Let C be the curve in R3 parameterized by r t sin t 2t cos t for 0 t 2 a Compute the length of C 3 points Length b Evaluate the integral C x 2 z ds 4 points C x 2 z ds c What is the average value of f x y z x 2 z on the curve C 1 point Average 5 Let C be the curve in R2 parameterized by r t t 2 t for 1 t 0 a Mark the picture of C from among the choices below 1 point 1 1 y 1 1 x y y x y x x 1 1 b For the vector field F y x 3 directly calculate C 1 1 F d r using the given parameterization 4 points C c The vector field F is conservative Find f R2 R with f F 2 points f x y d Use your answer in part c to check your answer from part b 2 points F dr 6 a Mark the picture of the curve in R3 parameterized by r t t cos t t sin t t for 0 t 4 2 points z z z z y x x y x y y x b Consider the three vector fields E F and G on R2 shown below E G F i One of these vector fields is yi xj Circle its name here E F ii Exactly one of these vector fields is not conservative Circle it here G E 1 points F G 1 points iii Exactly one of the following is a flowline also called a streamline or integral curve for G parameterized by time for 0 t 1 Circle it 1 point r t e t e t r t t 1 t c Consider the curve C and vector field H shown at right Is the integral H d r C positive negative zero 1 points r t t t r t 1 t 1 t 7 Find a parameterization r t of the curve of intersection of the cylinder x 1 2 y 2 1 and the plane x y z 1 specify the range of the parameter t 4 points r t for t 8 Consider the function f x y on the rectangle D 0 x 3 and 0 y 2 whose contours are shown below right For each part circle the best answer 1 point each a The value of D u f P is negative zero positive 2 0 2 f 2 f 1 f 0 f 5 C b The number of critical points of f in D is 0 1 2 c The integral negative d The integral 4 2 3 C zero C 0 f 3 f 2 1 0 1 f ds is positive f 1 u 0 5 f 1 0 5 f d r is 2 f 4 1 5 4 e Mark the plot below of the gradient vector field f P 1 0 1 1 5 2 0 2 f 0 f 1 f 2 f 3 2 5 3 0 3
View Full Document
Unlocking...