MIT OpenCourseWare http ocw mit edu 18 02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use visit http ocw mit edu terms 18 02 Problem Set 5 Due Thursday 10 11 07 12 45 pm 8 points Part A Hand in the underlined problems only the others are for more practice Lecture 13 Thu Oct 4 Gradient directional derivative tangent plane Read 13 8 Work 2D 1abe 2b 3ac 8 9 2E 7 Lecture 14 Fri Oct 5 Lagrange multipliers Read 13 9 through the top of p 922 one constraint only Work 2I 1 3 19 points Part B Directions Attempt to solve each part of each problem yourself If you collaborate solutions must be written up independently It is illegal to consult materials from previous semesters With each problem is the day it can be done Write the names of all the people you consulted or with whom you collaborated and the resources you used Problem 1 Thursday 8 points 2 1 1 4 As in PS3 Problem 3 consider the function f x y x3 xy 2 4x2 3x x2 y df a Find the maximum and minimum values of the directional derivative at 1 2 1 ds u as u varies b Say in which directions u the maximum and minimum occur df c Find the direction s u for which 0 ds u d Go to the applet http math mit edu 18 02 applets and select directional derivatives from the Show menu Click to move the point in the contour plot as close as you can to 1 2 1 0 51 1 01 is ok but record the actual values of x and y that you use Rotate the direction u hcos sin i and nd the numerical values the applet computes for df ds and answering questions a c above In b and c state the geometric relationship between the yellow direction of the directional derivative u the blue contour line and the purple gradient vector on the contour plot The value of the directional derivative is on the lower right in red Problem 2 Thursday 5 points 2 3 a Find the direction from 2 1 1 in which g x y z x2 yz decreases fastest b Follow the line in the direction you found in part a to estimate using linear ap proximation the location of the point closest to 2 1 1 at which g 2 Do not use a calculator Express your answer using fractions Next use a calculator to evaluate g at your point The value should be reasonably close to 2 1 Problem 3 Friday 6 points 2 4 a Now we seek an exact answer to Problem 2b Use the method of Lagrange multipliers to write down the system of equations satis ed by the point closest to 2 1 1 at which x2 yz 2 Hint it is easier to minimize the square of the distance b Solve the equations to get a numerical approximation to four decimal places for the location of the point Compare your answers with the approximation in 2 b Was each coordinate of the approximate answer in 2 b within 1 100 of the exact answer Suggestion by considering the sum of the second and third equations show that either z y or 2 Assuming z y use the multiplier equations to express x y z in terms of then substitute these formulas into the constraint equation After clearing the denominator you should obtain a degree 4 equation for Use a calculator to solve it Alternatively in Matlab roots a4 a3 a2 a1 a0 nds the roots of the polynomial a4 4 a3 3 a2 2 a1 a0 0 There should be two real roots Finally check that the other case 2 yields solutions x y z that are further away from 2 1 1 2
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