{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

10c_04f_2bs - 1 = βˆ‡ f(0 1 Β u = βˆ’ 2 √ 5 3 The...

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Math 10C Second Midterm Version B Solutions November 17, 2004 1. (a) This can be done in various ways: Best is to take a value on each side of the (50,10) entry and form the finite difference: w s (50 , 10) 37 21 60 40 = 16 20 and w t (50 , 10) 36 19 15 5 = 17 10 . Not as good is to take a value at the point and a point on one side (but this is still acceptable): w s (50 , 10) 37 29 60 50 = 8 10 or w s (50 , 10) 29 21 50 40 = 8 10 and w t (50 , 10) 36 29 15 10 = 7 5 or w t (50 , 10) 29 19 10 5 = 10 5 . (b) The units of w s are feet per knot and those of w t feet per hour. If you have singular or plural where you should not (foot, knots, hours), you will still receive credit. (c) The answer is w (50 , 10) + w s (50 , 10) × ( 1) + w t (50 , 15) × 1. You should plug in the numbers you got in (a). 2. By the chain rule f (1) = g x ( x (1) , y (1)) x (1) + g y ( x (1) , y (1)) y (1) = 1 × 1 + ( 2) × 3 = 5 . (b) Since f xy = f yx , we’ll compute f x first. It is 3 x 2 y . Thus f xy = (3 x 2 y ) /∂y = 3 x 2 . (c) |( 1 , 2 )| = 1 2 + 2 2 = 5. Thus u = ( 1 / 5 , 2 / 5 ) . Since f = ( 2 x 2 y, 2 x ) , we have f (0 , 1) = (− 2 , 0 ) Finally D u f
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: , 1) = βˆ‡ f (0 , 1) Β· u = βˆ’ 2 / √ 5. 3. The gradient is a 6 x, 2 y, 4 z A , which equals a 6 , βˆ’ 8 , 4 A at (1 , βˆ’ 4 , 1). Thus the equation of the plane is 0 = a 6 , βˆ’ 8 , 4 A Β· a x βˆ’ 1 , y + 4 , z βˆ’ 1 A = 6 x βˆ’ 8 y + 4 z βˆ’ 42 , which can be rewritten as 3 x βˆ’ 4 y + 2 z = 21. Any of these forms, including the dot product form, is acceptable. 4. Since we are given the critical points, we only need to compute f xx , f xy , f yy and D = f xx f yy βˆ’ ( f xy ) 2 there. quantity at ( x, y ) at (0 , 0) at (1 , √ 2) at (1 , βˆ’ √ 2) f xx βˆ’ 2 βˆ’ 2 βˆ’ 2 βˆ’ 2 f xy βˆ’ 2 y βˆ’ 2 √ 2 2 √ 2 f yy 2 x βˆ’ 2 βˆ’ 2 D 4(1 βˆ’ x βˆ’ y 2 ) 4 βˆ’ 8 βˆ’ 8 By the second derivative test, (0 , 0) is a local maximum and the other two are saddle points....
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business β€˜17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania β€˜17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University β€˜16, Course Hero Intern