Now we must solve the system f x y 4 λx 0 1 f y x 6

Info icon This preview shows pages 6–7. Sign up to view the full content.

Now we must solve the system F x = y + 4 λx = 0 (1) F y = x + 6 λy = 0 (2) F λ = 2 x 2 + 3 y 2 9 = 0 . (3) Multiplying (1) by 3 y and (2) by 2 x , we get the two equations F x = 3 y 2 + 12 λxy = 0 (4) F y = 2 x 2 12 λxy = 0 . (5) Adding (4) and (5), we get the new equation 3 y 2 2 x 2 = 0 3 y 2 = 2 x 2 . (6) Substituting (6) into (3), we have 3 y 2 + 3 y 2 9 = 0 6 y 2 = 9 y = ± radicalbigg 3 2 . If y = radicalbig 3 / 2 then (6) gives x = ± 3 2 . Likewise, if y = radicalbig 3 / 2 then (6) gives x = ± 3 2 . Thus we have four critical points: parenleftBigg 3 2 , radicalbigg 3 2 parenrightBigg , parenleftBigg 3 2 , radicalbigg 3 2 parenrightBigg , parenleftBigg 3 2 , radicalbigg 3 2 parenrightBigg , parenleftBigg 3 2 , radicalbigg 3 2 parenrightBigg . The function values for each of the critical points above are, respectively, 3 2 radicalbigg 3 2 , 3 2 radicalbigg 3 2 , 3 2 radicalbigg 3 2 , 3 2 radicalbigg 3 2 . Thus 3 / 2 radicalbig 3 / 2 is our maximum and 3 / 2 radicalbig 3 / 2 is our minimum. 6
Image of page 6

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

9. You appreciate your calculus instructor so much that you wish to set up a savings account from which he/she can withdraw $2,000 per month. Supposing that the savings account earns 10%/year compounded continuously and that your calculus instructor lives forever, how much money would you need to put into that account right now to make your wish come true. Since we wish to withdraw money from the account forever, this is an example of a perpetuity. We want $2,000 per month, so P = 2000 and m = 12. The interest rate is r = 0 . 1. Thus the present value of this perpetuity is V = (12)(2000) 0 . 1 = $240 , 000 . 10. Evaluate the following improper integrals whenever they are convergent. a. integraldisplay 1 1 x + 2 dx integraldisplay 1 1 x + 2 dx = lim b →∞ integraldisplay b 1 1 x + 2 dx ( u -substitution with u = x + 2) = lim b →∞ ln | x + 2 | vextendsingle vextendsingle vextendsingle b 1 = lim b →∞ ln | b + 2 | − ln | 1 + 2 | This limit diverges since ln( b ) goes to infinity when b goes to infinity.
Image of page 7
This is the end of the preview. Sign up to access the rest of the 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 ‘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 ‘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 ‘16, Course Hero Intern