Review for Final Exam—Part 2

  1. Compute the partial derivatives \partial z/\partial x and \partial z/\partial y if x, y and z are related by the equation yz = \ln (x+z).
  2. Find a linear approximation to the function f(x,y) = \sqrt{17-x^2-4y^2}. Use it to approximate the value of f(1.92,0.91).
  3. Find an equation of the tangent plane to the surface of z=5x^2-y^2+5y^2 at the point (-1,5,5).
  4. Use the chain rule to compute dz/dt if z=x^2+y^2+xy, x=\sin t, y=e^t.
  5. Find the directional derivative at (2,1) in the direction of vector \boldsymbol{v} = 10 \boldsymbol{i} + 5\boldsymbol{j} for the function g(r,s) = \tan^{-1}(rs).
  6. For the function f(x,y)=\sqrt{64-4x^2-4y^2},
    • Find domain an range
    • Sketch the level curves for k=0,1,4.
  7. Use differentials to estimate the amount of metal in a closed cylindrical can that is 26 cm high and 6 cm in diameter, if the metal in the top and the bottom is 0.3 cm thick, and the metal in the side is 0.05 cm thick.
  8. Find the maximum rate of change of f(x,y,z)= \displaystyle{\frac{x+5y}{z}} at the point (4,1,-1).
  9. Find all second partial derivatives of the function f(x,y)=\sin^2(mx+ny).
  10. Find all the extreme values of f(x,y)=x^2+2y^2 on the disk x^2+y^2 \leq 1.
  11. Find all maxima, minima and saddle points of the function f(x,y)=2x^3+xy^2+5x^2+y^2+8.
  12. Find the absolute maximum and absolute minimum values of f(x,y)=4x+6y-x^2-y^2+7 on the set D= \big\{ (x,y) : 0 \leq x \leq 4, 0 \leq y \leq 5 \big\}.
  1. Paul Townsend
    December 5, 2012 at 1:48 pm

    I’m sorry for all the confusion, but there seems to be some information missing from problem #2. The question should read: “Find the linear approximation to the function f(x,y) (given in the actual problem) at the point (2,1). Then use this approximation to find f(1.92, 0.91).”

  2. Miles Reese
    December 5, 2012 at 4:04 pm

    for #5 do you have to change the vector to a unit vector before you do the problem?

    • Gaston Jenkins
      December 5, 2012 at 5:17 pm

      Yeah its fine to do the unit vector or the gradient of g(r,s) first. It really doesn’t matter which one you do first because you will have to find both in the problem

  3. December 5, 2012 at 10:20 pm

    For #7 do we just take v=pi*r^2*h and find the differentials (with respect to r and h) and add them together and then set (h=26,r=3) and dr=0.05, dh=.03??

    • Gaston Jenkins
      December 5, 2012 at 11:20 pm

      Yes Tina taking the differentials adding them is correct. In the problem it says that the metal in the top and bottom is 0.3 cm thick though so your dh should equal 0.6 cm because the top is 0.3cm and the bottom is 0.3 cm so you have to add them together to get the dh

  4. briana glander
    December 5, 2012 at 10:41 pm

    Hey Gaston… did you ever figure out what was going on with the linear approximation problem?

    • briana glander
      December 5, 2012 at 10:43 pm

      nevermind i just saw pauls response.. but with the f(1.92, .92) do you plug that into the x and y that you get when you get the liear approximation problem?

      • Gaston Jenkins
        December 6, 2012 at 7:37 pm

        Yep Briana that sounds correct!

  5. December 6, 2012 at 3:15 pm

    I’m having a problem with trying to figure out how to start number 8. Could someone drop a little hint??

    • Alex Cartledge
      December 6, 2012 at 11:38 pm

      Tina, for #8 find the gradient of the function and use the given point to find the direction. Then use that to find the max. rate of change.

      • sean borgsteede
        December 7, 2012 at 10:17 am

        For 8, to find the max rate of change, dont we just find the magnitude of the directional vector?

        • Alex Cartledge
          December 7, 2012 at 12:51 pm

          Correct. It will be the directional vector over its magnitude.

  6. Ryan Meier
    December 6, 2012 at 6:53 pm

    Find the gradient of f(x,y,z), then plug in the given values to start.

  7. Hoke Ward
    December 6, 2012 at 11:29 pm

    Do you use the half angle formula for number 9 to make it easier to differentiate ?

    • Anonymous
      December 7, 2012 at 12:39 pm

      Yes, you can use half angle but it would make the problem more complicated. Try using chain rule

    • Nick Anderson
      December 7, 2012 at 12:42 pm

      For number 9, you can look at it like (sin(mx+ny))^2

  8. Robert Ashworth
    December 7, 2012 at 10:09 am

    Okay, so dumb question…. But with the changes being made to number 2, are these numbers always going to be given? I know I was discussing it with someone during the class section, and they mentioned something about error calculation, so is it a formula we need to know or will these be given values?

    • Nick Anderson
      December 7, 2012 at 12:48 pm

      Yes, these numbers will always be given, there is no formula to get them another way.

  9. Robert Ashworth
    December 7, 2012 at 10:11 am

    & for number 6, would the domain simply be 64>=4x^2+4y^2, or would we need to simplify further?

  10. Ryan Meier
    December 7, 2012 at 10:27 am

    The numbers should be given for number two. I don’t remember dealing with a problem that hasn’t. There is an error calculation formula but that is not what we are using here. For number 6, I would simplify just in case.

  11. Ryan Meier
    December 7, 2012 at 10:46 am

    The numbers should be given for number 2. I don’t remember dealing with a problem that hasn’t. There is an error calculation formula but that is not what we are using here. For number 6, I would simplify just in case.

  12. Charles Reese
    December 7, 2012 at 11:29 am

    Is there some kind of short cut for number 12 that we can use?

    • Gaston Jenkins
      December 9, 2012 at 11:02 pm

      nah charles sorry there isn’t one that i know of. Its just a long problem

  13. December 10, 2012 at 10:56 pm

    For 11, the first partials are 6x^2+y^2+10x=0 and 2xy+2y=0.

    What is the best way of finding your “candidates”?

    • Gaston Jenkins
      December 11, 2012 at 4:17 pm

      Coner, you have two equations two unknowns so i would set one of the equations to y and then plug what you get for y in the other equation for y and solve for x. This would give you values for x and then you should be able to get the values for y.

  14. Hoke Ward
    December 11, 2012 at 3:20 am

    Will there be saddle points for number 12?

    • Gaston Jenkins
      December 11, 2012 at 4:13 pm

      Hoke, there isn’t any saddle points for number 12. There are only saddle points when you are finding local not absolute.

  15. Sarah Thomas
    December 12, 2012 at 4:38 pm

    What is easiest way to integrate ln(x^2-1)dx? This integral isn’t on the review but is on the integrations review

    • Abel Jose
      December 12, 2012 at 6:44 pm

      I used integration by parts and had u=ln(x^2 + 1) and v=x. Then i got dv=dx and du=2x/(x^2+1) I’m not sure if that’s right but that’s how I did it and I think I got the right answer. Hope it helps

  16. Hoke Ward
    December 12, 2012 at 6:43 pm

    What is the most effective way of finding level curves?

  17. December 12, 2012 at 6:57 pm

    for #6 I honestly have issues figuring out the domain and range in terms of two (or more) variables. If someone could give me tips or walk me through the process of solving this question, I would appreciate it a lot.

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