Review for Final Exam—Part 2
- Compute the partial derivatives
and
if
and
are related by the equation
- Find a linear approximation to the function
Use it to approximate the value of
- Find an equation of the tangent plane to the surface of
at the point
- Use the chain rule to compute
if
- Find the directional derivative at
in the direction of vector
for the function
- For the function
- Find domain an range
- Sketch the level curves for
- 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.
- Find the maximum rate of change of
at the point
- Find all second partial derivatives of the function
- Find all the extreme values of
on the disk
- Find all maxima, minima and saddle points of the function
- Find the absolute maximum and absolute minimum values of
on the set
Francisco Blanco-Silva 
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).”
Thanks! I will update the problem accordingly.
for #5 do you have to change the vector to a unit vector before you do the problem?
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
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??
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
Hey Gaston… did you ever figure out what was going on with the linear approximation problem?
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?
Yep Briana that sounds correct!
I’m having a problem with trying to figure out how to start number 8. Could someone drop a little hint??
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.
For 8, to find the max rate of change, dont we just find the magnitude of the directional vector?
Correct. It will be the directional vector over its magnitude.
Find the gradient of f(x,y,z), then plug in the given values to start.
Do you use the half angle formula for number 9 to make it easier to differentiate ?
Yes, you can use half angle but it would make the problem more complicated. Try using chain rule
For number 9, you can look at it like (sin(mx+ny))^2
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?
Yes, these numbers will always be given, there is no formula to get them another way.
& for number 6, would the domain simply be 64>=4x^2+4y^2, or would we need to simplify further?
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.
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.
Is there some kind of short cut for number 12 that we can use?
nah charles sorry there isn’t one that i know of. Its just a long problem
That’s cold, man
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”?
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.
Will there be saddle points for number 12?
Hoke, there isn’t any saddle points for number 12. There are only saddle points when you are finding local not absolute.
What is easiest way to integrate ln(x^2-1)dx? This integral isn’t on the review but is on the integrations review
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
What is the most effective way of finding level curves?
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.