Review for final exam (MA241 Spring 2013)
- Find the angle between the vectors
and
- Find the curvature at
of the curve
- Find parametric equations for the tangent line at the point
to the curve with parametric equations
- Find the length of the curve
for
- Find an equation of the plane that contains the point
and is perpendicular to the line with equation
- Find an equation of the plane that contains the line
parallel to the plane
- Find the distance from the point
to each of the following:
- The
plane
- The
axis.
- The
- Find the volume of a parallelepiped with adjacent edges
where
- At what points does the helix
intersects the sphere
- Find a non-zero vector orthogonal to the plane through the points
- Compute the partial derivatives
and
if
and
are related by the equation
- Find a linear approximation to the function
at the point
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
- Find the mass and center of mass of the lamina with density
that occupies the region
- Evaluate
where
is the region that lies between the spheres
, and above the cone
- Evaluate
with domain
being the region bounded by
and
- Evaluate the triple integral
where
above the plane
and between the planes
and
- Is the vector field
conservative? If it is, find a function
so that
- Evaluate the triple integral
where
and the plane
- Evaluate the integral
where
is the circle
- Prove that the Jacobian to change from Cartesian to Spherical coordinates is
.
- Find the area of the region
bounded by the circles
in the half-plane
- Evaluate the line integral
where
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Francisco Blanco-Silva 
I know this is probably a super simple question but I am spacing on the beginning material from the semester–for number 5, how do you calculate a perpendicular vector? Do you use the cross product? And if so, what’s the second vector you use?
For number 5 its asking for the equation of a plane. You know that the formula for the equation of a plane is: a(x-x0) + b(y-y0) + c(z-z0)=0 where is a normal vector to the plane ( aka perpendicular vector) and (x0,y0,z0) is just a point on the plane. So because the question gives you a point on the plane (1,3,5), all you need is a normal vector to plug into the formula and you have the answer. You can get this normal vector by recognizing that the parametric equations the problem gives you for a line perpendicular to the plane: x=3+t , y=3t, and z=5-2t contains a normal vector for the plane.
I think you don’t need to calculate a perpendicular vector. The perpendicular vector of the plane is the same as the vector of the line given in the problem
EDIT: 3rd line from top, it should read where (a,b,c) is a normal vector…
Okay awesome, I thought so, just was spacing out. Thanks, guys!
Also for number 6, if the plane contains the line, can I just use the vector from the plane and the point from the line and write the equation that way? Or does the vector in the line have a factor in the equation of the plane as well?
For this one recognize that you can use the normal vector given by 3x+6y-3z=18 to form the equation for the other plane because the planes are parallel. so the normal vector for one plane will also be a normal vector to the other plane.
Yes, I think you use the vector from the plane since it is parallel and the point from the line.
This problem is very similar to HW 2, #13 if you are still having problems on how to go about it.
For number 11, do we have to find an equation in which z is in terms of x and y, and if so, how does that work with the ln?
For this one you use implicit differentiation.
No, you don’t need to find an equation in which z is in terms of x &y. You need to solve for \partial z/\partial x by taking the partial derivative of x in both sides and by keeping in mind that z is a function of x & y.
So what would be the partial in terms of x of ln(x+z)? Because z isn’t in terms of x and y at that point. Same with yz. I don’t understand how you can take the partial derivative of a function without knowing that function in the variables x and y.
implicit differentiation is hard for me to explain. this link may help. Its example 4 near the bottom of the page http://tutorial.math.lamar.edu/Classes/CalcIII/PartialDerivatives.aspx
To find dz/dx: I first got y(dz/dx)= (1+(dz/dx))/(x+z); then I got the (dz/dx) terms to one side which leads you to:
(dz/dx)((x+z)y-1)=1; which leads to the final answer
dz/dx = 1/((x+z)y-1)
For dz/dy just do the same thing: I got
dz/dy = (z(x+z))/(1-y(x+z))
Ahhhhhhhh thank you. Makes way more sense.
For dz/dx, you differentiate implicitly with respect to x. You treat y as a constant. So for the left side yz, you would get y*dz/dx. Am I on the right track? I implicitly differentiated both sides and then solved for dz/dx.
For number 13 is one of the y^2 terms supposed to be just a y?
No this is correct. Like usual just find f(sub x) and f(sub y). Then plug in the values of x and y into those equations. I got fx(-1,5)= -10; fy(-1,5)=40. Then just plug these values into the equation of the tangent plane. So its: z-5 = -10(x+1) + 40(y-5)… simplify, solve for z and you get the answer.
I think you are right, one of y^2 terms should be y because then the point (-1,5,5) is not on the plane and we can’t find the tangent line equation of a point that is not in the plane.
For 16 I am getting that the domain lies outside of the circle–does this work? Or does the domain need to be enclosed meaning I did something wrong?
I think it should be enclosed in this problem. Try it again setting Sqrt(64-x^2-y^2) >= 0 and get 4x^2+4y^2 by itself on one side then 64 on the other. if the x^2 + y^2 term is <= the term on the other side then the domain is going to lie within the circle.
You’re right–makes way more sense. My inequality was backwards from the beginning.
For the range–do I plug in the lower and upper bound for the domain and solve from there or just say [0, infinity) because that’s the capability of z?
The range is all the values that z can be for any value of x and y. The low end of your range [0 is right but z cannot go to infinity in this case.
You are right about the minimum of 0 limit but the range maximum is not infinity. When you found the domain, you should have got the equation x^2+y^2<=16; therefore the max value of x or y is the square root of 16=4. Range = [0,4]
Wouldn’t the range here be [0,8] since the lowest point we have in the domain is (0,0). Whenever we plug in (0,0) in the function we get f(x,y)=sqrt(64)=8
For the level curves, did anyone get circles with r=4 for k=0 , r=sqrt(63)/2 for k=1 , and r=sqrt(15) for k=4 ?
You are right, I believe. The range is [0,8]. what did you get for your equation for the level curves? I got r=sqrt(16-(k^2/4))
For # 12, do we need a specific point to find the linear approximation and then plug in the given point?
Yeah i think you would have to be provided a point
The specific point is (1.92, .91) and that goes in the (Xo, Yo) portion of the equation and the plain X and Y stay as variables. You should get an equation with a constant, an X term and a Y term.
No, actually you do need a point. You can’t find a linear approximation without one. The equation is not a linear approximation.
So do you take the partial derivatives and then plug the approximate value of x into the partial derivative of x, and approx. value of y into the partial derivative of y to find your answers?
This is throwing me off also because in all of the examples in the book, you are given an indicated point to find the linear approximation at, then you use the approximation to find the value of say f(1.92,0.91). Can someone please explain this?
Prof. Blanco-Silva said it is at the point (1,0) today in class.
I got my final answer to be .23, can anyone confirm this?
I got my answer to be 3.77. But I might be wrong!
Did you get f(1,0)=16 , fx(1,0)=-1/4 , and fy(1,0)=0 ?
L(x,y)=(-1/4)x+(17/4)?
whats the formula you are using? Because I have the same values as you do for f(1,0) fx(1,0) and fy(1,0)
Collin, I have used L(x,y)=4+(-1/4)(x-1)+0(y-0)
for #25 should the first integral be from x^3 to sqrt(x) or from sqrt(x) to x^3? I eventually came up with a negative answer so I’m doing something wrong. Also the second part goes from 0 to 1 right (where it converges)?
it would be from x^3 to sqrt(x) because x^3<=y<=sqrt(x)
If you graph it: you can see that sqrt(x) is above x^3 from 0 to 1; so sqrt(x) should be on top. All i did was a double integral from {0 to 1 dx {x^3 to sqrt(x) dy of the terms 4xy-y^2. I got 19/60 but not sure if right.
For number 26 should the question read ‘below the plane z=8’ instead of above?
Yes!
Did anyone else get (19.5pi)cm^3 for number 17?
You should be getting around 33 cm^3
I got (12.5pi)cm^3, V=(pi)(r^2)h is the equation for a cylinder.
Getting the differential you get
Delta V = 2pi(r)dr(h) +pi(r^2)(dh)
for dr= .05; dh =0.3; h=26; r=3
Thanks, Jan–for some reason I was trying to use x, y, and z to solve it.
Also just wanted to let you know I got (10.5pi)cm^3–I did the calculations a couple times and derived the same equation you did. Is this a mistake on my part?
Yea, my bad. It is 10.5pi.
dh should be equal to .6 because the thickness change is on the top and bottom making the answer 13.2pi
For 20 when I set the gradients equal to each other, I get 2=1…but 2 does not equal 1 and that’s a dead end for finding extreme values. Am I off in left field?
You have to find the critical points. After finding the partial derivatives, you should set them equal together 2x=4y; thus one critical point is (0,0). From the equation of the disk; you can obtain 4 more critical points (0,+-1) and (+-1,0). Plugging these values into the original equation: max=2; min=0
So in this problem we can’t use Lagrange multiplier?!
For 21, I got that the minimum is (0,0) and (-8/5, 4/5) is a saddle point. Is this a common answer? Everything made sense in my calculations, but I’m slightly confused as to the properties–do you have to have a max?
I found there to be 2 local minimums: (0,0) and (-5/3,0). I found there to be 2 saddle points as well as (-1,2) and (-1,-2). I also did not find a maxima. Since my answers are different than yours I can run through how I got there and then compare. I began by first finding fsubx and fsuby and set both equal to 0 and found the first critical point at (0,0). While setting fsuby equla to zero I also found x to possible be x=-1. I tested this in and found two critical points at (-1,-2) and (-1,2) after plugging both x=-1 and y=0 into fsubx. The setting y=0 is fsubx, I found another critical point at (-5/3,0) which may be where me and you differ. Then finalized by using the second derivative test to find 2 local minima and 2 saddle points.
I found the same critical points as Collin, but I had different conclusions! (-1,2) is a saddle because Hess(-1,2) 0 and second partial derivative 0 and second partial derivative > 0
(-1,2) saddle , (0,0) min , (-1,-2) & (-5/3,0) are max
For #18, once you get the magnitude of the gradient at the point, do you have to divide your new vector by the unit vector?
Nevermind, i made a careless error. I am pretty sure you divide your gradient by the magnitude of your gradient to give you the maximum rate of change. Somebody correct me if I am wrong
I think the maximum rate of change is just the gradient, but from what I understand, it’s just an accepted practice to turn it into its unit vector form.
Oh, ok. Thanks, that makes sense.
I think the maximum rate of change is the magnitude of the gradient according to theorem 15 in page 916. The gradient at the point ( after you divide it be the magnitude) is just the direction where the maximum change occur. Please correct me if I’m wrong.
I found the maximum rate of change to be sqrt(198). Did y’all get the same answer?
In # 15, do we need to change the vector to a unit vector? if yes, did anyone get 3/sqrt(125) as answer?
and if we want to find the direction of where the max change occur, Is it the vector given in the problem or the maximum occurs in the unit vector of the gradient?
For #15, I found the gradient and the unit vector first. Then I computed the dot product of the gradient and the unit vector to give me the directional derivative. I ended up with 4/sqrt(125), but I may have made a mistake somewhere that I have not found yet.
For the gradient, I got and for the unit vector I got .
EDIT: gradient = (1/5,2/5)
U = (10/sqrt(125), 5/sqrt(125))
I got the same thing. Thanks. and for my second question I think the maximum change happens in (1,2) which is the unit vector of (1/5,2/5) ( this is not required for question # 15 )
For number 17, I saw many of you guys came up with an answer at 10.5pi. Going back through the equation yall were using however, when you found dh to be .3, you didn’t factor in that you need to multiply it by 2 since it is on the top and bottom. So therefore instead of .3=dh, it would be .6=dh. Is this correct or do you only factor in the .3 once?
You are absolutely right. it has to be multiplied by 2 because it is in the top and bottom and I got 12.3 pi. You can see the solution for similar problem in webassign HW # 7 problem # 17.
Why do we chose dh=.3 and dr=.05 and not the opposite ?
For #24, I missed part of this on the last exam. If E lies above the cone phi = pi/3, then what are the boundaries of phi when you compute the triple integral? If someone could explain how they got this, I would really appreciate it.
Phi in spherical method starts at the z-axis where Phi=0 and doesn’t go further than Pi. In this problem since we are at the region above the cone we have 0<=phi<=pi/3 because our Phi can't go further than Pi/3.
Thanks, that cleared it up for me.
Did you all end up with the integral with 0<=theta<=2pi sin(theta)cos(theta) which ends up being 0??
For #18 I know this may be a simple question but I found the Gradient Vector, then evaluated it at each point. Is the maximum rate of change just the magnitude of it?
Yes I think so. I got sqrt(198). did you get the same thing?
I got sqrt(97) as the maximum rate of change. You need to start out with the gradient of f, then plug in the point (which is (4, 1, -1) in this problem). Afterwards, do |gradient f(4, 1, -1)| (| = absolute value mark) and you should get your answer for max rate of change.
Thanks. I probably have gotten the wrong gradient then. Can you please tell me what you got for the gradient?
I got =
I got the sqrt(107) but I think I am messing up on taking the derivative of x,y,z,. When you take the derivative of fx do you treat z as a constant or is the answer for fx 1/z?
I got the sqrt(107) as well.
fx is 1/z and z is treated as a constant because you are differentiating with respect to x.
For 22, I found absolute maximum to be 20 at (2,3) and an absolute minimum to be -32 at (8,-1). Can anyone confirm this or show me what they did instead if you got a different answer? Im skeptical of mine.
Nevermind just saw that I used the bounds given in problem 23 on accident.
I got max of 20 at (2,3) and min of 7 at (0,0) and (4,0). Did you get the same thing?
Im still getting different answers. My critical points are (2,3) (0,4) (4,-4) (6,0) (-4,5). Can you go through how you got your answer?
For # 19, did you get (partial f / partial x)= 2sin(mx+ny)(m*cos(mx+ny)) and then we need to use the product rule to find all second partial derivatives?
correct. I found fxx to be 2m^2cos(2mx+2ny), fyy to be 2n^2cos(2mx+2ny), and fxy to be 2mncos(2mx+2ny)
I found the same answer for the first derivative, and (partial f/ partial y)=2nsin(mx+ny)cos(mx+ny). I am not sure about the product rule because mine is coming up as a mess and I know there is a mistake in it but have not found it yet?
Is #31 supposed to be solved through green’s theorem? If so is the question missing a detail? How would I set up a problem such as this?
Or is it a simple double integral?
I think it should be solved by using double integral.
If we need to use a double integral then what are the bounds?
I know the bounds of theta and rho for the double integral but what do I put underneath the integral? I’m drawing a blank
Is it just x^2 + y^2 converted to polar coordinates?
In # 26 , do we use polar coordinates ?
Because its a triple integral and a paraboloid is involved, I think you would use cylindrical coordinates.
Hey, if you don’t mind, can you elaborate more on what you did on #26?
What limits did you use?
What bounds did yall get for 24?
I got 2 <= p <= 4, 0 =< theta =< 2pi, 0 =< phi =< pi/3.
2 < p < 4
0 < phi < pi/3
0 < theta < 2pi
I was hoping to be able to get some help on a few questions I am iffy about. Can anyone post their answers for 3, 9, 15, and 27?
3) x=5+t;y=2t;z=t^3+t
9)It was on homework 3 # 9
15) I am not sure but I got 4/(5*sqrt(5))
On number 3; I got z=2+4t.
for 3 I got x=5+t, y=2t, z=2+4t
for 9 (sin3, cos3, 9) and (sin(-3),cos(-3), 9)
for 15 i got 5
and for 27i got ye^x-cos(x)cos(y)
These could be wrong, but thats what I have
For 31 I know it’s supposed to be a double integral but what do I put under the double integral? I know 0<=theta<=pi and 2<=r<=4 but what's my equation?
Since we are trying to find the area, we don’t need to have an equation to integrate. We just integrate 1 and if you use polar which I used, you will integrate rho. Or without integrating you can just find the area of half the circles and find the difference between them. 1/2 pi r^2.
Is #26 a Type II triple integral?
The reason I ask is because I am having trouble finding the x boundaries. This one has me stumped.
Shouldnt problem 28 have another boundary for x?
No, because it is actually a cylinder not a paraboloid.
Any one getting 4*sqrt(35) as answer for #32
Yes, I got the same answer.
Can someone tell me what are the boundaries used for # 26. and is the answer = 0. ?
Ive been having trouble with it as well. What boundaries did you use? I used for y 0 to 2, x 0 to sqrt(4-y), and for z(iffy about) I have 2x^2+2y^2 to 8.
Can someone explain their way through number 9?