Ben

Capacitor C_1 is initially charged so that it has a potential difference of 12V. At time t=0, Switch S_1 is closed, allowing the capacitor to discharge through resistor R_1. At t=5 seconds, the potential across the capacitor has fallen to 6V. AT what time will the potential across the capacitor reach 3V?

The schematic is such that the capacitor runs first to the switch, and then to the Resister. I'm lost at even where to being...

Prove that if u and v are nonzero orthogonal vectors in R^n they are linearly independent.

Now I know that u and v are orthogonal when u dot v =0. I know that when a set of vectors are linearly independent when c_1u + c_2v = 0 only when c_1 = c_2 = 0.

I don't however, seem to know how to combine everything into a proof. I started trying to show that:

c_1u + c_2v = c_1(u_1, ... , u_n) + c_2(v_1, ... , v_n)

and try and end with =0, however I'm not sure how to involve that u dot v = 0.

Help?

Prove that the set of 2X2 invertible matrices is not closed under addition and scalar multiplication. (find a counter example)

I've been able to find a counter example for the first part:

[1 0] [2 0] [3 0]

[0 1] + [0-1] = [0 0]

Where the first two are invertible however the sum is not, therefore we are not closed under addition, I'm struggling with the second half, help? Thanks!

So my linear algebra class is giving us a gentle introduction to proof. I'm struggling with some of the ideas, specifically here:

Prove that a scalar multiple of a symmetric matrix is also symmetric.

Now I know that for a matrix to be symmetric then A (which is a nxn matrix) must equal A^T so I believe that I have to prove that:

c(A^T) = (cA)^T

I believe I can go as follows but something doesn't feel right...

c(A) = (cA)^T

= (cA)_ji (def of transpose)

= (ca)_ji (def of scalar multiplication of real numbers)

= c(a_ji) (associative property of multiplication of real numbers (?????))

= c(A_ji) (no idea what would go here...)

= c(A^T) (definition of transpose)

If someone could help clarify/edit this proof I would very much appreciate it, still very green when it comes to writing these guys...

Given the set {(3, 4, 1), (2, 1, 0), (9, 7, 1)}

Prove that the set is linearly dependent, then find a linearly indepentent subset.

I can prove that it is linearly dependent no problem, I am struggling with finding the subset.

c_3=r

c_1=-r

c_2=-3r

Shows that we have a non-trivial solution - how do I find the subset???

Thanks

Show that it is possible to have a set of vectors that are lineraly independent but do not span R^3.

I'm struggling here to try and come up with something, any help is appreciated!

Evaluate: Double integral F*ndS if:

F = <xy, z, yz-3>

S: Portion of the paraboloid z=2+x^2+y^2, z</= 6

Okay, so I started with finding n. I got 1/sqrt(2)<-2x, -2y, 1> - This was from writing the paraboloid as a function of (x, y, z) and taking the gradient, then dividing by the magnitude and using the fact that my function of (x, y, z) = 2.

After doting F * n I get: 1/(sqrt2) [ -2x^2y - yz - 3]

Okay, now for dS I get dS= sqrt(4x^2 + 4y^2 + 1) dA

Now if these calculations are correct, I'm really struggling with setting up with integrals. The projection onto the xy plane is a circle of radius 2 but after converting everything into polar, I'm not exactly sure I went in the right direction. Any help would be greatly appreciated.

Double Integral over S of (z + x^2y)dS where S is the part of the cylinder y^2 + z^2 = 1 that lies between x=0 and x=3.

I've parametrized the surface with respect to X and theta (cylindrical coordinates) taken the partial derrivatives with respect to x and theta, taken their cross product to reveal that dS = 1dA. If this is correct, where I am making my mistake is somewhere within setting up the bounds for dA. Help?

Okay find the volume of the solid enclosed by

the cone z=(x^2+y^2)^1/2 and

the sphere x^2 + y^2 + z^2 =2

Everytime i work this out I end up with zero so I think I'm doing something wrong in the setup.

Here is how I set up the triple integral, maybe you can catch my problem?

Converting to polar coordinates I get z=r and z=(2- r^2)^1/2

My inequalities are as follows: (all include or equal to)

r < z < (2 - r^2)^1/2

0 < r < (2)^1/2

pi/4 < theta < (3pi)/4

I then integrate with respect to z first followed by r and theta and yes, I remember to include the extra r as a part of the integration. Help?

find partial z by partial x at (-1, 1, 2)

z^2=x^2 - y^2cos(pi*z) + 3xz +10

Are there any points on the hyperboloid x^2 - y^2 - z^2 = 1 where the tangent plane is parallel to the plane z=x+y?

The temperature T in a metal ball is inversely proportional to the distance from the center of the ball (which we will take at the origin). The temperature at the point (1, 2, 2) is 120 degrees.

(a) - Find the rate of change T at (1, 2, 2) in the direction towards the point (2, 1, 3)

(b) - Show that at any point in the ball the direction of greatest increase in temperature is given by a vector that points toward the origin.

I feel like I have the tools to solve this if I were given a function for T, that is find the gradient of T, evaluate it at the point in question and then dot the evaluated point with a unit vector in the direction of our new point. I can't seem to get started here without a function... Help?

I need a little bit of direction. I'm taking a University (calculus based) Mechanical Physics course and have a question on getting started with the following derive problem..

A small block of mass=m sits at rest at the top of a frictionless hemispherical mound of ice of radius=R. The block is given a slight tap and begins to slide down the side of the mound. Find the height=H above the ground where the block leaves the ice. Express the height as a function of the radius.

So I believe that I'm looking for a point where the Normal Force is equal to zero - I think this is where the block would leave the ice but I'm not sure how to set things up. I also know I have an outside force of gravity (mg) and the centripetal force opposite the Normal force. I thought about creating a free body diagram but with the block moving, I'm not sure that a chance in coordinates helps much. I think I need to equate everything using the Conservation of Mechanical Energy Thm:

PEi + KEi = PEf + KEf. I think I use KEf as the point where the block leaves the ice? KEi would be zero as there is no movement?? Not entirely sure how to calculate the Potential Energy at either point, and again, don't know if this is the direction I should take. ANY help would be very much appreciated.

Is there away to calculute sums to infinity if they are non-geometric? I have the following:

E(from zero to infinity) [2^(n+1) + 3] / 4^n

Unless my calculations are off this series is not Geometric so I can't use a/(1-r) to find the sum. Any help is appreciated! This is review for a Calc 2 class...

So Sequence and Series was really glossed over in my pre calc class and now it is catching up to me... I'm able to find the general formula most of the time but I am struggling with sums of infinite series.

Find the Sum:

{4, -4/5, +4/25, -4/125 + ...} I know the general formula is: a_n=(-1)^n * 4/5^n but I have no idea how to calculuate the sum... Help?

Find the exact value of the sum:

{3(0.98)^2 + 3/2(0.98)^3 + 3/4((0.98)^4 + ...} Again, I know the general formula is:

a_n=[3(0.98)^(n+2)]/2^n but struggle with calculating the sum. Any help is VERY much appreciated!! Thanks!!!

Consider the region in the first quadrant bounded by the curves y=-x^2+6, y=x and x=0. Find the volume of the solid obtained by rotating the region about the y-axis.

Now I know I have to set this up with respect to y. My first integral will be bounded by 0 to 2 and my second integral will be bounded by 2 to 6 but I'm not 100% on how to set up the integrals. I know that the area of one "slice" is pi*r^2 and my radius is my "x" value, I just get stuck putting everything together. Thanks!!

2 of em that I'm struggling with:

A ball is thrown straight up with an initial velocity of 40.5 m/s, at the same instant a ball is dropped from the roof of a building 18.5 meters high.

After how long will the balls be at the same height? The acceleration of gravity is 10 m/s^2

and

Mary spots a plant falling outside her window at 1.14 m/s. Further down the same building, her friend clocks the same pot at 73 m/s. How far apart are the friends (in meters). Assume 9.8 m/s^2

A ball is dropped from rest at point O. After

falling for some time, it passes by a window

of height 2.2 m and it does so in 0.2 s. The

ball accelerates all the way down; let vA be its

speed as it passes the window’s top A and vB

its speed as it passes the window’s bottom B.

How much did the ball speed up as it passed

the window; i.e., calculate
vdown = vB−vA ?

The acceleration of gravity is 9.8 m/s2 .

Answer in units of m/s

Trying to review for calc 2:

indefitine integral of (x-1)/x^1/2dx

I think this goes as follows:

integral of x^1/2 - x^-1/2dx

= 2/3x^3/2 + 2x^-1/2 + c

Confirmation? Thanks!!