Math Is Fun Forum

If anyone could explain how the following problem is done, it would be greatly appreciated!

The rate at which people get sick during a flu epidemic is modeled by the function R(t)=1000 * t * e^-0.5t, where R(t) is measured in people per day and t is measured in days since the start of the epidemic.

Use an improper integral to find out how many people get sick altogether.

I found this calc problem in a chapter on improper integration. If anyone could explain how it is done, I would greatly appreciate it!

A car is travelling at 55 mph. The driver sees a traffic jam ahead and hits the brakes. Brakes apply friction. The car's velocity satisfies the differential equation v' = -1080v miles per hour per hour. How far does the car go after the brake is applied?

This problem is puzzling me so any help would be greatly appreciated!

Find the interval and the radius of convergence for the the following series.
a) The Maclaurin series for f(x)=xe^x

So far, I've multiplied the series for e^x by x and I've gotten:

x+x^2 +x^3/2! + x^4/3!...

However, I'm confused about what to do next....

I was confused as to how to do this, so any help would be appreciated

I have to see if the following series converges and justify my answer.

Ok, I'm given

g(x) = cos(2x), a=0, (-2pi, 2pi)

Using this information, I have to answer the following questions.
a) Find the Taylor series T for the given function g based at the given point a.
b) Find, if possible, an error-bound for the difference between g(x) and its degree (n-1) Taylor approximation T sub n-1 of x on the given interval.
c) On the basis of the answer to (b), can you conclude that T=g on the interval given?

I'm confused as to how to go about this, so someone can explain how this is done, I would REALLY appreciate it!

Thanks in advance.

Suppose we have a 3 by 3 grid (used for perhaps playing some weird kind of hopscotch.) A child is standing in a square on the grid (namely x). Once time per second, he jumps to another square that is adjacent to x (that includes diagonally as well).

I was wondering:
1) how one would find the transition matrix for the associated Markov chain.

and

2) what happens in the long term? if the markov chain does not have a steady-state vector, then explain what happens instead.

I'll show you my work so far:

I numbered the grid 1-9 like so:

1 2 3
4 5 6
7 8 9

Each number is in an individual box, but since I don't know how to draw boxes on this thing, let's imagine that they are!

I concluded that from square 1, the hopper can go to 2, 4, and 5. Thus, the probability of going to each of those from 1 is 1/3.

From square 2, the hopper can go to 1, 3, 4, 5, and 6. Therefore, the probability of going to each of those from 2 is 1/5.

And so on. As you can see, I've already gotten the probabilities. But how do I represent them in a transition matrix?

So I'm having trouble coming up with this proof, which I need to know for my quiz tomorrow, so any help would be greatly appreciated. 

An orthogonal matrix is defined as one for which A tranpose = inverse of A

In other words A^T = A^-1

Prove if a matrix A is orthogonal, then any two distinct columns of A have dot product zero.  Is the converse true?  Justify your answer.