# Math problems to tell where you are with your knowledge of math

1. Dec 28, 2011

### 2slowtogofast

Sort of a strange topic But I was wondering if some of you could suggest some problems for me to try to see how much math I understand. My background is I am a Mech Eng student in my last year of my bachelors degree. So I have taken calc 1-3 Diff Eq, Linear and a Applied math course for engineers. I guess I just would like to get a good idea of what I know. I do problems online and out of text books but I don't really know how hard or easy they are compared to what other people do. I am not asking for impossible questions just some questions that some one with my background should be able to do that are interesting or challenging. Thanks

2. Dec 28, 2011

### micromass

1) Find $\int \log(x)dx$.

2) Find $\int_0^\infty \int_0^\infty e^{-x^2-y^2}dxdy$ and deduce $\int_0^\infty e^{-x^2}dx$.

3) Does the series $\sum{\frac{n^2}{3^n}}$ converge?? Find its limit.

4) Find the Fourier series of the function f(x)=x if $x\in [0,\pi]$ and $f(x)=0$ if $x\in [-\pi,0]$.

5) Is the series $\sum{ \frac{1}{3^n}\sin(nx)}$ the Fourier series of some function?? Of which one??

6) Take the function $f(x)=\frac{1}{x^2}$ for x>1 and rotate it around the x-axis. What is it's volume and surface area??

7) Let $A=\left(\begin{array}{ccc} -1 & 0 & -1\\ -1 & 0 & 0\\ 2 & 1 & 2\\\end{array}\right)$. Find $A^{1000}$.

8) Solve the following system

$$\left\{\begin{array}{cc} u^\prime= 3u+4v\\ v^\prime= u+v\end{array}\right.$$

9) At time t=0 a tank contains Q0kg of salt of salt dissolved in 100l of water. Assume that water containing 1/4 kg of salt/l is entering the tank at a rate of r l/min and that the well-stirred mixture is draining from the tank at the same rate. Find the limiting amount of salt that is present after a long time. After which time is the salt level within 2% of the limiting amount? What flow rate would be required to obtain a salt level within 2% if it is required that the time does not exceed 45 min.

Something more theoretical:

10) Classify all the 6x6 matrices A such that the column space of A equal the nullspace of A.

3. Dec 28, 2011

### 2slowtogofast

Thank you this was what I am looking for. I am going to start working through these tommorow and I will post my solutions and hopefully you guys can let me know if they are correct. Thanks

4. Dec 29, 2011

### 2slowtogofast

1.) I always forget if you write log without a base is that base e or 10? I went with e here is my solution

∫logx dx = uv - ∫v du

u = logx du = 1/x
dv = dx v = x

plug in

∫logx dx = xlogx - ∫x (1/x) dx

∫logx dx = xlogx - ∫ dx

∫logx dx = xlogx - x

5. Dec 29, 2011

### micromass

Correct, but you forgot the integration constant...

6. Dec 29, 2011

### 2slowtogofast

2.)

$\int^{∞}_{0}\int^{∞}_{0}$e-x2-y2dxdy

Change to Polar Co-ords

-x2-y2=r2cos$\theta$2-r2sin$\theta$2

Factor the r2 and sin2+cos2 goes to 1

-x2-y2=-r2

Rewrite the integral

$\int^{∞}_{0}\int^{\pi}_{0}$e-r2rd$\theta$dr

Since there is no theta everthing is treated as a constant Evaluating the inner intergral

$\pi$$\int^{∞}_{0}$e-r2rdr

U substitution: u = r2 After finding du and solving for dx then plugging in we get

$\frac{\pi}{2}$$\int^{∞}_{0}$e-udr

=$\frac{\pi}{2}$(-e-∞+e0)

=$\frac{\pi}{2}$

I skipped a few steps when typing this because i am not very familar with latex. This problem took a while but i think I got it spent alot of time going through my calc book. For the second part is it just half of the answer so pi/4 because it is the same integral with the same limits repeated twice?

Anyway I have a question on the next one let me know if i am on the right track. I used the integral test to show that it converges but i can not find what it goes to. I tried taking the lim as n goes to inf and then using l'hospitals rule but no luck.

7. Dec 29, 2011

### micromass

The inner integral should go from 0 to $\pi/2$.

It's not half the answer but rather the square root of the answer. Do you see why??

The trick is to start from

$$\frac{1}{1-x}=\sum {x^n}$$

and to differentiate both sides.

8. Dec 29, 2011

### 2slowtogofast

I see why its sqd now b/c you can write it as (e^-x2)(e^-y2) so since the limits are the same the only thing different is the x and y should have seen that.

I dont see why the limit is pi/2 though if the original int was -inf to inf then it would be 0 to 2pi right? so if that is true would't this case be 0 to pi because the integral goes from 0 to inf. I am on my phone so i couldt type this out in latex sorry

9. Dec 29, 2011

### micromass

The area that you integrate over is the first quadrant. If you want to cover that area with polar coordinates, then your angle can only go from 0 to pi/2. An angle higher than pi/2 would go out of the first quadrant.

10. Dec 31, 2011

### 2slowtogofast

I am taking a break from 3 here is 8.

Solve the system:

u' = 3u + 4v
v' = u + v

Solve v' for u

u = v' - v

Differentiate both sides

u' = v'' - v'

Sub u and u' into the first eqn to get an eqn only in v

v'' - v' = 3(v' - v) + 4v

This will simplify to

v'' - 4v' - 1 = 0
Write the C.E.
s2 - 4s - 1 = 0

The roots are 2+/- √(5)

v(t) = C1e-(2- √(5)t) + C2e-(2+ √(5)t)

Take the derivative of v(t)

v'(t) = -(2 - √(5))C1e-(2- √(5)t) - (2+ √(5))C2e-(2+ √(5)t)

Use u = v' - v to solve fot u(t)

After substitution of v' and v we get

u(t) = (-3 + √(5) C1e-(2- √(5)t) + (-3 - √(5) )C2e-(2+ √(5)t)

Since no I.C.'s can not solve for constants