Exam/Question Deck Flashcards
2016 - Q4 b)
Answer ii)
Topic 3&4: If you see a boundary condition for the derivative of the main function which you are trying to solve, what do you need to think of?
To use the boundary condition, you need to remember to take the derivative of the ODEs for the three cases.
Any coefficients values you obtain from this expression need to be plugged back into the original function to obtain the solution for this boundary condition, not the derivative!
Can the hyperbolic cosine function (cosh) ever be 0?
The cosh function is never zero for real arguments.
What would this function look like graphically?
Seems like when they put cos t2 they mean cos(t2)
What does full width half maximum mean?
width of the curve at half the maximum height.
What is the thing you need to remember about doing guassian elimination?
Add row sums while you are doing the calculations
What do YOU need to remember about how to do eigenvector questions?
You’ve forgotten to express the eigenvectors as matrices in the end before.
Make sure you express the eigenvectors you find as matrices, these are easy points you cannot afford to drop.
What does a 90% confidence interval mean for the area of the right tail, if it is a one tail test?
That means that 0.1 will be above your z score threshold, and 0.4 below towards the mean.
If you are given two samples, with the mean and the standard deviation, and you are asked to find out which sample is statisically closer to a certain value, how should you approach the question?
Just because you are given two samples, and respective means and standard deviations, does not mean you need to do a two sample test.
Here they are asking how close the samples are to a certain value, not if they are statistically the same or if one is greater than the other.
You can just use the regular test statistic
If you are given a sample in the paper, what should you state about it?
When given a sample(s) you should state whether or not the sample has enough terms so that we can assume that the sample standard deviation is the same as the true standard deviation.
How do you have to use the variable P in your PDE calculations?
You replace √k by P, so that you don’t have to deal with the square root for the majority of the calculations. Bring K back at the end.
What is the equation for solving a ODE for the case K>0?…
Where P = sqrt(k)
PDEs: Should you use different constants for all the cases when solving the ODEs?
Yes you should. At least the answers do.
For a PDE question, if you get a first order ODE, i.e. g’(x)/g(x) = K), what is the solution to this ODE?
y = ekx + c = Cekt
PDEs: What should you think of when you see one condition for a ODE function?
e.g. the attached condition
If they give you only one condition for a function, don’t do the regular BC calculation, this value is a initial condition.
If you get a general solution for a PDE that has a trigonometric function, how do you solve for the coefficients of the trigonometric general solution?
First of all you need a IC to work with.
Once you get a solution from that IC, combine the coefficients into one term and use fourier series to calculate the coefficients.
Specifically you will need to use the equatoin to calculate the coefficient of a fourier series, because that is what you are doing.
Remember that when you aer solving this integral, that you need to integrate over a BC and the term in front of the integral should be the reciprocal of half the distance you are integrating over. So if you’re integrating over a distance of 0 to L, the term of front of the integral should be 2/L.
What is the procedure for solving a PDE question?
- Take note of the boundary conditions, which one is easier to solve, are there any derivative boundary conditions, are there any unconstrained conditions like intial conditions.
- Separate the variables if you need to.
- For the ODE function that has the easiest BCs, start going through the three cases and solving for each of the BC. You should get a solution for this function, which you can plug into the equation for the variable you are solving for to get a general solution.
- Since this ODE is equal to the same K values obtained you already know all the admissable values for the other one. Thus this limits the number of cases we need to solve for.
- After solving the last BC you want to add your solution to the general solution and then start solving for the coefficients. This is done through recognising that it is most likely a fourier series and you can use the integral that calculates fourier series coefficients to get the equation representing your coefficients. Make sure to combine all the coefficients into one.
- Any solution you get for the coefficients needs to be simplified as much as possible. With a trigonometric solution you need see if the arguments of the trig function lead to a patterned result , e.g.1 when n is even, -1 when odd, this can be simplified to -1n, which is 1 n is even and -1 when odd.
- Plug all solutions into general solution.
If you get this as a solution to a boundary condition, can you get any non-trivial solution for the general solution?
Options 1,2,4 all lead to trivial solutions whereby f(x) = 0.
The third solution of cosh(√𝐾. 𝐿) = 0 isn’t possible as there is no real value of K for which this is tru.
Recall that, for real values of its argument, the hyperbolic cosine function is never less than unity.
If you get a line integral like this what do you need to remember to do?
You solve the line integral in multiple parts.
You need to indentify which delta changes from one point to another and also what values stay constant during that path.
For example, when q moves from (1,2) to (1,1) you would integrate with respect to dy and but also note that the x value stays constant at x = 1 in this case.
If you are given these vectors what direction should you assume?
How do you check whether a line integral is independent?
Main thing to remember is that you need to take the term in front of dx and take the derivative wrt to y now (i.e. dy). Vice versa for the dy term.
Q6 from topic 5 worksheet
You need to use arc length.
Show that I is indenpent of the path taken…
Note that after verfying that I is independent of the path taken. They integrate 𝛿P with respect to the 𝛿x, obtain a arbitrary function of y to represent the constant you get from integrating and
Determine the region of the bounded region…
Why is 4 to 4.7 micro seconds one cycle?
Because as you can see the graph has been shifted upwards by 1.
If you bring it down by 1 you can more clearly see one cycle.
You need to remember that a cycle does not have to be a trough and a crest.
A cycle need to be one unique movement.
In this case if you wanted the traditional ‘cycle’, then all you would need to do is slightly shift the timeframe of 4 to 4.7 to the left.
Do part b) of this question
Bear in mind they squared the Kn in the exp term, which is wrong.
You need to use L’Hopitals rule, when there is a limit!
In this case only the trigonometric term has an ‘r’.
As r goes to 0 in the cosine, the term goes to 1.
Solve i) and ii)
What do you need to remember here?
That the constant D for some reason sticks with the lower order term. Likely because it is easier to get the general solution that way, that would make sense right…
Wha property does this matrix have?
It is orthonormal or orthogonal or symmetric
Orthonormal means that when you multiply the matrix by its tranpose then it is equal to the identity,
What is a really important thing you need to remember?
Specifically related to part iii)
You need to make sure you read the question properly…
they don’t say atleast… they say ONLY 3.
Solve this
Answer this question
Answer ii)
seems like when 1/2 gives the frequency threshold. Any frequency above this value is removed in this question.
Matrices question
Fourier Transform
