Monday, November 17, 2008

Differential Equations Project Tips

As some questions are asked more regularly, I thought I'd provide some general discussion here.

(1) Start with the proposed form of X(t).  Compute X'(t) and X''(t) based on that form.  Then use those calculations to discover when X'' + k/m X = 0.

(2) The question "mean physically about the mass on a spring" is not asking you to think about the mass (as in measurement) but is asking you to think about what the statement X(0)=1 means about the state of the mass at time t=0 and what the statement X'(0)=0 means about the state of the mass at time t=0.

(3) X(0) is a constant and has derivative of d/dt[X(0)] = 0.  Recall that dX/dt > 0 implies that X is increasing, dX/dt < 0 implies that X is stationary (instantaneous rate = 0)

(4) An arbitrary quantity A is proportional to some other quantity B if it is always that case that A = k B for some constant value k (the constant of proportionality).  Now interpret the statements to identify what pieces of the equation are proportional to what.

(5) Although you have studied ex in precalculus, you will not use the logarithm at all in this work.  Instead, I just want you to consider some function that has the special property that exp' = exp.  (This sentence is analogous to sin'=cos and cos'=-sin.)  However, you do need to think about the chain rule: X(t) = A exp(rt)  (Since the argument is not simply t, you must use the chain rule.)  This problem is exactly analogous to Step 1.

(6) You will get something like
dX/dt = "formula involving X and a, b, and m"
X is increasing when "formula" > 0, decreasing when "formula" < 0, and stationary when "formula" = 0.  So use your skills with algebra (think sign analysis) to find conditions when these are the case.

(7) You need to understand the relationship between a rate of change and an actual change.  To understand this as well as possible, see the section we skipped in Chapter 3 (last section).  But what you essentially need is that we will follow the tangent line for the time increment Δt.  How much change is there when the rate of change and the duration of time are both known?

(8) The new version of Excel has some unanticipated differences from what I had when I wrote the project.  The labels are not assigned from a menu anymore.  Instead of the 3-step process that is described, you just click in the label field in the header section of Excel and type in the new label and then hit enter.

The calculations you see in the first few lines should exactly match your hand calculations in part (7).  

Do not print the spreadsheet (it takes WAY too many pages).  That is why I ask you to submit your spreadsheet on Blackboard as part of the project.

(9) I must receive a print out of the graph -- hand drawn figures are not acceptable.   Ideally, this entire project report would be typed (perhaps using Equation Editor for the equations), with the figures naturally fitting in.

(10) Make hypotheses and test your hypotheses.

Exponential Project Tips

As some questions are asked more regularly, I thought I'd provide some general discussion here.

(1) exp is the name of the function, just as sin and cos are names of functions. From calculus, you learn that sin'=cos and cos'=-sin.  This step shows that exp_b' = ln b * exp_b. (That is, it leaves the function alone except for a constant multiple. (But be careful where the chain rule is needed!)

(2) You do not need to use the limit definition (epsilons and deltas).  Instead, for perhaps the easiest solution, you should think about how to finish the statement:
lim b^x = lim [(b^x-1)/x ... ]
That is, if you start with (b^x-1)/x, what do you do to that expression to leave only b^x.  Then use elementary limit rules to compute your resulting limit.

(3) One method is to use the method of substitution for limits (change of variables) and then use an identity for the function so that the result of Step 2 is applied --- this method mimics what is done to show that sin x is continuous everywhere.  A second method is to use a general theorem that makes continuity an obvious conclusion of the results from Step 1.

(4) You must start with a statement like:
ln (1/b) = lim_{x → 0} [(1/b)^x - 1]/x
There are two easy approaches: (1) Find a common denominator to rewrite this as a simple fraction before continuing or (2) Think of (1/b)^x as b to some appropriate power and then use a limit substitution.

(5) Since you do not know the derivative of ln x, it is incorrect to use the Mean Value Theorem applied to the logarithm.  Instead, you should apply the MVT to the function exp_b(x) on an interval so that b^a and b^b are incredibly easy and where it is clear which value is larger (so that you know if the average rate of change is positive or negative).  You may use the fact that b^x is positive for all values of x.

(6) The function fb(x) is a linear function. You should write it in slope-intercept form (e.g., mx+b).

(7) Do not attempt to solve the equation fb(x) = exp_b(x).  There is one obvious solution from the definition: x=0.  But the formulas themselves do not explain where there would not be more solutions.  Instead, you should define a function (perhaps g) so that
g(x) = exp_b(x) - fb(x).
You know that g(0) = 0.  You need to show that g(x)>0 for all x ≠ 0.  My hint suggested Rolle's theorem, but I have since found that the Mean Value Theorem helps even more.  Use the Mean Value Theorem to show that for x>0, the average rate of change between 0 and x must be positive.  What about x<0? x="0?">

(8) You may not use a limit form of the type b. You may take a limit of the function fb(x) because that is of a form we know how to work with. Then you should use the result of (7) to conclude what the limit of exp_b(x) must be.

(9) and (10) put all of the previous steps together to perform analysis similar to Sections 4.2 and 4.4 to understand the graph.