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Friday, September 17, 2010

Introductory Maths and Stats: Single Variable Calculus...

As a first year Phd student I will be teaching introductory maths and stats to the MSc students this year. I am going through the lecture notes and making little notes for myself about things that I think should be emphasized (or de-emphaszied) in the turorials as well as some little tricks that I have picked up along the way that should be helpful for the incoming MSc students.  The following are my notes to myself on single variable calculus...

Rules of Differentiation: The derivtive is a linear operator.  Mathematically this means that d/dx(f(x) + g(x))=d(f(x)) + d(g(x)) and d/dx(t*f(x))=t*d/dx(f(x)).  In words this means that the derivative of any linear combination of well-behaved functions is equal to the same linear combination of the derivatives of the individual functions.  Note that if you remember that the derivative is a linear operator then you automatically know how to take derivatives of sums and differences of functions.

Other Rules I remember:
  1. Constant: The derivative of a constant is always zero.
  2. Powers: If f(x)=x^k, then df(x)/dx=kx^(k-1)
  3. The Chain Rule: NEVER forget the chain rule! d/dx(f(g(x)))=df/dg*dg/dx.  Most simple mistakes in taking a derivative come from forgetting about the chain rule.
  4. Derivative of the exponential and the natural logarithm functions: Easy...d/dx(e^x)=e^x (this result is one of the reasons that exponential functions turn up so often in the general solutions to differential equations), and d/dx(ln(x)=1/x.  Maybe review some basic properties of logarithms and exponentials...
  5. Product Rule: d/dx(f(x)*g(x))=d/dx(f(x))*g(x) + f(x)*d/dx(g(x))
Rules I never remember:
  1. Quotient Rule: Why? Because the quotient rule is simply an application of the product rule and the chain rule.
  2. Rule for d/dx(a^x): Why? Because it is better to just take natural logarithms and the differentiate.  For example, if f(x)=a^x then ln(f(x))=ln(a^x) and because I know the my properties of logarithms, the rule for taking d/dx(ln(x)) and the chain rule this becomes d/dx(ln(f(x))=[1/f(x)]*d/dx(f(x))=d/dx(ln(a^x))=ln(a) and finally d/dx(f(x))=ln(a)*a^x
l'Hopital's Rule: Unbelievably useful for taking limits.  Basically if you are ever in the case where the lim of two functions turns out to be 0/0 or inf/inf, then take deriviates of the top and bottom and take limits again...

At this point the lecture notes have a discussion of 1st order differential equations that is out of place.  These equations have not been covered yet in the notes, and even though this discussion is brief it detracts from more important material.  Lecture notes also have a long digression on stock returns, capital gains, and dividends.  This is an important economic application of the material being taught, but should be covered by tutors in the QM0 tutorials where it can be gone over at a slower pace...

Optimization: Recall geometric interpretation of derivative: the value of a derivative at a given point tells you whether the function is increasing of decreasing at the point:
  • If df/dx>0, then the function is increasing
  • If df/dx<0, then the function is decreasing
  • If df/dx=0, then f has a critical point
At this point I like to draw pictures to help me remember that d^2f(x)/dx^2>0 (<0) implies that the function is convex (concave). Which leads to the corresponding definitions of maximums and minimums of functions.  I like to point out the main ideas both graphically and in terms of the FOC and SOC on derivatives.

Taylor Expansions: Important topic that the lecturer should spend more time laying out the details.  Re-empahsis should be placed on the Taylor Expansion in the tutorials.

Concavity/Convexity and Quasi-concavity/Quasi-convexity of Functions: I never remember the  derivative or algebraic definitions for these terms.  Best to draw pictures!  Three functions to remember f=x^2 (convex), f=ln(x) (concave), and f=x^3 (quasi-convex and quasi-concave)

Rules of Integration: Emphasize the area under the curve interpretation of an integral.  The rules for integration are easy IF you know your rules for differentiation.  The two processes work in reverse.  When taking an integral of f(x), think what function would I need to take the derivative of the get the function f(x).  Don't forget about the arbitrary constant!

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