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Pretzel Logic
Taylor & McLaurin Series

 

 There are two basic types of functions.

 One type is called algebraic functions.
 Stuff like ...
 

F(X) = 3X5 - 2X3 + X2 - 32

 
 The other type is called transcendental functions.
 Stuff like ...
 

 F(X) = cos23X

 
F(X) = eX
 

F(X) = ln(2X)

 
 Doing math with algebraic functions,
 is usually a lot easier than math with transcendental functions.
 Some math types realized that if they could find
 algebraic functions that graphed like transcendental functions,
 they might be able to do math with them instead.
 That would make calculations easier.
 
 Suppose we have some curvy transcendental function ...
 

 
 Now we want to find an algebraic function that looks just like it.
 
 We start with a straight line that we "tack"
 to the transcendental function at some X value.
 Call it "a", that is, X = a.
 We make the slope of the straight line
 the same as the slope of the transcendental function
 at the X value.
 Remember, the slope is the derivative.
 

 
 The first question is, 
 what is the slope of the straight line?
 
 Back in algebra, you probably saw something
 called the point slope equation for a line.
 It looks like this ...
 

Y - Y1 = m(X - X1)

 
 We are going to use this with a couple of changes of symbols.
 The slope of the straight line "m",
 is also the derivative of the transcendental function
 at the point where X = a, that means m = F'(a).
 The coordinates (X1, Y1) are any point on the line, 
 so we can say X1 = a and Y1 = F(a).
 To recap ...
 
m  = F'(a)
   
X1  = a
   
Y1  = F(a)
 
 Doing the substitution we get ...
 

Y - Y1 = m(X - X1)

 

Y - F(a) = F'(a)(X - a)

 
 Solving this equation for Y, we get ...
 

Y = F(a) + F'(a)(X - a)

 
 The only place that Y and F(X) are the same value
 is at the point X = a.
 

 
 We need to bend the straight line equation Y,
 to match the scary function equation F(X).
 
 Swell, how do we do it?
 
 We do it by more terms to the "Y = " equation.
 The second derivative tells us the concavity of the function
 so this is a good place to start.
 That is, with F'(a).
 We need to multiply it by something that's derivative is (X - a).
 In other words, the integral of (X - a) ...
 

 
 So the next term we add is ...
 

 
 which is usually written as ...
 

 
 This makes the whole equation to here ...
 

 
 The next term is the derivative of F''(X)
 multiplied by the integral of (X - a) 2.
 That works out to be ...
 

 
 When we have higher derivatives, 
 we sometimes write the exponents like F(3)(X)
 rather than F'''(X).
 Just don't mistake that (3) for an exponent!
 F(X) to the third power would be written (F(X))3.
 
 Each of these terms has bent the "Y" function
 closer to the transcendental function.
 

 
 The closer we are to the point X = a, 
 the closer the value of our substitute function is to our nasty function.
 
 Writing these terms in a slightly different,
 but totally equal form might show you a pattern to them ...
 

 (in case you forgot, 0! = 1)
 
 Each term in this is ...
 

 
 These terms can go on forever.
 Each one would add little changes to the graph of the substitute function.
 At some point we have to say "Hey, enough is enough!"
 and stop adding terms.
 
 The question is, how do we know when to stop adding terms.
 
 After we are past the first few terms, 
 the value of all of the remaining terms added together
 is less than the value of the last term that we just added.
 For example, if the 6th term adds .0005 to the function value,
 all of the terms from the 7th and beyond will add less than that.
 If you can live with an error of no more than .0005,
 6 terms is enough.
 If you need to be more accurate, keep on adding terms
 until the value of the last term that you add 
 is less than the amount of error that you can accept.
 
 Let's try one ...
 
 Example:
 
 Find the algebraic function that is approximately equivalent to ...
 

F(X) = eX

 
 We can choose any value that we want for a.
 The error we get is smaller for X values close to a,
 but we usually choose a value for a that is easy to calculate.
 a = 0 is a good choice for anything except logs and tangents
 and other equations where X = 0 is undefined.
 X = 0 IS defined for eX, so we can use it for a.
 

 
 So ...
 

 
 That simplifies to ...
 

 
 We can use this to evaluate eX for any value of X.
 Just substitute the value for X and add up the terms.
 
 For Example ...
 

 
 The last term we added was 0.0889 so our error can be no more than this.
 Actually, e2 to 4 decimal places is 7.3891 so the error is only 0.0335.
 
 Let's do another one ...
 
 Example:
 

 F(X) = sinX

 
 Again, a = 0 is an easy value to use ...
 

 F(0) = sin0 = 0 so that means F'(0) = cos0 = 1

 
 So ...
 

 
 As you can see above, every other term equals zero.
 That leaves us with ...
 

 
 So sin(p/4) is approximately ...
 

 
 The actual value of sin(p/4)  is about .70711 to 5 digits.
 We have an error of 0.0003.
 The last term we added was -.00249,
 so we knew that the error would be no more than that.
 
 This stuff may not seem too exciting,
 now that we have big time calculators and Excel and what not.
 
 POP QUIZ:
 How do you think the calculators and Excel
 and what not calculate the answers?
 
 We could write these things using summation notation.
 In fact, we almost did before when we discovered the pattern.
 The full official summation notation is ...
 

 
 This particular series gives good approximations of transcendental functions.
 
 Here are some of the more often used series,
 at a = 0.
 ( That makes them McLaurin Series [Big deal eh?])
 

 
 We can find the integral or derivative of any of these functions 
 term by term to as much accuracy as we want.
 
 We can find the product of any pair of functions 
 by multiplying them together term by term.
 Be very careful about how much accuracy you need when you do this.
 Multiplying n terms together from each of two functions,
 gives you n n terms.
 You WILL be able to combine lots of terms after it's all multiplied out,
 but in the mean time you will have a big mess.
 
 We can divide one series by another using long division.
 Just pick an accuracy level you want and divide.
 
 All of these are called Taylor series.
 The ones where a = 0 are called McLaurin series.
 (McLaurin is a subgroup of Taylor)
 
 This is OK and all, but what we really need are a set of general rules 
 that tells us if a series that goes on forever adds up to some number
 and what that number is, or infinity
 
 You'll never guess what the next chapter is about!
 

   copyright 2005 Bruce Kirkpatrick

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