Assignment #4  Due Tuesday, October 14

Read Chapters 5 and get started on reading Chapter 6.  Although Chapter 5 is largely a warm-up for Chapter 6, the ideas in Chapter 5 are of independent interest and have applications of many sorts.

Do the following:

1. Exercise 1, p. 81  In addition to checking formula (2) in the text, also obtain the relationship between Taylor polynomials around a for a function f and Taylor polynomials around a for any anti-derivative F of f.

2  Exercise 3, p. 85

3. Exercise 6, p.86  This is pretty easy but often useful.  It's the justification for many variations on the idea of interpreting various integrals of a function against a non-negative function as giving weighted average values of the function.

4. Exercise 2, p. 88  This exercise goes through the slow way of establishing that for various rapidly vanishing functions defined using exponentials, their Taylor polynomials are all zero.  The fast way is as in class--using the characterization of the nth order Taylor polynomial for f around a certain point "a" as the unique polynomial p of degree less than or equal to n for which
f(x) - p(x) = |x-a|^n  (function going to zero as x approaches a).   For practice in induction arguments, carry out  Exercise 2 on the form of the kth derivative of f but don't try to get an explicit formula for the polynomial called P(x) in the problem--it's a mess!  Set up your induction argument in such a way that you can say what the degree of P is in terms of the order k of differentiation.

5.  One of the most useful applications of Taylor series is for expansions around some point of non-integer powers of a function.
      (i)  With p any non-zero exponent (possibly even irrational) what is the nth Taylor polynomial around 0 for the function
             f(x) = (1+x)^p?
      (ii)  Use a substitution method to obtain  from (i) the nth Taylor polynomial around 0 for the function g(x) =(x +b)^p with b any non-zero real number.    Putting x=a, to what extent  does the resulting formula resemble the binomial theorem--i.e. compare the pattern of coefficients for any p with those given by the binomial theorem when p is a positive integer?
       (iii)  Use (i) to get an expansion through 6th order terms in v for h(v)= sqrt(1-v^2/c^2) where c is a non-zero number.  Special relativity calculations regularly use such Taylor expansions for this fuction h with c being the speed of light and v the velocity of an object traveling through space.

6.  This exercise involves using tricks to quickly pass from one variable Taylor polynomials to n-variable Taylor polynomials.  Later we will go over the proofs of the chain rule for differentiable n-variable functions and the proof of the equality of mixed partial derivatives under the assumption that the pertinent derivatives not only exist but are continuous in some open ball about the evaluation point.  For now, assume these things.
         Assume f is a function from an open set S in R^n (open means that for every point a in S, some ball about a is contained in S) and a is a point in S with B a fixed ball of some radius r>0 about a contained in S.  Also assume that, for some N >= 1, all Nth order mixed partial derivatives of f exist and are continuous on B--automatically, all lower order mixed partials of f also exist and are continuous on B.  Let D_i be the operator taking the  first order partial derivative of a function with respect to the ith variable
         Fix b in B and let g(t) = f(a+t(b-a) for t in [0,1].  Thus g(1) = f(b) and g(0) = f(a).   By the n-variable chain rule,
g'(t) = Df(a+t(b-a)) where D is the sum of (b_i -a_i)D_i for i from 1 to n.  Then the kth derivative of g at t is (D^k f )(a + t(b-a)) where D^k f means  D is applied k times to f.   It follows that the evaluation at t=0 of the Nth Taylor polynomial for g around 0
is the sum of terms (D^k f )(a)/k! for k between 0 and N.  We define this sum to be evaluation at x=b of the Nth Taylor polynomial for f about a.
            (i)  Justify writing D^k/k! as the sum over all multi-indices I =(i_1, i_2,....i_n) satisfying i_1+...i_n = k
of the terms (b-a)^I  D^I / I! where (b-a)^I means the products of the quantities (b_j -a_j)^i_j , D^I means the product of the operators D_j^i_j, and I! means the product of the factorials (i_j)!--with each such product, j goes from 1 to n.  DON'T reprove the multinomial theorem.  Instead, "justify" means arguing that one can simply borrow the theorem on the grounds that all of the objects in question can be "multiplied" against each other in any order.
           (ii)  Use (i) to write down a formula for the Nth order Taylor polynomial of f which is strikingly like the one-variable formula with multi-index sums replacing ordinary sums.
            (iii)  Use the Taylor remainder formula for g  to get a Taylor remainder formula for f.
             (iv)  Now replace b by the variable x in  your Taylor formulas.  Why does it follow from (iii) that
f(x) = (Nth Taylor polynomial for f around a)(x) + |x-a|^n (function going to 0 as x approaches a)?   DON'T try to reprove this "from scratch"--instead just use your remainder formula.