Series 215 problem 07.mws

1. Using Maple to find Taylor polynomials .

Maple can compute Taylor Series and MacLaurin polynomials with the series command. For example, to compute the 5th order MacLaurin polynomial for the function

> f :=x->sqrt(x+1);

f := proc (x) options operator, arrow; sqrt(x+1) en...

we enter the commands below.

> taylor(f(x),x=0,6);

series(1+1/2*x-1/8*x^2+1/16*x^3-5/128*x^4+7/256*x^5...

> convert(%,polynom);

1+1/2*x-1/8*x^2+1/16*x^3-5/128*x^4+7/256*x^5

In order to understand where these coefficients are coming from, let us apply the equation

f(x) = Sum(`@@`(D,n)(f(a))/n!*(x-a)^n) = f(0)+D(f)(a)/1!*(x-a)+`@@`(D,2)(f(a))/2!*(x-a)^2 + . . .

where a = 0 .

We compute some of the derivatives of our function at zero, to see the pattern.

> f(0),D(f)(0),(D@@2)(f)(0),(D@@3)(f)(0),(D@@4)(f)(0),(D@@5)(f)(0);

1, 1/2, -1/4, 3/8, -15/16, 105/32

The pattern is not too difficult to determine. The first coefficient is 1/2, the second is (3-2*2)/2 times the first, the third is the (3-2*3)/2 times the second, and the k-th coefficient is given by multiplying the previous coefficient by (3-2*k)/2.

The actual coefficients in the power series are given by dividing these derivatives by the appropriate factorial. Thus we have

> f(0)/0!,D(f)(0)/1!,(D@@2)(f)(0)/2!,(D@@3)(f)(0)/3!,(D@@4)(f)(0)/4!,(D@@5)(f)(0)/5!;

1, 1/2, -1/8, 1/16, -5/128, 7/256

Note that these are the first six coefficients in the Maclaurin series.

Submission:

For the functions below, do the following,

a) Find the 5th order Taylor polynomial about the point x = a .

b) Use Taylor's theorem to find the coefficients as is illustrated above, and verify that they are the same as you

obtained in part a).

c) Plot the function and its 5th order Taylor polynomial n some interval containing the point x = a .

Submission worksheet:

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2. Using Maple to find more Taylor polynomials.

Recall the Taylor series (in Maple's notation) is given by

Sum((`@@`(D,n)(f)(x)/n!)(x-a)^n,n = 0 .. infinity)

Maple can compute Taylor Series and MacLaurin polynomials with the series command. For example, to compute the 5th order MacLaurin polynomial for the function

> f :=x->sqrt(x+1);

f := proc (x) options operator, arrow; sqrt(x+1) en...

we enter the commands below.

> taylor(f(x),x=0,6);

series(1+1/2*x-1/8*x^2+1/16*x^3-5/128*x^4+7/256*x^5...

> convert(%,polynom);

1+1/2*x-1/8*x^2+1/16*x^3-5/128*x^4+7/256*x^5

In order to understand where these coefficients are coming from, we compute some of the derivatives of our function at zero, to see the pattern.

> D(f)(0),(D@@2)(f)(0),(D@@3)(f)(0),(D@@4)(f)(0),(D@@5)(f)(0);

1/2, -1/4, 3/8, -15/16, 105/32

The pattern is not too difficult to determine. The first coefficient is 1/2, the second is (3-2*2)/2 times the first, the third is the (3-2*3)/2 times the second, and the k-th coefficient is given by multiplying the previous coefficient by (3-2*k)/2.

Submission:

(a) Find the Taylor polynomial for f(x) = sec(x) at a = 0 (that is find the McLauren polynomial) for n = 2, 4, 6, 8 .

Then graph the four polynomials and the secant function on the same set of axes.

(b) Find the Taylor polynomial for f(x) = tan(x) at a = 0 for n = 1, 3, 5, 7, 9 . Then graph the five polynomials

and the tangent function on the same set of axes. Designate the color of each graph so that you will know which graph corresponds to which polynomial.

(c) Write a short paragraph about what you observe.

Submission worksheet:

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3. Applications of Taylor polynomials.

The resistivity rho of a conducting wire is the reciprocal of the conductivity and is measured in units of ohm-meters ( Omega -m). The resistivity of a given metal depends on the temperature according to the equation

rho(t) = rho[20]*exp(alpha*(t-20))

where t is tempeature in Celsius, alpha is the temperature coefficient, and rho[0] is the resistivity at 20 degrees celsius.

This problem contains a theme common in many scientific problems. In this case resistivity depends exponentially on temperature. However, near 20 degrees C, we may approximate this relationship with a linear or quadratic one, which is usually easier to analyze. The linear and quadratic approximations are given by the first and second order Taylor polynomials centered at 20.

Submission:

(a) Find expressions for these linear and quadratic approximations.

(b) For copper we have alpha = .39e-2 per deg C and rho[20] = 1.7*x*10^(-8) Omega -m. Graph the resistivity of copper and

the lilnear and quadratic approximations for t between -250 and 1000 degrees celsius.

(c) For what values of t does the linear approximation agree with the exponential expression to within one percent?

Submission worksheet:

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