# Calculus Book 3: Approximation

A third book in Calculus. Splines, Expansions, Taylor’s Formula, Convergence, Power Series, and more!

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## Syllabus

**3.01 Splines**

*Mathematics.*

Remarkable plots explained by order of contact. Splining for smoothness at the knots.

*Science and math experience.*

Experiments geared at discovering that the smoother the transition from one curve to another at a knot, the better both curves approximate each other near the knot. Splining functions and polynomials. Splines in road design. Landing an airplane. The natural cubic spline. Order of contact for derivatives and integrals.

**3.02 Expansions in Powers of x**

*Mathematics.*

The expansion of a function f[x] in powers of x as a file of polynomials with higher and higher orders of contact with f[x] at x=0. The expansions every literate calculus person knows:

1/(1 – x), E^{x}, Sin[x], and Cos[x].

Converting known expansions to others via change of variable.

Expansions for approximations.

*Science and math experience.*

Experiments geared toward discovering that using more and more of the expansion results in better and better approximation. Halley’s way of calculating accurate decimals of π. Expansions by substitution. Expansions by differentiation. Expansions by integration. Recognition of expansions. Expansions that satisfy a priori error bounds.

**3.03 Using Expansions**

*Mathematics.*

The expansion of a function f[x] in powers of (x-b) as a file of polynomials with higher and higher orders of contact with f[x] at x=b. Netwon’s method. Multiplying and dividing expansions. Using expansions to help calculate limits at a point. Expansions and the complex exponential function. Using expansions to help to get precise estimates of some integrals.

*Science and math experience.*

Centering expansions for good approximation. Newton’s method for root finding. Successes and failures of Newton’s method. Using the complex exponential to generate trigonometric identities. Comparing reflecting properties of spherical mirrors and the reflecting properties of parabolic mirrors. Using expansions to see why spherical mirrors have limited ability to concentrate light rays. Behavior of expansions very close to 0. Behavior of expansions far away from 0.

**3.04 Taylor’s Formula**

*Mathematics.*

Taylor’s formula for expansions in powers of (x-b).

*Science and math experience.*

Euler, Midpoint and Runge-Kutta approximations of f[x] given f'[x]. Experiments comparing the quality of Midpoint and Runge-Kutta approximations. Adaption of Euler, Midpoint and Runge-Kutta approximations to approximating the plots of the differential equation y'[x]=f[x, y[x]], with y[a] given. Taylor’s formula in reverse. L’Hospital’s rule by dividing the leading term of the expansion of the denominator into the leading term of the expansion of the numerator. Centering the expansion for best approximation. Experiments comparing the derivative of the expansion and the expansion of the derivative.

**3.05 Barriers to Convergence**

*Mathematics.*

Barriers and complex singularities. The convergence interval of an expansion as the interval between the barriers. Why some functions such as 1/(1 + x^2) have convergence barriers and others such as E^{x} and Sin[x] do not. Why functions such as x^{1/3} and Log[x] do not have expansions in powers of x but do have expansions in powers of (x-b) for b>0. Why the convergence intervals for f[x], f'[x] and are the same.

*Science and math experience.*

Shortcuts based on the expansion of 1/(1 – x) in powers of x. Using the expansion of 1/(1 – x) in powers of x for drug dosing. Infinite sums of numbers resulting from expansions. Barriers resulting from splines. Infinite sums and decimals. Experiments relating expansions in powers of x to interpolating polynomials. Runge’s disaster.

**3.06 Power Series**

*Mathematics.*

Functions defined by a power series. Functions defined by power series via differential equations. The power series convergence principle, which says that if for some positive number r the infinite list {a_{0}, a_{1} r, a_{2} r^{2}, a_{3} r^{3}, … , a_{n} r^{n}, … } is bounded, then the power series a_{0} + a_{1} r + a_{2} r^{2} + a_{3} r^{3} + … + a_{n} r^{n} + … converges for -r<x<r.

*Science and math experience.*

Experiments in trying to plot functions defined by power series. Experiments in plotting a function defined by a power series via a differential equation versus plotting the same function directly through Mathematica’s numerical differential equation solver. The ratio test for power series as a consequence of the power series convergence principle.

The functions E^{x}, Sin[x] and Cos[x] from the viewpoint of power series. Experiments in truncation of power series. The Airy function as a function defined by a power series.