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Calculus & Mathematica: 153 Course Description

Approximations

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

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]
• cos[x]

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 Pi. 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 to 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 mirror do 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 approximations. Adaption of Euler, midpoint and Runge approximations to approximating the plots of the differential equation y'[x] = f[x,y[x]] with y[a] = b. 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 like 1/(1 + x²) have barriers and others like e^x and sin[x] do not. Why functions like x^5/2 do not have expansions in powers of x but do have expansions in powers of (x - b) for b 0. Why the barriers for f[t], f'[t] and f[x] dx 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

a₀, a₁r, a₂r², a₃r³, . . ., a_kr^k, . . .

is bounded , then the power series

a₀ + a₁x + a₂x² + a₃x³ + . . . + a_kx^k + . . .

converges for -r < 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.

Significant Changes From the Traditional Course

• Student-produced splines are used for motivation.
• Series of numbers are de-emphasized in favor of using series of functions for approximation.
• For the first time, the complex singularity criterion for convergence of expansions is revealed to calculus students.
• Taylor's formula comes near the end, where it can be appreciated instead of at the beginning.
• Students find out what expansions are and what they are good for by visualization instead of hearing about them in a lecture.
• Convergence of expansions at the "endpoints" is not treated.
• The traditional collection of convergence tests is gone.
• Limits of quotients at a point are calculated by dividing the leading term of the expansion of the denominator into the leading term of the expansion of the numerator -- just the way most scientists and mathematicians do this.
• L'Hospital's rule is put in its proper calculus context as an easy consequence of Taylor's formula.
• Treatment of functions defined by a power series via a differential equation makes it possible to eliminate this topic from follow-up courses in differential equations.