Warning: include(../prefs.php) [function.include]: failed to open stream: No such file or directory in /usr/www/users/webprim4/rcurtis2/suffolk/courses/calcIII.php on line 3

Warning: include() [function.include]: Failed opening '../prefs.php' for inclusion (include_path='.:/usr/local/lib/php') in /usr/www/users/webprim4/rcurtis2/suffolk/courses/calcIII.php on line 3

Warning: include() [function.include]: Failed opening '' for inclusion (include_path='.:/usr/local/lib/php') in /usr/www/users/webprim4/rcurtis2/suffolk/courses/calcIII.php on line 4
Calculus III
(Series)
Detailed Syllabus

Mathematics

• Remarkable plots explained by order of contact. Splining for smoothness at the knots.
• 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), ex, sin[x] and cos[x]).
• Expansions for approximations.
• 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.
• Newton'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

• 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.
• 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.
• Expansions by substitution.
• Expansions by differentiation.
• Expansions by integration.
• Recognition of expansions.
• Expansions that satisfy a priori error bounds.
• 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 relecting 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.

Mathematics

• Taylor's formula for expansions in powers of (x - b).
• 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[x],f'[x] and f[t] dt are the same.
• 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, a₁ r, a₂ r², a₃ r³,. . . , a_k r^k, . . .} is bounded , then the power series a₀ + a₁ x + a₂ x² + a₃ x³ + . . . + a_k x^k + . . . converges for -r < x < r.

Science and Math Experience

• Euler, Midpoint and Runge 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.
• 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.
• 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.

Warning: include() [function.include]: Failed opening '' for inclusion (include_path='.:/usr/local/lib/php') in /usr/www/users/webprim4/rcurtis2/suffolk/courses/calcIII.php on line 91