# Calc-50 programming and graphics

David M. Smith
Professor of Mathematics (Emeritus)
Loyola Marymount University
Los Angeles, CA Here are some examples that use Calc-50 programming and graphics.

Program Fourier series using the sum key and display with plot.
Fourier series

Evaluate infinite sums to high precision.
Infinite sums

Analyze the rate of convergence of an infinite sum.
Arctan sum

Evaluate an infinite nested sequence of radicals.
Use the sum key as a loop control and the select key as an if statement.

Evaluate the limit of a sequence using extrapolation.
Approximate infinite sums, products or limits.
Extrapolation

Evaluate an infinite sum with only prime terms.
Define the prime counting function and try extrapolation to get more accuracy from a sum.
Prime sum

Derive a Guass quadrature integration rule.
Evaluate Legendre polynomials, and use combinations, derivatives, sums, equation solving.

Solve an equation where the function involves an integral.
The solve key calls a function which in turn calls the integrate function.
Solve an integral equation

Solve an ordinary differential equation and do a list plot of the solution.
Call the ode function repeatedly to make a list of points y(x(i)), and then plot the list.
Solve a differential equation

Use ode called from the solve function for a boundary value problem.
Define a function with a root that gives the solution to a boundary value problem.
Solve a boundary value problem

Estimate values for partial sums of a divergent series.
Derive an asymptotic formula to approximate a finite sum.
Asymptotics

Estimate the value of a double sum.
Use the sum key with a function that also uses the sum key.
Generate partial sums and then extrapolate.
Double sum

Estimate the number of primes less than a large number.
Compare the prime number theorem, log integral, and Riemann's prime counting function.
Define a function for the Moebius function from number theory.
Prime counting

Compute the integral of a function that has infinitely many oscillations with
ever increasing amplitude as x approaches zero.
Oscillating integrals

Use the complex arithmetic screen to sum a complex series.
Complex sum

Use the complex arithmetic screen to search for complex roots of an equation.
One example uses complex secant iteration for approximating roots.
Another example evaluates complex line integrals to count the number of roots within a
specified circle in the complex plane.
Complex roots

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