## Four Lectures on the Application of Calculus

 Dates: 4/22-29, 2013, Course: Math1a, Harvard University Lecture by: Oliver Knill
 These lectures were taped during a single variable calculus course taught in the spring of 2013. The aim of these application lectures is to review the material taught earlier in the course and to show the power and breath of calculus in other fields. This page is online since September 23, 2013.

## Music, April 22, 2013

We review functions in the context of music. The topic allows to see how functions are built up from basic functions, review the exponential function, the trig functions. We learn what it means to change the frequency c and amplitude A in A sin(c x) or how to implement amplitude or frequency modulation.

## Statistics, April 24, 2013

This lecture introduces probability density functions, cumulative distribution functions, and the expectation. Examples cover the exponential distribution, the normal distribution and the arc-sin distribution. This lecture aims to review integration techniques like trig substitution and integration by parts.

## Economics, April 26, 2013

This lecture reviews what it means to have a maximum or minimum and derives a theorem in economics which deals with marginal cost f(x), total cost F(x) and average cost g(x) F(x)/x. This theorem tells that the break-even point f(x) = g(x) is an equilibrium point g'(x) = 0. The theorem is illustrated with strawberries.

## Computer Science, April 29, 2013

This lecture deals with integration of functions. We find a way to generate random functions and then differentiate them to get integration problems. We also discuss some general difficulties of artificial intelligence like finding patterns or errors in proofs.
Thanks to the Harvard Bok Center for teaching and learning for videotaping these four lectures . The recording was intended to be part of the HILT project "Calculus practitioner series: Meeting the changing needs of our students and client disciplines" which had the goal:

 "Awardees plan to design and record a multimedia series on the interdisciplinary nature of calculus with speakers from Harvard faculty from the STEM and quantitative social science fields." . The project was funded by a Hauser grant .