Introduction: Why study calculus? Examples: Mathematical modeling of diseases, motion of stock prices, the calculation of pi.
(1.2) Mathematical modelling
(2.1) Tangent and velocity problems; limits
(2.8) Derivatives
(2.9) The derivative as a function
(3.1) Derivatives of polynomial and exponential functions. Why is the exponential function useful?
Various populations (yeast, e. coli, people) can be modelled by exponential growth. (We may do this later in the semester)
C-14 dating (Stonehenge, the Lascaux cave paintings, the Shroud of Turin. (We may do this later in the semester)
(3.2) Product and quotient rules
(3.3) Rates of change in the natural and social sciences
Why trigonometry? (The daily average temperature follows a sine curve; used by the Greeks to measure the size of the moon and sun and their distances from the earth)
(3.4) Derivatives of trigonometric functions.
(3.5) Chain rule
(3.6) Implicit differentiation
(3.7) Higher derivatives
(3.8) Derivatives of logarithmic functions. Why are logarithms useful?
(3.10) Related rates
(4.1) Maximum and minimum values.
(4.3) How derivatives affect the shape of a graph.
Applications of maxima and minima: The bee's eye; Elvis, the dog who knows calculus.
(4.10) Newton's method
Newton's method generates fractal patterns and chaos
(5.4) Antidifferentiation
(5.1) Areas and distances
(5.2) The definite integral
(5.3) Fundamental Theorem of Calculus
(5.5) Substitution rule
In addition, we will do some exercises from a very early calculus
book written by Maria Agnesi in 1748. She was a professor of
mathematics at the University in Bologna, Italy, and the first woman in
the western world who can legitimately be called a mathematician. Her
text on calculus was aimed specifically at Italian students. There is a
copy of the book in our rare book room. Portions of this book have been
scanned and are on the web, along with the first English translation,
which is in the Smith library.