7. Calculus I
Course Description:
In Calculus we learn about the mathematical tools needed to study changing systems and dynamic processes. For example, what path does a baseball follow when it is hit for a long home run? How can we model the population growth of a species (for example, moose in some population) under pressures due to predation and limited natural resources in the environment? Without Calculus, these questions (and many others) are impossible to tackle mathematically. Calculus adds two main mathematical concepts to the toolbox you've already developed in order to allow us to study these sorts of questions:
1. Derivatives - the derivative of a function measures the rate at which the function is changing.
2. Integrals - this is the opposite of the derivative. If we're given the rate at which a function is changing, the integral allows us to work backwards and figure out what the original function was.
Surprisingly, derivatives and integrals have a huge number of applications to situations that superficially have nothing to do with rates of change. For example, how do you find the area underneath a curve or the volume enclosed by a region? The answer involves integrals. How can you approximate the value of the square root of 2 to 10 decimal places without using a calculator (for that matter, how does your calculator approximate the value of the square root of 2)? The answer involves derivatives.
The textbook is:
Stewart: Calculus, Early Transcendentals 6th edition (Covering Chapter 1-6, 9)
The teacher is:
Detailed Course Topic List:
0. Review of some Precalculus topics
Modeling the real world using functions
Some function review: linear functions, trigonometric functions
Limits and continuity
Motivation for the concept of a limit
Determining limits graphically and numerically
Limit laws and analytic calculation of limits
Continuity and the Intermediate Value Theorem
Derivatives
Derivative as slope of tangent line and as a limit
Tangent line and application to linear approximation
Derivative as the rate of change of a function
Calculating derivatives: product rule, quotient rule, chain rule
Implicit differentiation
Applications of derivatives
Derivative as measure of rate of increase/decrease
Second derivative as measure of concave up/concave down
Application to finding maximum and minimum values of functions
Using derivatives to understand the graphs of functions
Real-world applications: related rates of change
Real-world applications: optimization
(If time) Real-world applications: business and economics
Integration
Integration as opposite of differentiation
Review of summation notation
Riemann sums and interpretation via areas
Fundamental Theorem of Calculus
Integration techniques: pattern recognition, u-substitution
Applications of Integration
Exponential and logarithmic functions: definition via integrals
Exponential growth
Other special functions: inverse trig
Differential equations: slope fields, Euler's method, separation of variables
Differential equations: applications to physics, chemistry, biology, economics
Area between curves
Volume of a solid of revolution
Other applications as time permits (Arc length, surface area, probability)