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:

Peter Mannisto

Detailed Course Topic List:

0. Review of some Precalculus topics

• • 1. Modeling the real world using functions

    2. Some function review: linear functions, trigonometric functions

• 1. Limits and continuity

     1. Motivation for the concept of a limit

     2. Determining limits graphically and numerically

     3. Limit laws and analytic calculation of limits

     4. Continuity and the Intermediate Value Theorem

  2. Derivatives

• • 1. Derivative as slope of tangent line and as a limit

    2. Tangent line and application to linear approximation

    3. Derivative as the rate of change of a function

    4. Calculating derivatives: product rule, quotient rule, chain rule

    5. Implicit differentiation

• 3. Applications of derivatives

• • 1. Derivative as measure of rate of increase/decrease

    2. Second derivative as measure of concave up/concave down

    3. Application to finding maximum and minimum values of functions

    4. Using derivatives to understand the graphs of functions

    5. Real-world applications: related rates of change

    6. Real-world applications: optimization

    7. (If time) Real-world applications: business and economics

• 4. Integration

• • 1. Integration as opposite of differentiation

    2. Review of summation notation

    3. Riemann sums and interpretation via areas

    4. Fundamental Theorem of Calculus

    5. Integration techniques: pattern recognition, u-substitution

• 5. Applications of Integration

• • 1. Exponential and logarithmic functions: definition via integrals

    2. Exponential growth

    3. Other special functions: inverse trig

    4. Differential equations: slope fields, Euler's method, separation of variables

    5. Differential equations: applications to physics, chemistry, biology, economics

    6. Area between curves

    7. Volume of a solid of revolution

    8. Other applications as time permits (Arc length, surface area, probability)