Takeaways
On this page, we will collect key course “takeaways” i.e. some of the most important mathematical ideas which the course explores.
- Trigonometric functions are defined first on $[0, \pi/2]$ in terms of triangles, then on $[0, 2\pi]$ via the unit circle, and then finally to all of $(-\infty, \infty)$ via periodicity.
- A radian is a measurement of the size of an angle, and 1 radian is the size of the angle which cuts out an arc of length 1 on the unit circle.
- When computing a derivative (or anti-derivative) it is sometimes more efficient to do some simplifying algebra before computing the derivative (or anti-derivative).
- Calculating complicated areas (e.g. the area bounded by a curve and an axis) is challenging, so we begin by approximating the area with a bunch of rectangles (whose areas we know how to compute), and then take the number of rectangles to infinity to compute the exact area (calculus is the mathematics of limits!).
- A definite integral represents the exact area under a curve and -- via the Fundamental Theorem of Calculus -- we can use them to calculate the areas of complicated two-dimensional regions.
- Sometimes it is useful -- for example, when integrating -- to introduce a new variable "u" and make substitutions to simplify an expression before making computations.
- The Fundamental Theorem of Calculus (Parts I and II) together make precise the idea that integration and differentiation are "inverse" operations.
- When we can't solve a differential equation exactly (for example, an inseparable differential equation), we can still study the solutions using slope fields or Euler's method.
- The equilibrium solutions of a differential equation (DE) provide insight into the various types of solutions to the DE, and the equilibria are calculated by finding the values of the dependent variable which make the derivative equal zero.
- Applications of differential equations include modelling radioactive decay, substances in a chemical reaction, temperature of an object, concentrations of chemical solutions, and population growth (including exponential and logistic growth, with or without harvesting or migration)!