In this interlude, we will investigate a famous and simple way to model temperature using differential equations, known as “Newton’s Law of Cooling”. Although the situation is much more complicated in reality (see Heat Equation), we can already understand the calculus behind Newton’s Law of Cooling, which is a very useful approximation in many situations….
Tag: handout
Interlude 3 – Slope Fields
For any differential equation of the form dy/dx = f(x,y), we can construct a “slope field” for the differential equation, which allows us to understand the qualitative behavior of the solutions without needing to know a formula for the solutions. The following set of notes introduces the main ideas, and has some exercises for you…
Interlude 2 – Introduction to Differential Equations
Starting with Interlude 2, and continuing through Interludes 3 and 4, we will investigate how our ability to understand and calculate derivatives lends itself to some powerful modeling applications in the real world. These notes for Interlude 2 introduce “differential equations” (DE’s) and “initial value problems” (IVP’s), and the important idea that even though solving…
The Definition of ‘e’ (0.4, 2.5)
In this handout, we investigate an alternative way of defining the number e, as the base of the exponential function which is its own derivative. This Desmos graph lets you experiment with different bases of an exponential function, and observe the connection between the exponential function and its derivative.
Implicit Differentiation (2.4)
These notes contain a summary of the important ideas behind implicit differentiation, its connection to the Chain Rule, and how it may be used to find tangent lines to general equations. You can use this Desmos graph to look at the graphs produced by various equations, as well as to check that your answers to…
Interlude 1 – Newton’s Method
The handout below contains a summary of the important points about Newton’s Method, as well as introduces the Bisection Method for comparison. There are some exercises at the end for you to test your understanding of some of the important ideas. This Desmos graph gives a visual representation of what is going on during each…
Derivative Rules Handout
The above PDF handout has a comprehensive list of the derivatives of the essential elementary functions, as well as the rules, from which the derivative of any elementary function can be calculated by hand. Although we will derive almost all of these statements in class, they are – perhaps more than anything else in the…
Precalculus Review Problems
The above PDF handout has a list of problems that should help you recall and review important ideas and methods from algebra, trigonometry, and precalculus generally. Please consult Chapter 0 of the text and seek help during your instructor’s office hours or at the Math Tutoring Center whenever you have questions.