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….

# Category: Desmos Links

## 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…

## 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…

## Linear approximations and their derivatives (2.2)

This graph shows how we can make an approximation to the graph of a function using a piecewise function where each piece is a secant line. For this piecewise function — since lines are easy to deal with — we can compute the piecewise derivative and see how this compares with the graph of the…

## Definition of the derivative (2.2)

This Desmos graph shows how the derivative represents the slope of the tangent line to the graph, and can be calculated as the limit of the slopes of secant lines. In the same way that the slope of a secant line represents an “average rate of change”, the slope of the tangent line represents an…

## Epsilon-Delta Definition of Limit (1.2)

This Desmos graph allows you graphically investigate the relationship between epsilon and delta when considering the formal definition of limit. There are two accompanying videos on the YouTube page: Video 1.2a explains the interpretation of epsilon, delta, and how you can make sense of the formal definition of limit. Video 1.2b shows how to solve…

## Limits from a graph (1.1)

This Desmos link shows what is happening graphically when you construct a table of values to investigate a limit. Desmos is also very useful for quickly evaluating functions at several inputs (just define f(x)=……, and then type f(0.01), f(0.001), etc. in lines below it).

## Average Rate of Change (AROC) (0.1)

This Desmos graph gives an interactive way to think about average rates of change in the context of examples from physics and economics. You should try changing the definition of the function, and see how you can interpret the examples in that case.