12 Differentiation & Integration

This week our goals are to be able to:

Differentiation:

  • Understand the concept of numerical differentiation and its application in solving differential equations.
  • Apply numerical methods, such as finite differences, to calculate derivatives from experimental data.
  • Explain the process of numerical integration and its significance in science and engineering.
  • Apply numerical integration techniques, such as the trapezoidal rule, to calculate approximate areas under functions or data.
  • Evaluate the limitations and associated error of basic numerical differentiation and integration methods and the advantages of using smoothing techniques, such as splines.

Now that we’ve got some tools from earlier in this semester, we’re going to focus on differentiation & integration. Specifically, we will look at examples that apply these calculus methods to real life scenarios. This topic will be particularly useful in future job positions. Being able to automate workflows that can quickly load data and analyze rates & variable relationships has normalized educated & efficient decision-making in the workplace.

There will first be a differentiation problem & then an integration problem - that integrates in 2 ways. We’ll use mathematical integration to quantify mass/volume under a curve. If you have mass/time on the y-axis, and time on the x-axis, then you can integrate under the observational points & obtain the total mass that has passed through or accumulated in a system over a specific time period.

We’ll start the HW assignment in class - it’s more like a case-study this week. You’ll draw on previous content you’ve learned, and use a new function for integration - trapz again from the pracma package.

12.1 Biggest takeaway - workflow

The biggest thing I want you to get out of this week: workflow. That is, think about how you are going to accomplish this problem, break it down into steps, and then implement those steps.

12.2 Reading