Date Range: December 5 – December 20
Hours Spent:
20 hours
- 8 hours on lectures and notes Professor Leonard – Calculus 2 Lecture 10.2: Introduction to Parametric Equations & Professor Leonard Calculus 2 Lecture 10.3 Calculus of Parametric Equations \
- 12 hours on exercises (Thomas 13th ed, 11.1 questions 1 – 12, 19 – 27; 11.2 1 – 7; 15 – 16)
Summary:
This last leg of study has been slower than usual, as I’ve been involved in an international move (not so uncommon for me) and spinning more plates than usual. My record-keeping for the current topic hasn’t been as tight as usual, but I did it. And, thankfully, I’ve moved forward.
OK now, parametric equations. Parametric equations are a useful way of relating x and y variables to a 3rd common variable, often t, which stands in well for ‘time’. More formally, we say that x = f(t) and y = g(t). We use parametric equations to describe curves on a plane together with the directions and speed of traversal across said curve. They provide means of describing curves that are awkward or unnatural to describe using a single Cartesian equation. Think of a bow-tie pattern on a two-dimensional space. Or maybe the path of a fly crawling on a wall. We can perform calculus operations on parametric equations with respect to the third parameter (often t) and translate the results back to our traditional Cartesian (x, y) coordinates. Such common operations include
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- Differentiation
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- Identifying horizontal and vertical tangents/asymptotes
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- Calculating lengths of a curve
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- Calculator surface areas of a surface of revolution
In doing so, we learn about the path that a particle, or value at t, travels. We can glean insights and gather data that we wouldn’t otherwise capture via Cartesian equations, such as
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- Direction
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- Retracing
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- Cusps
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- Loops
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- Multiple passes over the same curve
Personal Observations:
- Given the prevalence of such phenomena in two-dimensional change, parametric equations are obviously useful. I imagine they show up across a breadth of engineering applications.
- Interestingly, some arcs with a clear initial point, and an infinite terminal point (that’s not really terminal, is it?) will have particles that travel in a circular pattern, such as this parametric equation in which the particle reverses its direction at the origin
- The exercises for these, particularly the “calculus” problems, were largely a tedious nightmare. I found myself spending an hour on a single problem, slogging through trig transformations, or equations plagued by fractional radicals such as this:
- Every so often I run into sections in the textbook where problems seem more like time heavy, algebra and trig busy-work than immediately relevant Calculus. My time is scarce and precious. I invariably end up wasting it on such problems because I’m determined to learn and don’t want to risk missing something important
- I need to better identify when I encounter such problems, and move on. Like I ultimately did with Parametric Equations – albeit later than I should have.
Reflections:
In working through some of the early calculus problems, like good old fashioned differentiation, I’m impressed by how far I’ve come, and at the same time peeved by how quickly memory fades. Use it or lose is more relevant than ever as I speed towards 50 (I just turned 48, by the way), learning math that students half my age have difficulty grappling with. But it’s all good – it keeps me sharp! And I’m nearly done with Calculus 2. I can’t wait to be officially done. It’s been fun, but I’m ready for the next subject.