
Explore how calculus foundations—limits, differentiation, and integration—unlock trigonometry through Taylor series and Fourier analysis, building a sine relationship from scratch and applying it to sound analysis with Excel.
Explore how Taylor series and differentiation yield the first level velocity with respect to angle, in radians, revealing that instantaneous velocity equals cosine theta on a unit circle.
Explore how breaking time into small intervals and summing constant-velocity distances leads to the integration concept, handling acceleration and limits to relate distance and time.
Explore Fourier analysis, a key tool in sound analysis, separating signals into independent sine-wave frequencies and revealing each frequency component.
Demonstrates creating sine waves for different frequencies, applies Fourier analysis, and explains how to wrap the Taylor series using a remainder modulo two pi to compute sine values accurately.
Learn how to compute binomial coefficients by counting selections and using factorials, and see how the binomial theorem applies to any two components a and b.
This course will teach you Calculus in a brand new way! Traditional way to learn Calculus follows a sequential model: algebra-geometry-limit-differentiation-integration. Each piece was taught on an isolated basis. Hence it’s very difficult for students to make connections. Without being able to connecting the dots, it’s impossible to obtain a deep understanding about Calculus. Therefore most students walk away with random bits of memory about Calculus, they don’t really have a coherent and systematic understanding of Calculus.
This course will do something totally different. It starts with the ultimate use of Calculus, and approach the core of Calculus step by step in a logical sequence. We wouldn’t throw you a concept out of blue, instead you will be prompted with a real problem, a problem that would encourage you to think proactively what to do next. You won’t be prompted with limit tool day 1, because there was no need at that time. Instead you will face a real problem: what should we do in order to understand sine? From there, you will see gradually the need to differentiate, then integrate. You will see why the limit tool would pop up. You would gain a thorough understanding of one of Calculus’ greatest tool: Taylor series expansion. You would also have hands-on experience with Fourier analysis.
We believe the worst way to learn math is to follow the assembly line model where things are thrown at you without you seeing the actual need. This will prevent you from engaging with the mechanics, and learning becomes a boring course of memorizing and blind practicing. We believe the best way to learn math is to think from the end use, from the real problems, then think backwards you will understand how each piece of math snap into the right place effortlessly. And this will make your math learning experience smooth and enriched.
Please join in this course if you want to have a systematic understanding of Calculus!