Lectures

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  1. A Competitive Edge in Math

  In our introductory lecture, Dr. Snellman highlights the practical value of applied mathematics, showing how understanding the principles behind algorithms and optimization can provide a competitive edge. We explore real-world applications such as business optimization, medical imaging, drug discovery, and autonomous vehicles, illustrating how applied mathematics is transforming these fields. The lecture concludes with inspiring examples of students who used their mathematical knowledge to launch exceptional careers, reinforcing that anyone can master these concepts to build impactful, marketable skills.
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  2. Derivatives in Action

  In lecture two, Dr. Snellman introduces the concept of derivatives, their computation rules, and their significance in modern mathematics and data science. The lecture covers various derivative rules including the power rule, chain rule, product rule, and quotient rule, while emphasizing their applications in optimization problems and machine learning. The discussion concludes with an explanation of Euler's number (e) and its unique property in exponential functions, demonstrating how mathematical concepts interconnect to solve complex problems.
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  3. Optimization in Action

  In lecture three, we explore the importance of derivatives in optimization problems, demonstrating how nearly everything in life can be viewed through the lens of optimization—from choosing lunch locations to taking exams. We learn that finding where derivatives equal zero helps identify peaks and valleys in functions, which correspond to maximum and minimum values crucial for real-world applications like minimizing error in machine learning models. Dr. Snellman introduces Newton's method, one of the most important algorithms in mathematics, and highlights cases where we must be careful using it.
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  4. Introducing Limits

  In lecture four, we learn about the concept of limits, their intuitive and rigorous definitions, and their application in understanding function behavior and continuity. We examine various examples of limit calculations, including indeterminate forms, and introduce L'Hôpital's rule as a powerful tool for evaluating limits. The lecture concludes by connecting limits to the concept of continuity and introducing the Intermediate Value Theorem, setting up the foundation for the bisection method of finding zeros of continuous functions.
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  5. Limits in Action

  In lecture five, we study the bisection method as an alternative to Newton's method for finding where functions equal zero, highlighting its advantages in avoiding pitfalls like derivative-related issues and oscillating values. The lecture demonstrates the method's application through examples, including approximating √2 and implementing binary search in ordered lists, emphasizing how the Intermediate Value Theorem forms the theoretical foundation for these techniques. The discussion concludes by addressing practical applications of calculus concepts, particularly limits, in modeling real-world phenomena like disease spread and drug metabolism.
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  6. First and Second Derivatives

  In lecture six, Dr. Snellman explains optimization fundamentals by focusing on critical points where derivatives vanish. We examine how the first derivative reveals when functions are increasing or decreasing, and how the second derivative distinguishes convexity from concavity. Using sign lines, we visualize these behaviors to identify local maxima and minima, and conclude with methods for constrained optimization on closed intervals.
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  7. Gradient Descent

  In lecture seven, we explore gradient descent, a powerful optimization algorithm used to find the minimum values of functions through iteration. We examine how to implement gradient descent through detailed examples, including a simple quadratic function and a more complex fourth-degree polynomial, demonstrating both its effectiveness and limitations in finding local versus global minima. The lecture concludes with discussions on the ethical implications of applied mathematics in AI and data science, and the importance of teaching students to ask meaningful questions when working with these tools.
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  8. Code for Calculus

  In our eighth and final lecture, Dr. Snellman demonstrates how to implement three fundamental optimization algorithms—Newton's method, bisection method, and gradient descent—using Python in Google Colab, illustrating their practical applications beyond manual calculations. We examine how to translate theoretical algorithms into working code, demonstrating their application on various functions, while comparing their efficiency and discussing when each method succeeds or fails. Dr. Snellman emphasizes the power of computational tools in solving complex mathematical problems that would be impossible to solve by hand.