Lectures

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  1. Understanding Functions

  In our introductory lecture, Dr. Snellman covers precalculus fundamentals, focusing on functions as input-output rules and their operations—addition, subtraction, multiplication, division, and composition. Together, we explore inverse functions as tools that reverse original functions, with examples using exponentials and logarithms. The lecture concludes by showing how inverse functions reflect across the line y = x, revealing their symmetrical relationship.
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  2. Complex Arithmetic

  In lecture two, we investigate complex numbers, tracing their origins to the 1500s as solutions to quadratics with negative discriminants, introducing the imaginary unit i. We cover arithmetic operations, the conjugate, modulus, and representations in standard and polar form using Euler’s formula. Finally, we apply polar form to solve polynomial equations, revealing how complex numbers and roots of unity are fundamental to advanced applications in fields like harmonic analysis and signal processing through the fast Fourier transform.
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  3. Matrix Arithmetic

  In lecture three, we delve into matrices and vectors as systematic tools for solving systems of linear equations, moving beyond graphical methods to handle multiple variables and equations simultaneously. We explore matrix arithmetic operations, then demonstrate how to represent systems of linear equations in matrix form and solve them using matrix inverses when they exist. Dr. Snellman concludes by connecting these concepts to linear regression, showing how matrices enable us to find the best-fit line through data points, which forms the fascinating foundation for machine learning, data science, and AI applications that are transforming our world today.
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  4. Sequences and Series

  In lecture four, we learn about sequences and series, exploring how sequences function as ordered lists of numbers with specific patterns, such as arithmetic versus geometric sequences. We examine the concept of series as infinite sums of sequence terms, demonstrating through examples like geometric series how these infinite sums can converge to finite values under certain conditions. The notion of limits is introduced as the foundation for understanding convergence and divergence, emphasizing how sequences and series naturally lead us toward the core principles of calculus.
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  5. Average Rates of Change

  In lecture five, we learn about average rates of change as a foundation for understanding calculus and derivatives. Using real-world examples like speedometers and heart rate monitors, we build the mathematical framework for calculating average change over an interval. We compute rates using data tables and functions, then interpret them graphically as slopes of secant lines. Dr. Snellman also discusses the idea of instantaneous change, leading naturally to the concept of limits in calculus, the topic of lecture six.
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  6. Understanding Limits

  In lecture six, we continue our exploration of limits, focusing on how a function approaches a value as its input nears a specific point. Through examples, we examine direct substitution, indeterminate forms, and one-sided limits. These tools reveal how functions behave near discontinuities. The lecture concludes by linking limits to features like vertical asymptotes, jumps, and holes, laying the groundwork for understanding continuity in calculus.
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  7. Limits and Continuities

  In lecture seven, we look at limits and their applications, exploring how limits serve as the foundation for understanding continuity and solving complex equations. We examine different types of discontinuities—removable, jump, and infinite—and introduce the intermediate value theorem, a powerful tool for proving the existence of solutions to equations that cannot be solved algebraically. The lecture concludes by connecting these ideas to the challenge of finding instantaneous rates of change, introducing the concept of the derivative.
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  8. Introducing Derivatives

  In our eighth and final lecture, Dr. Snellman explains the concept of derivatives, building on limits, secant lines, and average rates of change to define instantaneous rates of change. We explore practical applications, including computing derivatives using the limit definition, using tangent lines to estimate values, and introducing Newton’s method for finding equation roots. The lecture concludes by connecting derivatives to optimization problems, showing that points where the derivative equals zero are critical points, such as maxima and minima, and introducing gradient descent as a method for locating optimal values in real-world applications.