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Brilliant

Calculus Fundamentals

via Brilliant

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Overview

Calculus is the mathematical study of things that change: cars accelerating, planets moving around the sun, economies fluctuating. To study these changing quantities, a new set of tools - calculus - was developed in the 17th century, forever altering the course of math and science.

This course sets you on the path to calculus fluency. The first part provides a firm intuitive understanding of limits, the central idea underlying the entire subject. The second part applies limits to define derivatives, an indispensable tool for measuring change. By the end of the course you'll have practical calculus experience that any aspiring scientist, engineer, or mathematician needs.

Syllabus

  • Introduction: Learn about the basics of limits!
    • Indeterminate Forms: Examine the speed of motion from the perspective of calculus.
    • Limits Intuition: Take your first step towards mastering limits.
    • Limits of Functions: Learn the essentials of computing limits.
  • Computing Limits: Master the infinitesimal.
    • Infinite Limits: Discover how limits tame the infinite.
    • Continuity: When does a function's graph come in pieces?
    • Computing Limits I: Take limits of algebraic expressions.
    • Computing Limits II: Learn limit techniques to conquer derivatives.
    • The Squeeze Theorem: Use geometry to prove an important limit.
    • IVT Application: Root Finder: Explore the Intermediate Value Theorem and its uses.
    • Precise Definition of Limits: See limits from a mathematician's point of view.
  • Derivatives: Making quantitative sense of the moment when everything's changing.
    • The Derivative at a Point: Learn what derivatives can do for you.
    • First Examples of Derivatives: Gain hands-on experience with a few basic derivatives.
    • What Derivatives Tell Us: What's the point of taking a derivative, anyway?
    • The Second Derivative: Try taking a derivative's derivative, and explore what it can do for you.
    • Real-World Derivatives: See how derivatives are used in motion, geometry, and learning curve problems.
    • Average vs. Instantaneous: Compare tangent lines to secant lines and uncover the mean value theorem.
  • Computing Derivatives: The practitioner's toolkit: the product rule, chain rule, and more.
    • Polynomials: Apply your calculus skills to find shortcuts for polynomial derivatives.
    • Products, Reciprocals, and Quotients: Find the derivatives of compound functions that are made by multiplying and dividing simpler pieces.
    • Trigonometric Functions: Step away from algebraic expressions and into the world of trigonometric derivatives.
    • The Chain Rule: Break a complicated derivative problem into a chain of simpler ones.
    • Exponentials and Logarithms: Round out your tour of elementary derivatives with some transcendentals.
  • Linear Approximation and Applications: Linear approximation, implicit differentiation, L'Hôpital's Rule, and some physics!
    • Linear Approximation: Apply tangent lines to the classic root-finding problem.
    • Pendulums: An Application: Use rates of change to model the swing of a pendulum.
    • Optimization: Find extreme values with derivatives, one of the most important uses of calculus.
    • Implicit Differentiation: Compute rates of change of related quantities.
    • L'Hôpital's Rule: Come full circle and use derivatives to find difficult limits.
    • Antiderivatives & Integrals: Glimpse the next step on the calculus journey.

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