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Brilliant

Calculus in a Nutshell

via Brilliant

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Overview

Calculus has such a wide scope and depth of application that it's easy to lose sight of the forest for the trees.

This course takes a bird's-eye view, using visual and physical intuition to present the major pillars of calculus: limits, derivatives, integrals, and infinite sums. You'll walk away with a clear sense of what calculus is and what it can do.
Calculus in a Nutshell is a short course with only 19 quizzes. If you want to quickly learn an overview of calculus or review the foundational principles after a long hiatus from the subject, this course ought to be perfect.

Calculus Fundamentals and Integral Calculus are the two courses that can follow next in the Calculus sequence. If/when you want to go into more depth and learn a wide spread of specific techniques in differential calculus and integral calculus respectively, that's where you should look. For example, integration techniques like "integration by parts" are only in the Integral Calculus course.

Syllabus

  • Derivatives in a Nutshell: Rates of change, tangent lines, and applications.
    • Rate of Change: Use graphs and rates to understand how things change over time.
    • Rate of Change: Practice Questions:
    • Rate of Change: Challenge Questions:
    • Instantaneous Rate of Change: Zoom in on change at an instant in time.
    • Instantaneous Rate of Change: Practice Questions:
    • Instantaneous Rate of Change: Challenge Questions:
    • The Derivative: Find the slope of the tangent line at any point.
    • The Derivative: Practice Questions:
    • The Derivative: Challenge Questions:
    • Applying the Derivative: Optimization: Use the derivative to find a maximum value.
    • Optimization: Practice Questions:
    • Optimization: Challenge Questions:
  • More Derivatives: Extend your understanding of the derivative.
    • What is a Derivative?: Discover a crucial calculus idea with a microscope.
    • What are Derivatives for?: Explore the many uses for derivatives.
    • How are Derivatives found?: Practice fundamental skills for computing derivatives.
    • Application: Optimization: Use derivatives to solve a challenging real-world problem.
  • Integrals: Discover why reversing a derivative is important, too.
    • What is an Integral?: Explore integrals with an ancient geometry problem.
    • How are Integrals found?: Unlock integrals with the most important theorem in calculus.
    • Application: Gabriel's Horn: Explore a paradoxical shape with calculus!
  • Infinite Sums: Unending sums and their place in the real world.
    • Sequences: Learn to solve geometry and science problems with sequences.
    • What is an Infinite Sum?: Understand the meaning of infinite sums with thought experiments.
    • The Tower of Lire: What do divergent sums have to do with real life?
    • What are Infinite Sums for?: Solve a fundamental physics problem with Taylor series!
  • Advanced Topics: The farthest reaches of single-variable calculus.
    • Sine & Cosine: Explore the quintessential models of periodic change with calculus.
    • Euler's formula: Recreate one of the most beautiful results in all of math.
    • Taylor Series: Master an essential tool of applied calculus.
    • Limits: Dig deeper into the very foundations of calculus.
    • Continuity: Learn how to stop worrying and love non-smoothness.

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