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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

  • What is a Derivative?
    • The Rate Of Change: Use graphs and rates to understand how things change over time.
    • Tangents: Explore how to find the rate of change for a single point.
    • Maxima And Minima: Explore how derivatives help us identify function behaviors.
    • Critical Points: Investigate the behavior of functions when the derivative is zero.
    • Apply: Drop Testing: Apply what's been learned so far to a scenario with gravity.
  • Using Derivatives
    • Identifying Extremes: Learn how to sort critical points when they are minimums or maximums.
    • Test Limitations: Explore a few situations where the second derivative test doesn't work.
    • Higher-Order Derivatives: Explore derivatives of other derivatives.
    • Apply: From Distance To Jerk: See how higher-order derivatives can be used testing a car.
  • Finding Derivatives
    • Constant Multiples: Use the constant multiple rule to find derivatives.
    • Sums: Use the sum rule to find derivatives.
    • Products and Powers: Use product and power rules to find derivatives.
    • Chain Rule: Unlock the chain rule to find the derivative of composite functions.
    • Quotients: Use the quotient rule to find derivatives.
    • Apply: Optimization: Use derivative rules to find the perfect brewing time for coffee.
  • Integrals
    • The Area Problem: Learn to break down an area into smaller parts to find the total.
    • The Integral: Learn to approximate areas under curves.
    • The Fundamental Theorem: Explore net changes of a car's velocity to unlock the Fundamental Theorem.
    • Antiderivatives: Discover how to reverse differentiation with antiderivatives.
    • Apply: Falling objects: Apply integrals to find the equation for a falling object.
  • Three Dimensions
    • Volume With Integrals: Find the volume of a sphere by using integrals.
    • Surface Area via Integrals: Derive surface area from a volume using integrals.
    • Apply: Gabriel's Horn: Explore the volume and surface area of the rotated curve – Gabriel's Horn.
  • Sequences and Series
    • Sequences and Limits: Use sequences and limits to estimate the area of a circle.
    • Apply: Carbon Dating: Use sequences and their limits to understand carbon dating.
    • What is an Infinite Sum?: Explore an infinite sum with area.
    • Geometric Sums: Delve deeper into the geometric sum and discover when it converges.
    • Harmonic Sum: Explore the harmonic sum and its unique behavior.
    • Apply: The Tower of Lire: Use the harmonic sum to explore the Tower of Lire.
    • Quadratic Approximations: Approximate functions at a point using quadratic equations.
    • Apply: The Spring Equation: Use approximations to better understand the spring equation.
  • Limits and Continuity
    • Function Limits: Understand the underlying behavior of functions as they approach limits.
    • Limit Theorems: Learn rules that simplify finding limits.
    • Continuity: Investigate one of the most important ideas in all of calculus.
    • Smooth Functions: Explore what makes a function smooth.
    • Intermediate Value Theorem: See how a function reaches every value between its endpoints.
    • Extreme Value Theorem: Find the maximum and minimum values of a function on a closed interval.
  • Advanced Topics
    • Sine and Cosine: Explore sine and cosine and their derivatives.
    • The Exponential Function: Explore the derivative of exponential functions.
    • Taylor Series: Approximate functions at a point using polynomials.
    • Geometric Series Revisited: Explore convergence with the Geometric and Taylor series.
    • Apply: Normal Curves: See how the Taylor series can aid calculations in statistics.

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