Math · Level 5

5.1 Calculus in a Nutshell

Investigate the central ideas of calculus and learn how to put them to use.

42 Lessons

The Rate Of Change

Tangents

Maxima And Minima

Critical Points

Apply: Drop Testing

Identifying Extremes

Test Limitations

Higher-Order Derivatives

Apply: From Distance To Jerk

Constant Multiples

Sums

Products and Powers

Chain Rule

Quotients

Apply: Optimization

The Area Problem

The Integral

The Fundamental Theorem

Antiderivatives

Apply: Falling objects

Volume With Integrals

Surface Area via Integrals

Apply: Gabriel's Horn

Sequences and Limits

Apply: Carbon Dating

What is an Infinite Sum?

Geometric Sums

Harmonic Sum

Apply: The Tower of Lire

Quadratic Approximations

Apply: The Spring Equation

Function Limits

Limit Theorems

Continuity

Smooth Functions

Intermediate Value Theorem

Extreme Value Theorem

Sine and Cosine

The Exponential Function

Taylor Series

Geometric Series Revisited

Apply: Normal Curves

Course description

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.

Topics covered

  • Antiderivatives
  • Derivatives
  • Derivative rules
  • The Fundamental Theorem
  • Geometric applications
  • Geometric series
  • Infinite sums
  • Integrals
  • Limits
  • Riemann sums
  • Science applications
  • Tangent lines
  • Taylor series

Prerequisites and next steps

You’ll need an understanding of algebra and the basics of functions, such as domain and range, graphs, and intercepts.