Introduction

Notes on basic Calculus concepts.

Study resources

• Khan academy: AP Calculus BC

• Kristan King: Calculus I (Differential Calculus)

• Kristan King: Calculus II (Integral Calculus)

• Kristan King: Calculus III (Multivariable Calculus)

• The Organic Chemistry Tutor: New Calculus Playlist

• Mathispower4u: Calculus playlist

• Math@Xaktly

• Geoff(x): Teach math using Desmos

• MIT OCW 18.01SC: Single Variable Calculus

• MIT OCW 18.02SC: Multivariable Calculus

• Coursera: Mathematics for Machine Learning: Multivariate Calculus

Study Tools

• Symbolab

• Desmos

• GeoGebra

• MIT Mathlets

  • TAYLOR POLYNOMIALS

  • HARMONIC FREQUENCY RESPONSE: VARIABLE INPUT FREQUENCY

Practice To-do List (Linked with Unit tests)

• Limits and continuity

• Differentiation: definition and basic derivative rules

• Differentiation: composite, implicit, and inverse functions

• Contextual applications of differentiation

• Applying derivatives to analyze functions

• Integration and accumulation of change

• Differential equations

• Applications of integration

• Infinite sequences and series

Table of Contents

• ▶ Limit & Continuity

  • Limit properties & Limits of Combined Functions

  • Limits at infinity

  • All types of discontinuities

• ▶ Differential Calculus

  • Differentiability

  • Local linearity & Linear approximation

  • Basic Differential Rules

  • Chain Rule

  • Derivatives of Trig functions

  • Implicit differentiation

  • Higher Order Derivatives

  • Derivative of Inverse functions

  • Derivative of exponential functions

  • Existence Theorems

  • L'Hopital's Rule

  • Critical points

    • Extrema: Maxima & Minima

    • Concavity

    • Inflection Point

  • Second Derivative Test

  • Anti-derivative

  • Analyze Function Behaviors with Derivatives

  • Optimization

  • Applications of Derivatives

    • Motion problems

    • Planar motion

• ▶ Integral Calculus

  • Definite Integrals

  • Antiderivatives

  • Fundamental Theorem of Calculus (FTC)

  • Basic Integral Rules

  • Calculate Integrals

  • Integration using Trig identities

  • Improper Integral

  • U-substitution → Chain Rule

  • Integrate by Parts → Product Rule

  • Partial fractions → Log Rule

  • Trig-substitutions → Trig Rule

  • Average Value of Functions

• ▶ Differential Equations

  • Parametric Equations Differentiation

  • Separable Differential Equations

  • Specific antiderivatives

  • Polar Curve Functions (Differential Calc))

  • Logistic Growth Model

  • Slope Field

  • Euler's Method

• ▶ Applications of definite integrals

  • Area under Rate function

  • Motion problems

  • Planar motion

  • Function Area between curve & axes

  • Area of Polar Curves (Integral Calc)

  • Arc Length

  • Parametric Curve Arc Length

  • Volumes of 3D Solids

  • Disc Method

  • Washer Method

  • Shell Method

• ▶ Series (Calculus)

  • Infinite Seires

  • Infinite Geometric Series

  • Convergence Tests

    • nth Term Test

    • Integral Test

    • p-series Test

    • Comparison Test

    • Ratio Test

    • Root Test

    • Alternating Series Test

  • Absolute vs. Conditional Convergence

    • Error Estimation of Alternating Series

    • Error Estimation Theorem

    • Interval of Convergence

  • Power Series

    • Taylor Series

    • Maclaurin Series

    • Lagrange Error Bound

    • Finding Taylor series for a function

    • Function as a Geometric Series

    • Maclaurin Series of Common functions

    • Euler's Formula & Euler's Identity

• Multivariable functions

  • Parametric Functions

  • Partial derivatives

  • Gradient