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