Calculus Basics
  • Introduction
  • ▶️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
  • ▶️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
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  1. Differential Calculus

Analyze Function Behaviors with Derivatives

Analyzing function's behaviors is one of the Core Purposes of studying Calculus.

AND THE CORE PURPOSE OF ANALYZING FUNCTION, IS FOR COMPUTER TO UNDERSTAND IT "BLINDLY", OR SAY "ALGEBRAICALLY"! BECAUSE IT CAN'T BE LIKE HUMAN TO "EYE BALL" IT!

First Derivative

  • Function has critical points when f'(x) = 0

  • Function has relative extrema when f'(x) crosses X-axis:

    • It has relative maxima when f'(x) crosses UP.

    • It has relative minima when f'(x) crosses DOWN.

Second Derivative

  • Function has inflection points when f''(x)=0 and f''(x) crosses X-axis.

  • It's concave up if f''(x) > 0

  • It's concave down if f''(x) < 0

1st & 2nd Derivative

  • It has a relative maximum when f'(x)=0 & f''(x) < 0

  • It has a relative minimum when f'(x)=0 & f''(x) > 0

  • It has an inflection point when f'(x) changes direction, OR f''(x) changes sign.

  • It has a possible inflection point when f'(x) = 0 & f''(x) = 0.

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Last updated 6 years ago

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