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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On this page
  • Intermediate Value Theorem (IVT)
  • Find roots by using IVT
  • Example
  • Extreme Value Theorem (EVT)
  • Mean Value Theorem (MVT)
  • Example
  1. Differential Calculus

Existence Theorems

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Last updated 6 years ago

Existence theorems includes 3 theorems: Intermediate Value Theorem, Extreme Value Theorem, Mean Value Theorem.

Refer to Khan academy: Existence theorems intro

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Intermediate Value Theorem (IVT)

The IVT is saying:

When we have 2 points connected by a continuous curve: one point below the line, the other point above the line, then there will be at least one place where the curve crosses the line!

Refer to Maths if fun: Intermediate Value Theorem Refer to video: Intermediate Value Theorem Explained

Find roots by using IVT

IVT is often to find roots of a function, which means to find the x value when f(x)=0. So for finding a root, the definition will be:

If f(x) is continuous and has an interval [a, b], which leads the function that f(a)<0 & f(b)>0 , then it MUST has a point f(c)=0 between interval [a,b], which makes a root c.

Example

Tell whether the function f(x) = x² - x - 12 in interval [3,5] has a root. Solve:

  • We got that at both sides of intervals: f(3)=-6 < 0, and f(5)=8 > 0

  • So according to the Intermediate Value Theorem, there IS a root between [3,5].

  • Calculate f(c)=0 get the root c=4.

Extreme Value Theorem (EVT)

The EVT is saying:

There MUST BE a Max & Min value, if the function is continuous over the closed interval.

Refer to Khan lecture: Extreme value theorem Refer to video: Extreme Value Theorem

Mean Value Theorem (MVT)

Refer to Khan academy article: Establishing differentiability for MVT

The MVT is saying:

There MUST BE a tangent line that has the same slope with the Secant line, if the function is CONTINUOUS over [a,b] and DIFFERENTIABLE over (a,b).

Which also means that, if the conditions are satisfied, then there MUST BE a number c makes the derivative is equal to the Average Rate of Change between the two end points.

Conditions for applying MVT:

  • Continuous over interval (a, b)

  • Differentiable over interval [a, b]

Example

  • He's totally right.

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Equation
Graph

Solve:

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