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Continuity and Differentiability — Concept Maps

Punjab Board · Class 12 · Mathematics

3 concept maps of Continuity and Differentiability for Punjab Board Class 12 Mathematics, each also written out as a text outline.

44 questions25 flashcards8 formulas & key relations5 concepts

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A graph of a continuous function at a specific point, showing the limit approaching the function value, illustrating that the function is unbroken and defined at that point.
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3 Concept Maps

Chapter 5: Continuity and Differentiability – Complete Overview

Chapter 5: Continuity and Differentiability – Complete Overview

The map in words

  • Continuity and Differentiability
    • Continuity
      • Definition
        • lim f x equals f c
        • LHL equals RHL equals f c
      • Types of Functions
        • Polynomial always continuous
        • Rational continuous in domain
        • sin x and cos x always continuous
        • Greatest Integer discontinuous at integers
      • Algebra
        • Sum f plus g continuous
        • Product f times g continuous
        • Quotient f over g continuous if g nonzero
        • Composite fog continuous
      • Finding Discontinuity
        • Piecewise functions at boundary
        • Rational functions where q equals 0
    • Differentiability
      • Definition
        • Limit of difference quotient
        • LHD equals RHD
      • Key Theorem
        • Differentiable implies Continuous
        • Converse is FALSE
        • Counterexample mod x at zero
      • Standard Derivatives
        • Power Rule xn
        • Trig sin cos tan
        • Inverse Trig sin inverse cos inverse tan inverse
        • Exponential e power x and a power x
        • Logarithm log x
    • Differentiation Techniques
      • Chain Rule
        • Composite functions
        • dy over dx equals dy over dt times dt over dx
      • Implicit Differentiation
        • Differentiate both sides wrt x
        • dy over dx via chain rule on y terms
      • Logarithmic Differentiation
        • Take log both sides
        • For u power v type functions
        • For complex products
      • Parametric Differentiation
        • x equals f t and y equals g t
        • dy over dx equals dy over dt divided by dx over dt
    • Second Order Derivative
      • Notation d2y over dx2
      • Differentiate dy over dx again
      • Proof type questions
      • Differential equations

Decision Tree for Choosing Differentiation Method

Decision Tree for Choosing Differentiation Method

The map in words

  • Start: Differentiate f(x)
    • Is f a simple power, poly, trig, exp, log?
      • Yes: Use basic differentiation rules
        • Apply rule and simplify
          • Final Answer: dy/dx
      • No: Is f a composition (f ∘ g)?
        • Yes: Use Chain Rule: f'(g)·g
        • No: Is f a product or quotient?
          • Product: Use Product Rule: u'v + uv
          • Quotient: Use Quotient Rule: (u'v - uv')/v²
          • No: Is f implicit y = y(x)?
            • Yes: Implicit Differentiation: Differentiate both sides
            • No: Is f parametric x=x(t), y=y(t)?
              • Yes: Parametric Diff: dy/dx = (dy/dt)/(dx/dt)
              • No: Base and exponent both variables?
                • Yes: Logarithmic Diff: ln both sides

Continuity and Differentiability – Complete Chapter Overview

Continuity and Differentiability – Complete Chapter Overview

The map in words

  • Continuity and Differentiability
    • Continuity
      • Definition
        • LHL equals RHL equals f of c
        • Three conditions must hold
      • Types of Discontinuity
        • Removable
        • Jump
        • Infinite
      • Continuous Functions
        • Polynomials everywhere
        • Rationals except denom zero
        • sin x and cos x everywhere
        • Greatest Integer at integers
      • Algebra of Continuous Functions
        • Sum and Difference
        • Product
        • Quotient where denom nonzero
        • Composite functions
    • Differentiability
      • Definition
        • LHD equals RHD
        • First principles limit
      • Key Theorem
        • Differentiable implies Continuous
        • Continuous does not imply Differentiable
      • Example mod x
        • Continuous everywhere
        • Not differentiable at zero
    • Differentiation Techniques
      • Chain Rule
        • Composite functions
        • f of g x gives f prime g x times g prime x
      • Implicit Differentiation
        • d dx of y terms uses dy dx
        • Collect and solve
      • Logarithmic Differentiation
        • Variable base variable exponent
        • Take log then differentiate
      • Parametric Differentiation
        • dy dx equals dy dt divided by dx dt
        • Answer in terms of parameter
    • Special Functions
      • Inverse Trig Derivatives
        • sin inverse 1 over root 1 minus x squared
        • cos inverse negative 1 over root 1 minus x squared
        • tan inverse 1 over 1 plus x squared
      • Exponential and Log
        • d dx e to x equals e to x
        • d dx log x equals 1 over x
        • d dx a to x equals a to x log a
    • Second Order Derivatives
      • Definition d squared y over dx squared
      • Physical Meaning Acceleration
      • Proof Problems
    • Mean Value Theorems
      • Rolles Theorem
        • Continuity plus Differentiability plus equal endpoints
        • f prime c equals zero
      • LMVT
        • Continuity plus Differentiability
        • f prime c equals average rate of change

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Frequently Asked Questions

What are the important topics in Continuity and Differentiability for Punjab Board Class 12 Mathematics?
Key topics in Continuity and Differentiability include Continuity of a Function, Algebra of Continuous Functions, Differentiability, Chain Rule and Derivatives of Composite Functions. Study these first, then practise questions on each for the Punjab Board Class 12 board exam.
What do the concept maps for Continuity and Differentiability show?
The 3 maps show how the ideas in Continuity and Differentiability connect: Chapter 5: Continuity and Differentiability – Complete Overview; Decision Tree for Choosing Differentiation Method. Each map is also written out as an outline on this page.
How should I revise Continuity and Differentiability for the Punjab Board Class 12 board exam?
Learn the core ideas first, then work through the 44 practice questions on Continuity and Differentiability. Revise definitions regularly and use flashcards for quick recall before the exam.

Sources & Official References

Content is aligned to the official syllabus. Refer to the board website for the latest curriculum.

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All Punjab Board Class 12 Mathematics Concept Maps

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