5. Continuity and Differentiability

10 topics4 h

Chapter 5 of NCERT Mathematics for Class 12: Continuity and Differentiability. Three sections — Continuity, Differentiating harder functions and New functions and higher derivatives. Ten videos, 4 h in all.

Worked answers to the 137 exercise questions in this chapter: Exercise 5.1 · Exercise 5.2 · Exercise 5.3 · Exercise 5.4 · Exercise 5.5 · Exercise 5.6 · Exercise 5.7 · Miscellaneous Exercise

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Part 1

Continuity

16 minA function is continuous where its limit and its value agree16 min · 17 parts16 minWhere the standard functions stay unbroken, and where they jump16 min · 18 parts15 minCombining continuous functions, and why a composite survives too15 min · 18 parts

Part 2

Differentiating harder functions

16 minDifferentiability, and why it forces continuity while the reverse fails16 min · 17 parts19 minThe chain rule, and differentiating a relation without first solving it for y19 min · 20 parts19 minGetting the derivatives of the inverse trigonometric functions19 min · 19 parts

Part 3

New functions and higher derivatives

25 minExponential and logarithmic functions, and the two derivatives that make them worth having25 min · 19 parts24 minTaking logarithms first, when the base and the power both vary24 min · 19 parts25 minCurves given through a parameter, and the derivative recovered by the chain rule25 min · 20 parts36 minDifferentiating twice, and what the second derivative is for36 min · 20 parts

What you will be able to do

  • Split Definition 1 into its three separate demands and say which one a given function fails
  • Split the real line at every input where a piecewise rule changes, and decide the easy regions in one line each
  • State the four parts of Theorem 1 and identify which one carries a proviso
  • State the definition of the derivative at a point and say what the attached caution is guarding against
  • Recognise a composite and name its inner and outer functions before differentiating
  • Turn an inverse trigonometric equation into a relation between the two letters and differentiate it
  • Compare the growth of two whole powers at a common input and say which is steeper
  • Recognise the two shapes that call for logarithms before differentiating
  • Say what a parametric form is and identify the parameter in a stated pair
  • Differentiate a derivative and write the result in each of the five notations the chapter gives

What this chapter assumes you already know

  • Limits from Class XI: what a function approaches near an input, and the two one-sided versions of that question
  • The algebra of limits — sums, products and quotients of limits
  • Piecewise definitions, and reading which branch applies at a given input
  • Domain, range, and what a function returns at a given input
  • Polynomial, constant, identity and reciprocal functions and their graphs
  • Interval notation, and the difference between a closed and an open end
  • The idea that a symbol can be undefined at a point without the formula around it being wrong
  • Definition 1 and Definition 2 from the previous topic, and the three demands hidden in the first

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "Continuous means you can draw it without lifting the pen."
  • "If the limit exists, the function is continuous."
  • "If the value exists, the function is continuous."
  • "A hole in the domain is a discontinuity."
  • "The limit at zero is infinity, so the limit is infinity."
  • "Both one-sided limits agreeing settles it."
  • "At the end of a closed interval you compare both sides."
  • "There must be some constant that patches any join."