6. Application of Derivatives

9 topics3 h

Chapter 6 of NCERT Mathematics for Class 12: Application of Derivatives. Three sections — The derivative as a rate, Where a function rises and where it falls and Highest and lowest values. Nine videos, 3 h in all.

Worked answers to the 82 exercise questions in this chapter: Exercise 6.1 · Exercise 6.2 · Exercise 6.3 · Miscellaneous Exercise

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Part 1

The derivative as a rate

25 minReading a derivative as how fast one quantity answers another25 min · 19 parts23 minTwo quantities both changing with time, and the chain rule that links their rates23 min · 20 parts

Part 2

Where a function rises and where it falls

18 minWhat increasing and decreasing mean on an interval, before any calculus18 min · 18 parts19 minUsing the sign of the derivative to split the line into rising and falling intervals19 min · 19 parts

Part 3

Highest and lowest values

21 minLocal maxima and minima, and why critical points include the non-differentiable ones21 min · 20 parts21 minA sign change either side of a critical point decides which kind it is21 min · 16 parts18 minThe quicker second-derivative test, and the case where it tells you nothing18 min · 16 parts20 minLargest and smallest over a closed interval: candidates inside plus the two ends20 min · 16 parts21 minTurning a stated problem into one function of one variable to optimise21 min · 18 parts

What you will be able to do

  • Read a derivative aloud as a statement about two named quantities rather than as a computation
  • State the chain rule in the form §6.2 prints it, including the condition attached to it
  • Read rising and falling off a graph and off a table of values, and say why neither is a definition
  • State Theorem 1 with the inequalities the page prints, and say why they are the non-strict ones
  • Read the three motivating problems and say which single mathematical question all three ask
  • State all three parts of Theorem 3 and say which hypothesis each needs
  • State Theorem 4's three parts with the correct sign in each
  • Distinguish the largest value over a whole interval from a value that is only largest nearby, using the chapter's own four names
  • Identify, from a described problem, the quantity to be made largest or smallest

What this chapter assumes you already know

  • Differentiating a polynomial, and the derivative of a power, from Chapter 5
  • The two notations for a derivative — the ratio form and the primed form
  • Evaluating a derivative at a stated input
  • Area and circumference of a circle; volume and surface area of a cube
  • Units, and how a ratio of two units is read aloud
  • Function notation, and what it means to declare one quantity a function of another
  • Rounding a money value to two decimal places
  • The derivative as a rate, and the two notations for it, from the previous topic

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.

  • "A rate always means per second."
  • "The derivative and its value at a point are the same thing."
  • "Marginal cost is the cost of the next unit."
  • "The units look after themselves."
  • "Rates are about motion."
  • "This chapter will teach me tangents and normals, because the introduction says so."
  • "A negative rate means the quantity is negative."
  • "Pi has to be turned into a decimal."