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Application of Derivatives
Introduction
Rate of Change of Quantities
Practice: Rates of Change
Increasing and Decreasing Functions
Practice: Increasing and Decreasing Functions
Maxima and Minima
Practice: Maxima and Minima
Miscellaneous Examples
Chapter Review: Mixed Practice
Summary
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Application of DerivativesIntroduction

Reading time: ~10 min

It is sports day at Aman's school, and the final over of the students-versus-teachers cricket match. Aman gets under a slow full toss and skies it: the ball goes almost straight up. Its height above the ground, t seconds after the shot, is

h(t) = 20t - 5t^2 metres.

Three questions come up at once. How fast is the ball moving at a given moment? When is it going up, and when is it coming down? And how high does it get?

You already own the tool that answers all three: the derivative, which you met in Class XI and in the chapter on continuity and differentiability. Here h'(t) = 20 - 10t is the ball's velocity. On the graph, the orange dash is the tangent at the ball, and its steepness is h'(t).

Drag the clock and complete the three missions.

Time since the shot: ${t} s

Height h = 20t - 5t^2 m against time t s.

Three Questions, One Derivative

Look back at what the missions showed you.

While the ball was going up, its velocity h'(t) was . At the very top, h'(t) = . On the way down, h'(t) was .

So one derivative answered three different questions, and each question is one section of this chapter:

  1. How fast is it changing? The value of h'(t) is the rate of change of the height with time. That is Rate of Change of Quantities.
  2. Is it going up or down? The sign of h'(t) tells you: positive while the height increases, negative while it decreases. That is Increasing and Decreasing Functions.
  3. Where is the highest point? Where h'(t) = 0 and its sign switches from + to −. That is Maxima and Minima.

The idea that a highest or lowest point is where the tangent turns flat is older than calculus itself. Around 1636 the French lawyer and amateur mathematician Pierre de Fermat found maxima and minima by asking where a tiny change in x makes almost no change in the value, which in today's language means "where the derivative is zero". A few decades later Isaac Newton and Gottfried Leibniz turned such tricks into one general method. Today the same ideas help an engineer find the lightest strong beam, a shopkeeper find the price that earns the most, and a doctor work out how fast a medicine spreads through the body.

Sina