Chapter 6 ยท Introduction

Chapter 6 โ€” Application of Derivatives

A derivative is a rate of change. Read it that way and it tells you how fast a balloon grows, where a graph rises and falls, where it turns, and how to make a box, a can or a fence as good as it can possibly be.

A water tanker fills a village storage tank in Rajasthan at a steady $2$ litres per second. How fast is the water level rising โ€” and why does the answer change as the tank fills, if the tank is shaped like a cone? A drone camera climbs, hovers and descends: at which instant is it highest, and how can you find that instant from a formula without plotting a thousand points? A packaging company cuts squares from the corners of a sheet of cardboard and folds up the sides: how big should the squares be so that the box holds as much as possible? These three questions look different, but each one is answered by the same tool you built in Chapter 5 โ€” the derivative.

The key is to read $\dfrac{dy}{dx}$ as a rate: how many units $y$ changes per unit change in $x$, at that instant. In this chapter we use that one idea four ways. First, to compute rates of change, including rates linked through a formula such as $V=\tfrac43\pi r^3$. Second, to decide where a function is increasing or decreasing from the sign of $f'(x)$. Third, to find and classify the local maxima and minima โ€” the peaks and valleys of a graph โ€” with the first-derivative and second-derivative tests. Fourth, to find the absolute largest and smallest values on a closed interval, and to solve real optimisation problems. Throughout, $\log x$ means the natural logarithm $\log_e x$, as in Chapter 5.

Derivative = rateโ†’Linked ratesโ†’Sign of fโ€ฒ โ‡’ rising / fallingโ†’Critical pointsโ†’Local max / min testsโ†’Absolute extremaโ†’Optimisation
๐ŸŽฏ By the end of this chapter you canโ€ฆ
  • interpret $\dfrac{dy}{dx}$ as an instantaneous rate of change, with units, and compute marginal cost and marginal revenue;
  • solve linked-rate problems โ€” an inflating balloon, a sliding ladder, a filling tank, a moving shadow โ€” by differentiating a formula with respect to time;
  • define increasing, decreasing, strictly increasing and strictly decreasing functions, and find the intervals on which a function rises or falls from the sign of $f'(x)$;
  • find the critical points of a function and classify them as local maxima, local minima or neither by the first-derivative test and the second-derivative test;
  • find the absolute maximum and minimum values of a continuous function on a closed interval by comparing its values at the critical points and the end points;
  • set up and solve optimisation problems in geometry, packaging, business and engineering, and answer the Board 3- and 5-mark and the JEE questions on this chapter.
Teaser โ€” One Throw, Every Idea of the Chapter

A ball is thrown straight up from a height of $1.5$ m at $14.7$ m/s, so its height after $t$ seconds is $h(t)=1.5+14.7t-4.9t^2$ metres. The left panel shows the ball; the right panel its heightโ€“time graph. The short dark arrow on the graph has slope $h'(t)$, the rate of change of height โ€” the velocity. Watch the graph turn green while $h'(t)\gt0$ (rising) and red while $h'(t)\lt0$ (falling). At the top the rate is exactly zero: that is the maximum. Drag the time slider or press Play.

$t$ (s)โ€”
height $h$ (m)โ€”
rate $h'(t)$ (m/s)โ€”
the ball isโ€”

Sections in this chapter

6.1

Rate of Change of Quantities

The derivative as an instantaneous rate, with units; marginal cost and revenue; linked rates of balloons, ladders, tanks and shadows.

$\dfrac{dV}{dt}=4\pi r^2\,\dfrac{dr}{dt}$
Open section โ†’6.2

Increasing and Decreasing Functions

Increasing, decreasing and strictly monotonic functions; the sign of $f'$ test; intervals of increase and decrease from a sign chart.

$f'(x)\gt0$ on $(a,b)\Rightarrow f$ strictly increasing
Open section โ†’6.3

Maxima and Minima: Local Extrema and Derivative Tests

Local maxima and minima; critical points; the first-derivative test; the second-derivative test and when it fails.

$f'(c)=0,\ f''(c)\lt0\Rightarrow$ local max
Open section โ†’6.4

Absolute Extrema and Optimisation Problems

Absolute maximum and minimum on a closed interval; the critical-points-and-end-points table; optimisation of boxes, cans, fences and profits.

compare $f(a)$, $f(b)$, $f(c_i)$
Open section โ†’
โœ… Before you start
You need to differentiate fluently: the power, product and quotient rules and the derivatives of $\sin x$, $\cos x$ from Class 11 Mathematics, Ch. 12 โ€” Limits and Derivatives, and from Chapter 5 (Continuity and Differentiability) the chain rule, implicit differentiation, the derivatives of $e^x$, $\log x$ and the inverse trigonometric functions (see Section 2.1 for their principal value branches), and second-order derivatives. You also need the mensuration formulae for a sphere, cone and cylinder, and interval notation $(a,b)$, $[a,b]$. Quick self-test: differentiate $x^3-6x^2+9x$, $\dfrac{x}{x^2+1}$ and $e^{-x}\sin x$. (Answers: $3x^2-12x+9$, $\dfrac{1-x^2}{(x^2+1)^2}$, $e^{-x}(\cos x-\sin x)$.)
โ† 5.4 Parametric Forms and Second-Order Derivatives6.1 Rate of Change of Quantities โ†’