Related rates
Applications of Derivatives · Rate of Change of Quantities\[\frac{dy}{dt}=\frac{dy}{dx}\cdot\frac{dx}{dt}\]
Area of circle: dA/dt = 2πr dr/dt. Volume of sphere: dV/dt = 4πr² dr/dt. Cube: dV/dt = 3a² da/dt.
Chapter 6 · Calculus
Applications of Derivatives
Chapter Overview
Derivatives measure change. In this chapter we put them to work: finding how fast one quantity changes when another does (rates of change), deciding where a function rises or falls (increasing and decreasing functions), and locating the highest and lowest values of a function (maxima and minima) – the heart of optimisation problems in science, engineering and business.
Board focus: A long-answer optimisation problem (5 marks) and a case study on maxima/minima appear almost every year.
Topics
Key Concepts
- \(\frac{dy}{dx}\) is the rate of change of \(y\) with respect to \(x\); for related rates use \(\frac{dy}{dt}=\frac{dy}{dx}\cdot\frac{dx}{dt}\).
- \(f\,'(x)>0\) on an interval ⇒ \(f\) strictly increasing; \(f\,'(x)<0\) ⇒ strictly decreasing.
- Critical points: \(f\,'(c)=0\) (or undefined). Second-derivative test: \(f\,''(c)<0\) ⇒ local max, \(f\,''(c)>0\) ⇒ local min.
- Absolute extrema on [a, b]: compare f at critical points and at the end points.
Formulas
Second-derivative test
Applications of Derivatives · Maxima and Minima\[f'(c)=0:\quad f''(c)\lt 0\Rightarrow\text{local max},\qquad f''(c)\gt 0\Rightarrow\text{local min}\]
Key Points
Sign chart
Applications of Derivatives · Increasing and Decreasing FunctionsMark the zeros of \(f\,'\) on a number line and test one point in each interval; the sign of \(f\,'\) gives increasing (+) or decreasing (−).
Classic optimisation results
Applications of Derivatives · Maxima and Minima- Fixed perimeter ⇒ maximum area rectangle is a square.
- Closed cylinder of given volume with least surface area: h = 2r (height = diameter).
- Open box from an a × a sheet: cut squares of side a/6.
- Sum of two positive numbers fixed ⇒ product is greatest when they are equal.
Absolute extrema on [a, b]
Applications of Derivatives · Maxima and MinimaCompare the values at critical points inside the interval and at the two end points.
Common Mistakes
Substituting too early
Applications of Derivatives · Rate of Change of QuantitiesDifferentiate \(V=\frac43\pi r^3\) first, then put \(r=5\). If you substitute first, \(V\) becomes a constant with derivative 0.
Sign of a decreasing rate
Applications of Derivatives · Rate of Change of Quantities"Decreasing at 2 cm/s" means \(\frac{dx}{dt}=-2\).
Forgetting the end points
Applications of Derivatives · Maxima and MinimaThe absolute maximum of \(x^3-3x\) on \([0,3]\) is at the end point \(x=3\) (value 18), not at a critical point.
Local vs absolute
Applications of Derivatives · Maxima and MinimaA local maximum need not be the largest value overall; a local minimum value can even exceed a local maximum value.
Solved Examples
Intervals of monotonicity
Applications of Derivatives · Increasing and Decreasing Functions\(f(x)=2x^3-3x^2-12x+6\).
\(f\,'=6(x-2)(x+1)\): increasing on \((-\infty,-1]\cup[2,\infty)\), decreasing on \([-1,2]\).
Open box of maximum volume
Applications of Derivatives · Maxima and MinimaSquares of side \(x\) are cut from the corners of a 30 cm × 30 cm sheet and the sides folded up.
\(V=x(30-2x)^2\), \(V'=(30-2x)(30-6x)=0\Rightarrow x=5\) (\(x=15\) gives \(V=0\)). \(V_{\max}=5\cdot20^2=2000\) cm³.
Practice & Tests
Chapter Practice
Untimed mixed questions from all topics. Answers are revealed after you submit.
Start PracticeApplications of Derivatives – Chapter Test
A board-style chapter test drawn fresh from the question bank each time: MCQs, assertion-reason, numericals, written answers you self-check against model solutions, and a case study.
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