Chapter 3 · Introduction

Chapter 3 — Applications of Derivatives

Putting derivatives to work — tangent lines, motion and rates of change, linked rates, the shape of a graph, the best possible design, and tricky limits.

A rocket is being launched from the ground and cameras are recording the event. A video camera is located on the ground a certain distance from the launch pad. At what rate should the angle of inclination (the angle the camera makes with the ground) change to allow the camera to record the flight of the rocket as it heads upward?

A rocket launch involves two related quantities that change over time. Being able to solve this type of problem is just one application of derivatives introduced in this chapter. We begin with the most direct application of all — using the derivative to write the equation of the tangent line (and the normal line) to a curve, the best straight-line approximation to the curve near a point. Next we read the derivative as a rate of change: velocity and acceleration of a moving object, growth rates in biology, and marginal cost and profit in economics. We also look at how derivatives are used to find maximum and minimum values of functions. As a result, we will be able to solve applied optimisation problems, such as maximising revenue and minimising surface area.

Tangent & normal lines→Rates of change & motion→Related rates→Critical points & shape→Mean Value Theorem→Optimization→L'Hôpital's rule
🎯 By the end of this chapter you can…
  • write equations of tangent and normal lines (explicit, implicit and parametric curves) and use them for linear approximation;
  • interpret derivatives as rates of change with units, and analyse motion: position, velocity, speed and acceleration;
  • set up and solve related-rates problems with the chain rule;
  • find critical points, absolute and local extrema, and use the first and second derivative tests;
  • describe where a graph increases, decreases, is concave up or down, and find inflection points;
  • state and apply Rolle's theorem and the Mean Value Theorem;
  • solve optimisation problems and evaluate indeterminate limits with L’Hôpital’s rule.
Teaser — The Best Garden

100 ft of fence, a rock wall on one side (Example 1 of 3.6). The side perpendicular to the wall is $x$, so the area is $A(x)=x(100-2x)$. Drag $x$ and find the biggest garden — then see how calculus finds it without guessing.

the garden (wall on top)
$A(x) = x(100 - 2x)$
$x$—
parallel side $100 - 2x$—
area A—
$A'(x) = 100 - 4x$—

Sections in this chapter

3.1

Tangent and Normal Lines

Equations of tangent and normal lines for explicit, implicit and parametric curves — and where they are used: linear approximation, direction of motion, optics.

$y=f(a)+f'(a)(x-a)$
Open section →3.2

Derivatives as a Rate of Change

Position, velocity and acceleration; speeding up and slowing down; rates in biology and economics, and interpreting $f'(a)$ with units.

$v=s^\prime,\;a=v^\prime=s^{\prime\prime}$
Open section →3.3

Related Rates

Balloons, planes, rockets, funnels, ladders — rates linked through an equation.

$\tfrac{dV}{dt}=4\pi r^2\tfrac{dr}{dt}$
Open section →3.4

Maxima and Minima

Absolute and local extrema, critical points, first and second derivative tests, concavity.

$f'(c)=0$ or undefined
Open section →3.5

The Mean Value Theorem

Rolle’s theorem, the MVT and its corollaries about increasing functions.

$f'(c)=\tfrac{f(b)-f(a)}{b-a}$
Open section →3.6

Optimization

Gardens, boxes and travel time: the six-step optimisation strategy.

maximise $A(x)$ on $[0,50]$
Open section →3.7

L'Hospital's Rule

Evaluate $\frac{0}{0}$ and $\frac{\infty}{\infty}$ limits by differentiating top and bottom.

$\lim\tfrac fg=\lim\tfrac{f'}{g'}$
Open section →
✅ Before you start
Be fluent with the differentiation rules and the chain rule (Chapter 2), solving equations and sign charts (0.1 §5) and area/volume formulas (Formula Sheet).
← 2.5 Parametric Differentiation3.1 Tangent and Normal Lines →