Introduction

"He who hasn't tasted bitter things hasn't earned sweet things" — Gottfried Wilhelm Leibniz (1646–1716).

Real life constantly asks us to estimate how much a quantity changes when some input changes slightly, without recomputing everything from scratch:

• A thin circular metal plate is heated uniformly, so its radius (and hence its area) increases. Given only the approximate increase in the radius, how do we estimate the increase in area — without measuring the new area directly?
• Water fills an inverted right-circular-conical tank. As time passes, the height and radius of the water level both change, and so does the volume. Given the change in height and radius over a short interval, how do we estimate the change in volume?
• A satellite lifts off, tracked by a camera at a fixed distance. Given two consecutive angles of elevation measured a short time apart, how do we estimate the distance travelled by the satellite in that interval?

Each question has the same shape: a known small change in one or more inputs, and an unknown resulting change in an output that is expensive or impossible to measure directly. This chapter answers it using derivatives (for one input) and partial derivatives (for several inputs), through two connected ideas: linear approximation and the differential.

The starting observation is that linear functions $y=mx+b$ are trivially easy to evaluate at any point, while a general nonlinear function can be tedious or impossible to evaluate exactly by hand. For instance, given $f(x)=\sqrt{x+1}$ and $g(x)=x-7$, evaluating $g(3.25)$ is immediate while $f(3.25)$ is not. The resolution: accept a small, controlled error, and replace $f$ near a convenient point by the tangent line to its graph there — which is a genuine linear function, and (because the graph of $f$ looks almost straight in a small neighbourhood of the point of tangency) gives an excellent approximation to $f$ nearby. This "linearizing" idea, developed first for one variable, is then extended: to functions of two or three variables (where the graph is a surface, and derivatives with respect to one variable at a time — "partial derivatives" — measure the rate of change in that one direction), and to differentials, useful later for solving differential equations and for the substitution method in definite integrals.

Why more than one variable? A company producing two products (say pens and notebooks) wants to maximise profit; its revenue, cost and profit are all genuinely functions of two variables (units of each product). The volume of a box is a function of three variables (length, width, height). Even a country's economy depends on very many variables. So after building linear approximation and differentials for one variable, the chapter develops the same ideas — partial derivatives, linear approximation, and differentials — for real-valued functions of two and three real variables, including how to test whether such a function is homogeneous and how to apply Euler's theorem for homogeneous functions.

Learning Objectives

By the end of this chapter you should be able to:

• calculate the linear approximation of a function of one variable at a point;
• use that linear approximation to estimate a function's value without a calculator;
• calculate the differential of a function;
• apply linear approximation and differentials to real-life situations;
• find the partial derivatives of a function of more than one variable;
• calculate the linear approximation of a function of two or more variables;
• determine whether a function of several variables is homogeneous; and
• apply Euler's theorem for homogeneous functions.