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A guide for teachers - Years 11 and 12

1 2 3 4 5 6 7 8 9 1 0 1 1 1

2Supporting Australian Mathematics ProjectCalculus: Module 10

Introduction to dierential calculus

Full bibliographic details are available from Education Services Australia.

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Supporting Australian Mathematics Project

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Editor: Dr Jane Pitkethly, La Trobe University

Illustrations and web design: Catherine Tan, Michael Shaw Introduction to dierential calculus - A guide for teachers (Years 11-12) Principal author:Dr Daniel Mathews, Monash University

Peter Brown, University of NSW

Dr Michael Evans, AMSI

Associate Professor David Hunt, University of NSW

Assumed knowledge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 How fast are youreallygoing?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 The more things change .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 Content. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 The gradient of secants and tangents to a graph. . . . . . . . . . . . . . . . . . .7 Calculating the gradient ofyAEx2. . . . . . . . . . . . . . . . . . . . . . . . . . . .9 Definition of the derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11 Notation for the derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13 Some derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14 Properties of the derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16 The product, quotient and chain rules. . . . . . . . . . . . . . . . . . . . . . . . .19 Summary of differentiation rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . .26 The tangent line to a graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26 The second derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27 Differentiation of inverses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28 Implicit differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30 Links forward. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32 Higher derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32 Approximations of functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32 History and applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34 The discoverers of calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34 The Newton-Leibniz controversy. . . . . . . . . . . . . . . . . . . . . . . . . . . .34 Mathematically rigorous calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . .35 Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36 Functions differentiable and not. . . . . . . . . . . . . . . . . . . . . . . . . . . . .36 Answers to exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44

Introduction to

differential calculus

Assumed knowledge

The content of the modules:

•Coordinate geometry •The binomial theorem •Functions I •Functions II •Limits and continuity.

Motivation

How fast are youreallygoing?

You ride your bicycle down a straight bike path. As you proceed, you keep track of your exact position - so that, at every instant, you know exactly where you are.How fast were you going one second after you started? instant! Nonetheless, let us suspend disbelief and imagine you have this information. (Perhaps you have an extremely accurate GPS or a high-speed camera.) By considering this question, we are led to some important mathematical ideas.

First, recall the formula foraverage velocity:

vAE¢x¢t, where¢xis the change in your position and¢tis the time taken. Thus, average velocity is the rate of change of position with respect to time.

A guide for teachers - Years 11 and 12

•{5} Draw a graph of your positionx(t) at timetseconds. Connect two points on the graph, representingyourpositionattwodifferenttimes. Thegradientofthislineisyouraverage velocity over that time period.x t 0 txAverage velocityvAE¢x¢t. Trying to discover your velocity at the one-second mark (tAE1), you calculate youraver- age velocityover the period fromtAE1 to a slightly later timetAE1Å¢t. Trying to be more accurate, you look at shorter time intervals, with¢tsmaller and smaller. If you really knew your position at every single instant of time, then you could work out your aver- age velocity over any time interval, no matter how short. The three lines in the following diagram correspond to¢tAE2,¢tAE1 and¢tAE0.5. x t

011.523Average velocity over shorter time intervals.

As¢tapproaches 0, you obtain better and better estimates of your instantaneous veloc- ity at the instanttAE1. These estimates correspond to the gradients of lines connecting closer and closer points on the graph. {6}•Introduction to differential calculus

In the limit, as¢t!0,

lim

¢t!0¢x¢t

gives the precise value for the instantaneous velocity attAE1. This is also the gradient of thetangentto the graph attAE1.Instantaneous velocityis the instantaneous rate of change of position with respect to time. of how fast you were going attAE1. Given a functionx(t) describing your position at timet, you could calculate your exact velocity at timetAE1. In fact, it is possible to calculate the instantaneous velocity atanyvalue oft, obtaining a function which gives your instantaneous velocity at timet. This function is commonly denoted byx0(t) ordxdt , and is known as thederivativeofx(t) with respect tot.

In this module, we will discuss derivatives.

The more things change ...

Velocity is an important example of a derivative, but this is just one example. The world is full of quantities which change with respect to each other - and these rates of change can often be expressed as derivatives. It is often important to understand and predict how things will change, and so derivatives are important. Here are some examples of derivatives, illustrating the range of topics where derivatives are found: •Mechanics.We saw that the derivative of position with respect to time is velocity. Also, the derivative of velocity with respect to time isacceleration. And the derivative of momentum with respect to time is the (net)forceacting on an object. •Civil engineering, topography.Leth(x) be the height of a road, or the altitude of a mountain, as you move along a horizontal distancex. The derivativeh0(x) with respect to distance is thegradientof the road or mountain. •Population growth.Supposeapopulationhassizep(t)attimet. Thederivativep0(t) withrespecttotimeisthepopulationgrowthrate. Thegrowthratesofhuman, animal and cell populations are important in demography, ecology and biology, respectively. •Economics.In macroeconomics, the rate of change of the gross domestic product (GDP) of an economy with respect to time is known as theeconomic growth rate. It is often used by economists and politicians as a measure of progress.

A guide for teachers - Years 11 and 12

•{7} •Mechanical engineering.Suppose that the total amount of energy produced by an engineisE(t)attimet. ThederivativeE0(t)ofenergywithrespecttotimeisthepower of the engine. All of these examples arise from a more abstract question in mathematics: •Mathematics.Consider the graph of a functionyAEf(x), which is a curve in the plane. What is thegradientof a tangent to this graph at a point? Equivalently, what is the instantaneous rate of change ofywith respect tox? Inthismodule, wediscusspurelymathematicalquestionsaboutderivatives. Inthethree modulesApplications of differentiation,Growth and decayandMotion in a straight line, we discuss some real-world examples. is an important motivating example, we now concentrate on calculating the gradient of a tangent to a curve.

Content

The gradient of secants and tangents to a graph

Consider a functionf:R!Rand its graphyAEf(x), which is a curve in the plane. We wishtofindthegradientofthiscurveatapoint. Butfirstweneedtodefineproperlywhat we mean by the gradient of a curve at a point! The moduleCoordinate geometrydefines the gradient of a line in the plane: Given a non-vertical line and two points on it, thegradientis defined asriserun .run riseGradient of a line. {8}•Introduction to differential calculus Now, given acurvedefined byyAEf(x), and a pointpon the curve, consider another pointqon the curve nearp, and draw the linepqconnectingpandq. This line is called asecant line. We write the coordinates ofpas (x,y), and the coordinates ofqas (xÅ¢x,yÅ¢y). Here ¢xrepresents a small change inx, and¢yrepresents the corresponding small change iny.y x0 x x + x y f x p x y xq = (x + x, y + y) y f x x) f x )Secant connecting points on the graphyAEf(x)atxandxÅ¢x. As¢xbecomes smaller and smaller, the pointqapproachesp, and the secant linepq approaches a line called thetangentto the curve atp. We define thegradient of the curveatpto be the gradient of this tangent line. y x0 x y f x p x y )Secants onyAEf(x)approaching the tangent line atx. Note that, in this definition, the approximation of a tangent line by secant lines is just like the approximation of instantaneous velocity by average velocities, as discussed in theMotivationsection. With this definition, we now consider how to compute the gradient of the curveyAEf(x) at the pointpAE(x,y).

A guide for teachers - Years 11 and 12

•{9} TakingqAE(xÅ¢x,yÅ¢y) as above, the secant linepqhas gradient riserun respectively. You cannot cancel the¢"s! As¢x!0, the gradient of the tangent line is given by lim

We also denote this limit by

dydx lim

¢x!0¢y¢xAEdydx

The notation

dydx indicates the instantaneous rate of change ofywith respect tox, and is not a fraction. For our purposes, the expressionsdxanddyhave no meaning on their own, and thed"s do not cancel! The gradient of a secant is analogous to average velocity, and the gradient of a tangent is analogous to instantaneous velocity. Velocity is the instantaneous rate of change of position with respect to time, and the gradient of a tangent to the graphyAEf(x) is the instantaneous rate of change ofywith respect tox.

Calculating the gradient ofyAEx2

Let us consider a specific functionf(x)AEx2and its graphyAEf(x), which is the standard parabola. To illustrate the ideas in the previous section, we will calculate the gradient of this curve atxAE1. We first construct secant lines between the points on the graph atxAE1 andxAE1Å¢x, and calculate their gradients.y x011.523 y x x = 2, gradient = 4 x = 1, gradient = 3 x = 0.5, gradient = 2.5Gradients of secants fromxAE1toxAE1Å¢x. {10}•Introduction to differential calculus For instance, taking¢xAE2, we consider the secant connecting the points atxAE1 and xAE3. Between these two points,f(x) increases fromf(1)AE1 tof(3)AE9, giving¢yAE8, and hence

¢y¢xAE82

AE4. We compute gradients of secants for various values of¢xin the following table.

Secants of the parabolaf(x)AEx2Secant between points¢x¢yAEf(xÅ¢x)¡f(x) Gradient of secant¢y¢xxAE1,xAE3 2 8 4

xAE1,xAE2 1 3 3 xAE1,xAE1.5 0.5 1.25 2.5 xAE1,xAE1.1 0.1 0.21 2.1

xAE1,xAE1.001 0.001 0.002001 2.001As¢xapproaches 0, the gradients of the secants approach 2. It turns out that indeed the

gradient of the tangent atxAE1 is 2. To see why, consider the interval of length¢x, from xAE1 toxAE1Å¢x. We have

¢yAEf(1Å¢x)¡f(1)

AE(1Å¢x)2¡12

AE2¢xÅ(¢x)2,

so that

¢y¢xAE2¢xÅ(¢x)2¢x

AE2Å¢x.

In the limit, as¢x!0, we obtain the instantaneous rate of change dydx

AElim¢x!0¢y¢x

AElim¢x!0¡2Å¢x¢AE2.

(So, if you were riding your bike and your position wasf(x)AEx2metres afterxseconds, then your instantaneous velocity after 1 second would be 2 metres per second.)

A guide for teachers - Years 11 and 12

•{11} There"s nothing special about the pointxAE1 or the functionf(x)AEx2, as the following example illustrates.Example Letf(x)AEx3. What is the gradient of the tangent line to the graphyAEf(x) at the pointquotesdbs_dbs28.pdfusesText_34

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