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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 12

Applications of di?erentiation

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Editor: Dr Jane Pitkethly, La Trobe University

Illustrations and web design: Catherine Tan, Michael Shaw Applications of dierentiation - A guide for teachers (Years 11-12)

Principal author:? Dr Michael Evans, AMSI

Peter Brown, University of NSW

Associate Professor David Hunt, University of NSW

Dr Daniel Mathews, Monash University

Assumed knowledge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 Content. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 Graph sketching. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 More graph sketching. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18 Maxima and minima problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26 Related rates of change. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37 Links forward. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41 Critical points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41 Finding gradients on a parametric curve. . . . . . . . . . . . . . . . . . . . . . . .42 History and applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45 History. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46 Answers to exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .49

A guide for teachers - Years 11 and 12

•{5}

In this module, we consider three topics:

•graph sketching •maxima and minima problems •related rates. We will mainly focus on nicely behaved functions which are differentiable at each point of their domains. Some of the examples are very straightforward, while others are more difficult and require technical skills to arrive at a solution.

Content

Graph sketching

Increasing and decreasing functions

Letfbe some function defined on an interval.

Definition

The functionfisincreasingover this interval if, for all pointsx1andx2in the interval, x

1·x2AE)f(x1)·f(x2).

This means that the value of the function at a larger number is greater than or equal to the value of the function at a smaller number. The graph on the left shows a differentiable function. The graph on the right shows a piecewise-defined continuous function. Both these functions are increasing.y x 0y x

0Examples of increasing functions.

{6}•Applications of differentiation

Definition

The functionfisdecreasingover this interval if, for all pointsx1andx2in the interval, x

1·x2AE)f(x1)¸f(x2).

The following graph shows an example of a decreasing function.y x0Example of a decreasing function. Note that a function that is constant on the interval is both increasing and decreasing overthisinterval. Ifwewanttoexcludesuchcases,thenweomittheequalitycomponent in our definition, and we add the wordstrictly: •A function isstrictly increasingifx1Çx2impliesf(x1)Çf(x2). •A function isstrictly decreasingifx1Çx2impliesf(x1)Èf(x2). We will use the following results. These results refer to intervals where the function is differentiable. Issues such as endpoints have to be treated separately. •Iff0(x)È0 for allxin the interval, then the functionfis strictly increasing. •Iff0(x)Ç0 for allxin the interval, then the functionfis strictly decreasing. •Iff0(x)AE0 for allxin the interval, then the functionfis constant.

Stationary points

Definitions

Letfbe a differentiable function.

•Astationary pointoffis a numberxsuch thatf0(x)AE0. •The pointcis amaximum pointof the functionfif and only iff(c)¸f(x), for allx in the domain off. The valuef(c) of the function atcis called themaximum value of the function. •The pointcis aminimumpointof the functionfif and only iff(c)·f(x), for allxin the domain off. The valuef(c) of the function atcis called theminimum valueof the function.

A guide for teachers - Years 11 and 12

•{7}

Local maxima and minima

In the following diagram, the pointalooks like a maximum provided we stay close to it, and the pointblooks like a minimum provided we stay close to it.y x0 a bDefinitions withc2(a,b) such thatf(c)¸f(x), for allx2(a,b). •The pointcis alocalminimumpointof the functionfif there exists an interval (a,b) withc2(a,b) such thatf(c)·f(x), for allx2(a,b). These are sometimes calledrelative maximumandrelative minimumpoints. Local maxima and minima are often referred to asturning points. The following diagram shows the graph ofyAEf(x), wherefis a differentiable function. It appears from the diagram that the tangents to the graph at the points which are local maxima or minima are horizontal. That is, at a local maximum or minimum pointc, we havef0(c)AE0, and hence each local maximum or minimum point is a stationary point. y x0 a bThe result appears graphically obvious, but we will present a formal proof in the case of a local maximum. {8}•Applications of differentiation

Theorem

Letfbe a differentiable function. Ifcis a local maximum point, thenf0(c)AE0. Proof Consider the interval (c¡±,cű), with±È0 chosen so thatf(c)¸f(x) for all x2(c¡±,cű). For all positivehsuch that 0ÇhDZ, we havef(c)¸f(cÅh) and therefore f(cÅh)¡f(c)h

·0.

Hence,

f

0(c)AElimh!0Åf(cÅh)¡f(c)h

·0. (1)

For all negativehsuch that¡±ÇhÇ0, we havef(c)¸f(cÅh) and therefore f(cÅh)¡f(c)h

¸0.

Hence,

f

0(c)AElimh!0¡f(cÅh)¡f(c)h

¸0. (2)

From (1) and (2), it follows thatf0(c)AE0.The first derivative test for local maxima and minima The derivative of the function can be used to determine when a local maximum or local minimum occurs.

Theorem(First derivative test)

Letfbe a differentiable function. Suppose thatcis a stationary point, that is,f0(c)AE0. aIf there exists±È0such thatf0(x)È0, for allx2(c¡±,c), andf0(x)Ç0, for all x2(c,cű), thencis a local maximum point. bIf there exists±È0such thatf0(x)Ç0, for allx2(c¡±,c), andf0(x)È0, for all x2(c,cű), thencis a local minimum point. Proof aThe function is increasing on the interval (c¡±,c), and decreasing on the in- terval (c,cű). Hence,f(c)¸f(x) for allx2(c¡±,cű). bThe function is decreasing on the interval (c¡±,c), and increasing on the in- terval (c,cű). Hence,f(c)·f(x) for allx2(c¡±,cű).

A guide for teachers - Years 11 and 12

•{9} In simple language, the first derivative test says: •Iff0(c)AE0 withf0(x)È0 immediately to the left ofcandf0(x)Ç0 immediately to the right ofc, thencis a local maximum point.y x0 c f c ))We can also illustrate this with a gradient diagram.

Value ofxc

Sign off0(x)Å0¡

Slope of graphyAEf(x)-

•Iff0(c)AE0 withf0(x)Ç0 immediately to the left ofcandf0(x)È0 immediately to the right ofc, thencis a local minimum point. y x 0 c f c ))We can also illustrate this with a gradient diagram.

Value ofxc

Sign off0(x)¡0Å

Slope of graphyAEf(x)-

{10}•Applications of differentiation There is another important type of stationary point: •Iff0(c)AE0 withf0(x)È0 on both sides ofc, thencis astationary point of inflexion.y x 0 c f c ))Here is a gradient diagram for this situation.

Value ofxc

Sign off0(x)Å0Å

Slope of graphyAEf(x)-

•Iff0(c)AE0 withf0(x)Ç0 on both sides ofc, thencis astationary point of inflexion. y x 0 c f c ))Here is a gradient diagram for this situation.

Value ofxc

Sign off0(x)¡0¡

Slope of graphyAEf(x)-

A guide for teachers - Years 11 and 12

•{11}Example Find the stationary points off(x)AE3x4Å16x3Å24x2Å3, and determine their nature.

Solution

The derivative offis

f

0(x)AE12x3Å48x2Å48x

AE12x(x2Å4xÅ4)

AE12x(xÅ2)2.

Sof0(x)AE0 impliesxAE0 orxAE¡2.

•IfxÇ¡2, thenf0(x)Ç0. •If¡2ÇxÇ0, thenf0(x)Ç0. •IfxÈ0, thenf0(x)È0. We can represent this in a gradient diagram.Value ofx¡20

Sign off0(x)¡0¡0Å

Slope of graphyAEf(x)--

Hence, there are stationary points atxAE0 andxAE¡2: there is a local minimum atxAE0, and a stationary point of inflexion atxAE¡2. The graph ofyAEf(x) is shown in the following diagram, but not all the features of the graph have been carefully considered at this stage.y x 2,19) 0 ,3)y = 3x⎷ + 16x² + 24x + 3 {12}•Applications of differentiationExercise 1 Assume that the derivative of the functionfis given byf0(x)AE(x¡1)2(x¡3). Find the values ofxwhich are stationary points off, and state their nature.Exercise 2

Find the stationary points off(x)AEx3¡5x2Å3xÅ2, and determine their nature.Use of the second derivative

The second derivative is introduced in the moduleIntroduction to differential calculus. Using functional notation, the second derivative of the functionfis written asf00. Using Leibniz notation, the second derivative is written as d2ydx

2, whereyis a function ofx.

In the moduleMotion in a straight line, it is shown that the acceleration of a particle is the second derivative of its position with respect to time. That is, if the position of the particle at timetis denoted byx(t), then the acceleration of the particle is¨x(t).

Recall that, in kinematics,

xmeansdxdt and¨xmeansd2xdt 2.

Concave up and concave down

Letfbe a function defined on the interval (a,b), and assume thatf0andf00exist at all points in (a,b). We consider the shape of the curveyAEf(x). Iff00(x)È0, for allx2(a,b), then the slope of the curve is increasing in the interval (a,b).

The curve is said to beconcave up.y

x0 y x0 y x0Examples of concave-up curves.

A guide for teachers - Years 11 and 12

•{13}

The curve isconcave down.y

x0Example of a concave-down curve.

Inflexion points

A point where the curve changes from concave up to concave down, or from concave down to concave up, is called apoint of inflexion. In the following diagram, there are points of inflexion atxAEcandxAEd. y x c f c ))(d,f(d))Examples of points of inflexion. The graphs ofyAE(x¡2)3Å1 andyAE ¡(x¡2)3Å1 are shown below. The point (2,1) is a point of inflexion for each of these graphs. In fact, the point (2,1) is a stationary point of inflexion for each of these graphs. y x 0y x 0 (2,1)y = (x ? 2) 3 + 1y = ?(x ? 2) 3 + 1 (2,1) {14}•Applications of differentiation The graph ofyAEx3¡3x2Å4xÅ4 is as follows. It has a point of inflexion at (1,6), but this is not a stationary point. In fact, this function has dydx

È0, for allx.y

x y = x 3 x + 4 x 4 (1,6)Note.Clearly, a necessary condition for a twice-differentiable functionfto have a point of inflexion atxAEcis thatf00(c)AE0. We will see that this isnota sufficient condition, and care must be taken when using it to find an inflexion point. For there to be a point of inflexion, there must be achange of concavity.Example Find the inflexion point of the cubic functionf(x)AEx3¡3x2¡144x.

Solution

We find the first and second derivatives:

f

0(x)AE3x2¡6x¡144 andf00(x)AE6x¡6.

Thusf00(x)AE0 impliesxAE1. ForxÇ1, we havef00(x)Ç0. ForxÈ1, we havef00(x)È0. The curve changes fromconcave downtoconcave upatxAE1. Hence, there is a point of inflexion atxAE1. y x (1,

146)It is not hard to show that there is a local maximum atxAE ¡6 and a local minimum at

xAE8.

A guide for teachers - Years 11 and 12

•{15}Exercise 3 Find the inflexion points of the functionf(x)AEx4Å28x3Å10x.The second derivative test We have seen how to use the first derivative test to determine whether a stationary point isalocalmaximum, alocalminimumorneitherofthese. Thesecondderivativeprovides an alternative test.

Theorem(Second derivative test)

Supposefis twice differentiable at a stationary pointc. aIff00(c)Ç0, thenfhas a local maximum atc.y x0 y fquotesdbs_dbs10.pdfusesText_16

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