Mathematical Techniques: Revision Notes

Transcription

1 Differentiation Dr A. J. Bevan October 20, 200 Mathematical Techniques: Revision Notes Dr A. J. Bevan, These notes contain the core of the information conveed in the lectures. The are not a substitute for attending the lectures and none of the examples covered are reproduced here. Worked examples of the techniques described in this note can be found in the tutorial question/solutionmaterialprovidedonthe course web site. Rates of change The change in with respect to x between these two points on the curve is given b the ratio δ δx = 2 x 2 x, = (x + δx) (x). δx (.) δ x 2 δ x x x If we take the limit as δx ten to zero we obtain ( ) δ lim = δx 0 δx, = lim δx 0 Figure : The function = f(x). ( (x + δx) (x) δx ). (.2)

2 Differentiation Dr A. J. Bevan 2 which is exact. In general the derivative obtained will be a function of x and can itself be differentiated in order to obtain higher order derivatives.. Standard Derivatives It is tedious to work out the derivatives of common functions using Eq. (.2) each time, and in practice we rel on tables or books of standard derivatives. Table lists a number of useful results. Table : Table of standard derivatives. = f(x) x n e x e kx a x nx n e x ke kx a x ln a ln x x log a x x ln a sin x cos x cos x sin x tan x sec 2 x cot x cosec 2 x sec x sec x tan x cosecx cosecx cot x sinh x cosh x cosh x sinh x tanh x / cosh 2 x sin x x 2 cos x x 2 tan x +x 2 sinh x x2 + cosh x x 2 tanh x x 2 2 Chain Rule Consider the function = f(u), where u = g(x). Here is a function of u, whichitselfisafunctionofsome other variable x. Thechainrulestatesthatforsuchafunction = du du. (2.)

3 Differentiation Dr A. J. Bevan 3 3 Differentiation of a Products The rule for the differentiation of the product of two functions u(x) and v(x) is d du (uv) =v + u dv, (3.) where it is assumed that both u and v are differentiable with respect to x. 4 Differentiation of Quotients The rule for the differentiation of the ratio of two functions u(x) andv(x) is d ( u ) = v du u dv v v 2, (4.) where it is assumed that both u and v are differentiable with respect to x. 5 Logarithmic Differentiation If one has a more complicated function, such as = un (x)v m (x) h p, (x) (5.) while it is possible to recursivel use the quotient and product rules to compute the derivative, it is possible, and often convenient to use natural logarithms to simplif the problem at hand. Taking the log of both sides of Eq. (5.2) one obtains ln = n ln u(x)+m ln v(x) p ln h(x), (5.2) which can be differentiated, noting that is a function of x so that d (ln ) =, (5.3) thus = n du u + m v dv p dh h, (5.4) which can be used to determine the derivative as a function of x given the form of u, v and h. 6 Implicit Differentiation It is possible to differentiate functions that are implicitl relating to x using the chain rule (as with the previous example). For example, consider x = R 2, (6.)

4 Differentiation Dr A. J. Bevan 4 which describes a circle. The derivative of this equation is 2x +2 =0, (6.2) so = x. (6.3) 7 Parametric Functions If we have a function defined b = h(θ) an = g(θ), (7.) then we can compute the derivative of with respect to x b using the chain rule, and noting that = dθ dθ. (7.2) assuming that it is possible do differentiate both h and g with respect to θ. 8 Tangents and normals to a curve Consider a straight line where = mx + C, (8.) and m =, (8.2) which is the slope of the line. Given a point on the line (x, )itispossibletocomputetheequationofthe line b noting that = m(x x ), = mx mx +, (8.3) (8.4) which is the equation of the tangent to a curve with gradient m at the point (x, ). The equation of the normal to a curve at (x, )canbeobtainedbrecallingthatthegradientofthenormal to a curve is /m. Sotheequationofthenormaltoacurveisjust = m x + m x +. (8.5)

5 Differentiation Dr A. J. Bevan 5 9 Differentiating inverse trigonometric functions The table of standard derivatives (Table ) lists the derivatives of the trigonometric functions. These can be used in order to derive the rules for differentiating inverse trigonometric functions. If we consider the equation = arcsin(x), we are able to take the sine of both sides to obtain sin() = x. We can now differentiate x with respect to which iel = cos(), = x 2, using sin 2 ()+cos 2 () =. Wecannowreadilobtainthederivativeofarcsin(x) whichis = x 2 It is possible to show that the derivative of =arccos(x) is = x 2 and similar steps can be taken in order to show that for =arctan(x) = +x 2. 0 Radius of curvature AusefulpropertofacurveistheradiusofcurvatureR. The radius of curvature is related to the change in direction δθ of the curve as a function of the change in position δs along the curve. The meaning of this quantit can be seen from the following: In the limit that δs ten to zero, the arc segment of length for the function = f(x) formspartofthecircumferenceofacircle(seefig.2).theradius of this circle is R, and the center of the radius of curvature is the point (x c, c ). We can relate to R via the angle subtended b the arc so, dθ = R, dθ = R.

6 Differentiation Dr A. J. Bevan (x, ) c c.8.6 δ.4 R θ.2 δ s (x, ) x Figure 2: The function = f(x) showingtheradiusofcurvaturer and center of curvature (x c, c )ofthe function for a point (x, )onthecurve. Recall that =tanθ. If we differentiate both sides of the last equation with respect to the position along the curve s we obtain d ( ) d ( ) = d [tan θ] = d dθ [tan θ] dθ d 2 2 cos θ = sec2 θ dθ d 2 2 = sec 3 θ dθ If we recall that + tan 2 θ =sec 2 θ and that tan θ =,wecansubstituteforsec 3 θ to obtain d 2 2 = (+tan 2 θ) [ d 2 ( ) ] 2 3/2 dθ 2 = +. 3/2 dθ We can re-arrange the equation and substitute dθ R = [ + ( ) ] 2 3/2 d 2 2 for R to obtain (0.)

7 Differentiation Dr A. J. Bevan 7 On inspection of Eq. (0.) we can see that the sign of R is given b the sign of the second derivative of the function. Ifthesecondderivativeisnegativeatapoint(x, ), then the curve is convex at that point. Conversel if the second derivative is positive at (x, ) thenthecurveisconcaveatthatpoint. The point (x c, c )correspondingtothecenterofthecirclewithradiusofcurvature R is given b x c = x R sin θ, c = + R cos θ. Stationar points: Maxima, Minima and Inflections We are able to use the derivatives of a function to obtain more information about features of how the value of a function f changes with x. In particular we can identif the positions where the function f is locall at a maximum or minimum value as illustrated in Figure 3. Genericall we call maxima and minima turning points. Maxima have a positive gradient of f(x) forx<x maximum,andanegativegradientfor x>x maximum. At the point x = x maximum the gradient is zero. The minima of f(x) canbeidentifiedb noting that the gradient is negative for x<x minimum,zeroforx = x minimum and positive for x>x minimum. It is not sufficient to identif minima and maximum solel b =0,aswehavenotconsideredallofthe possibilities as to how the gradient changes. We know that the gradientischangingaswescanthrough aturningpoint,andthatthesignofthegradientchangessignaswedothis. Thus,thevalueof at a turning point is non-zero. In fact the value of at a turning point is negative for a maximum and positive for a minimum (x,) max (x,) inflection (x,) min x Figure 3: The function = f(x) illustratingalocalmaximum,minimum,andpointofinflection. If (x) =0forsomevalueofx, thenthispointiscalledapoint of inflection. In order for the second derivative to be zero at a point of inflection x I the value of the second derivative has to change sign when we scan from values of x<x I through x = x I to x>x I.Collectivelturningpointsandpointsofinflectionare called stationar points. The values of (x) an (x) inthevicinitofstationarpointsaresummarised in Table 2.

8 Differentiation Dr A. J. Bevan 8 Table 2: Derivative values near stationar points: positive values are indicated b + and negative values are indicated b. Maximum x<turning point + + or x at the turning point 0 x>turning point + or Maximum x<turning point + or x at the turning point 0 + x>turning point + + or Inflection x<turning point or + + or x at the turning point 0 0 x>turning point + or + or 2 Partial Differentiation Consider the function z = f(x, ), which can be represented as a surface in three dimensions (See Fig. 4). The variables x and are orthogonal, which is another wa of saing that the are independent. This means if we want to differentiate z with respect to x, wecanconsider as a constant while doing so. Similarl, if we want to differentiate z with respect to, wecanconsiderx as a constant. When we do this, we use slightl different notation than when differentiating a function of a single variable. We use the curl d smbol called del to indicate a partial derivative,i.e. that we are differentiating a function of more than one variable. So we can write z x as the partial derivative of z with respect to x. Thisquantitistherateofchangeofz with respect to x for aconstant. Similarl z is the partial derivative of z with respect to, ortherateofchangeofz with respect to for a constant x. Higher derivatives of z can be calculated, but it is possible to differentiate either with respect to or with respect to x, so we have four choices of the second derivative of z. If we differentiate z/ x we can obtain 2 z x 2, 2 z x, b differentiating with respect to x or, respectivel.similarl we obtain 2 z 2, 2 z x,

9 Differentiation Dr A. J. Bevan 9 z x Figure 4: The curve f(x, ) aboutasaddlepoint. b differentiating z/ with respect to or x, respectivel.aconsequenceoftheorthogonalitof and x is that the same result is obtained when differentiating z b then x, andwhendifferentiatingz b x then, toputthisanotherwa: 2 z x = 2 z x. NOTE: With the function = f(x) itispossibletoinvertaderivativeusingtherelation =. It can be convenient to use this relation if it is simpler to invert and differentiate a function with respect to than with respect to x. The same is not true with partial differentiation! We can see this b considering derivatives of the following function z = x 2 +. Thederivativeofz with respect to x is simpl 2x. However in order to differentiate x with respect to z, wemustfirstre-writethefunctionasx = z. Thederivative of x with respect to z is /(2 z ). 3 Revisiting stationar points We can extend the discussion in Section to functions of morethanonevariable.forafunctionf(x, ) it is possible to use the tests summarised in Table 2 to identif stationar point either in x or in. In order to identif local maximum we require that a function f(x, ) hasamaximuminbothx and. Similarlfor apointtobeatalocalminimumofthefunctionwerequirethatboth x and are at minimum. As shown in Figure 4 it is possible that a function has a minimum in x and a maximum in (or visa versa) at a given point (x s, s ). When this situation arises we call the point (x s, s )asaddlepoint.

10 Differentiation Dr A. J. Bevan 0 4 Total Derivative For some function z = f(x, ), it can be useful to know how z changes for small changes in both x and. If we recall that x and are independent variables, an small change δx in the variable x is independent of achangeδ in the variable. Let us consider the case where is constant, and x changes b δx. We can write the change in z simpl as δz = z x δx. If we now consider x to be constant, and allow to change b some small amount δ, weareabletowrite down the change in z as δz = z δ. Each of these solutions results from the fact that x and are independent, and z = f(x, )canbereducedinto aone-dimensionalproblem. Nowwearereatoconsiderthegeneral case where both x and independentl change b some small amount, and calculate the change in z which is δz = z z δx + x δ. (4.) This is called the total derivative, and it is the sum of the change in z expected as a result of a small change in x, andthechangeinz expected for a small change in Rates of Change with Respect to Time Consider the change z when x and change in an interval of time δt; Eq.(4.)becomes(dividingbδt) z = z x dt + z dt. In the limit where δt 0: dz dt = z x dt + z dt. (4.2) 5 Change of Variables So in general for some function z = f(x, ), where x = x(u), and = (u) dz du = z x du + z du, We can go a step further. If some function is given b z = f(x, ), where x = h(u, v), and = g(u, v), where u and v are two orthogonal variables, then z is also a function of u and v and we can write: z u = z x x u + z u, z = z x v x v + z u. (5.) (5.2)

11 Differentiation Dr A. J. Bevan 5. Changing between polar and cartesian co-ordinates We are able to write a function z = f(x, ) intermsofpolarcoordinatesintermsofthedistancefromthe origin r, andananglesubtendedbetweentheradialdistancetoapointonthefunction,theoriginandthe x axis θ. The angle θ has a value between zero and 2π. We can determine expressions for the cartesian coordinates x and in terms of r and θ. Thesearesimpl x = r cos(θ), = r sin(θ). We can write the derivative of z in terms of r and θ b noting that f r f θ = f x x r + f = f x x θ + f r, θ. Where, x r x θ = cosθ, = r sin θ, and, r θ = sinθ, = r cos θ. Using these results we can write f r f θ = cosθ f f +sinθ x, = r sin θ f f + r cos θ x. We are able to use this result to determine the derivative of f with respect to r and θ for an differentiable function f = f(x, ).