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(BQ) Part 2 book Higher engineering mathematics has contents: Differentiation of parametric equations, differentiation of implicit functions, partial differentiation, standard integration, integration using partial fractions, some applications of integration,...and other contents.
Differential calculus 29 Differentiation of parametric equations 29.1 Introduction to parametric equations Certain mathematical functions can be expressed more simply by expressing, say, x and y separately in terms of a third variable For example, y = r sin θ, x = r cos θ Then, any value given to θ will produce a pair of values for x and y, which may be plotted to provide a curve of y = f (x) The third variable, θ, is called a parameter and the two expressions for y and x are called parametric equations The above example of y = r sin θ and x = r cos θ are the parametric equations for a circle The equation of any point on a circle, centre at the origin and of radius r is given by: x + y2 = r , as shown in Chapter 14 To show that y = r sin θ and x = r cos θ are suitable parametric equations for such a circle: (e) Cardioid x = a (2 cos θ − cos 2θ), y = a (2 sin θ − sin 2θ) (f) Astroid (g) Cycloid x = a cos3 θ, y = a sin3 θ x = a (θ − sin θ) , y = a (1− cos θ) (a) Ellipse (b) Parabola (c) Hyperbola (d) Rectangular hyperbola (e) Cardioid (f) Astroid Left hand side of equation = x + y2 = (r cos θ)2 + (r sin θ)2 = r cos2 θ + r sin2 θ = r cos2 θ + sin2 θ = r = right hand side (since cos2 θ + sin2 θ = 1, as shown in Chapter 16) 29.2 Some common parametric equations The following are some of the most common parametric equations, and Figure 29.1 shows typical shapes of these curves x = a cos θ, y = b sin θ x = a t , y = 2a t x = a sec θ, y = b tan θ c (d) Rectangular x = c t, y = t hyperbola (a) Ellipse (b) Parabola (c) Hyperbola (g) Cycloid Figure 29.1 29.3 Differentiation in parameters When x and y are given in terms of a parameter, say θ, then by the function of a function rule of DIFFERENTIATION OF PARAMETRIC EQUATIONS dx = cos t dt From equation (1), differentiation (from Chapter 27): x = sin t, hence dy dθ dy = × dx dθ dx It may be shown that this can be written as: dy dy dθ = dx dx dθ (1) For the second differential, d d2 y = dx dx dy dx d = dθ dy dx dθ · dx d2 y = dx2 dy dx dx dθ dx i.e (2) Problem Given x = 5θ − and dy y = 2θ (θ − 1), determine in terms of θ dx x = 5θ − 1, hence dy =5 dθ y = 2θ(θ − 1) = 2θ − 2θ, dy hence = 4θ − = (2θ − 1) dθ From equation (1), dy dy 2(2θ − 1) dθ = = or (2θ − 1) dx dx 5 dθ Problem The parametric equations of a function are given by y = cos 2t, x = sin t dy d2 y Determine expressions for (a) (b) dx dx (a) y = cos 2t, hence dy = −6 sin 2t dt dy −6 sin 2t −6(2 sin t cos t) dy = dt = = dx dx cos t cos t dt from double angles, Chapter 18 dy = −6 sin t i.e dx (b) From equation (2), d2 y or d dθ 315 = d dt dy dx dx dt d (−6 sin t) −6 cos t = dt = cos t cos t d2 y = −3 dx2 Problem The equation of a tangent drawn to a curve at point (x1 , y1 ) is given by: y − y1 = dy1 (x − x1 ) dx1 Determine the equation of the tangent drawn to the parabola x = 2t , y = 4t at the point t dx1 = 4t dt dy1 =4 and y1 = 4t, hence dt From equation (1), At point t, x1 = 2t , hence Hence, the dy dy = dt = = dx dx 4t t dt equation of the tangent is: y − 4t = x − 2t t Problem The parametric equations of a cycloid are x = 4(θ − sin θ), y = 4(1 − cos θ) d2 y dy (b) Determine (a) dx dx G 316 DIFFERENTIAL CALCULUS (a) x = 4(θ − sin θ), dx hence = − cos θ = 4(1 − cos θ) dθ dy y = 4(1 − cos θ), hence = sin θ dθ From equation (1), dy dy sin θ sin θ = dθ = = dx dx 4(1 − cos θ) (1 − cos θ) dθ (b) From equation (2), d dy sin θ dx dθ − cos θ = = dx dx 4(1 − cos θ) dθ (1 − cos θ)(cos θ) − (sin θ)(sin θ) (1 − cos θ)2 = 4(1 − cos θ) d dθ d2 y = cos θ − cos2 θ − sin2 θ 4(1 − cos θ)3 = cos θ − cos2 θ + sin2 θ 4(1 − cos θ)3 = cos θ − 4(1 − cos θ)3 = −(1 − cos θ) −1 = 4(1 − cos θ)3 4(1 − cos θ)2 1 (a) − cot θ (b) − cosec3 θ 16 dy π Evaluate at θ = radians for the dx hyperbola whose parametric equations are x = sec θ, y = tan θ [4] The parametric equations for a rectangular dy hyperbola are x = 2t, y = Evaluate t dx when t = 0.40 [−6.25] The equation of a tangent drawn to a curve at point (x1 , y1 ) is given by: dy1 (x − x1 ) y − y1 = dx1 Use this in Problems and Determine the equation of the tangent drawn π to the ellipse x = cos θ, y = sin θ at θ = [y = −1.155x + 4] Determine the equation of the tangent drawn to the rectangular hyperbola x = 5t, y = at t t = y=− x+5 29.4 Further worked problems on differentiation of parametric equations Now try the following exercise Exercise 128 Further problems on differentiation of parametric equations Given x = 3t − and y = t(t − 1), determine dy in terms of t (2t − 1) dx A parabola has parametric equations: dy x = t , y = 2t Evaluate when t = 0.5 dx [2] The parametric equations for an ellipse dy are x = cos θ, y = sin θ Determine (a) dx d2 y (b) dx Problem The equation of the normal drawn to a curve at point (x1 , y1 ) is given by: y − y1 = − (x − x1 ) dy1 dx1 Determine the equation of the normal drawn to the astroid x = cos3 θ, y = sin3 θ at the point π θ= dx = −6 cos2 θ sin θ dθ dy = sin2 θ cos θ y = sin3 θ, hence dθ x = cos3 θ, hence DIFFERENTIATION OF PARAMETRIC EQUATIONS From equation (1), = dy sin2 θ cos θ sin θ dy dθ = = =− = −tanθ dx dx −6 cos θ sin θ cos θ dθ dy π π , = −tan = −1 dx π π x1 = cos3 = 0.7071 and y1 = sin3 = 0.7071 4 Hence, the equation of the normal is: −2 cosec θ cot θ sec θ tan θ − = When θ = (x − 0.7071) −1 y − 0.7071 = x − 0.7071 y =x y − 0.7071 = − i.e i.e Problem The parametric equations for a hyperbola are x = sec θ, y = tan θ Evaluate d2 y dy (b) , correct to significant figures, (a) dx dx when θ = radian (a) x = sec θ, hence =− d dy d2 y dθ dx = dx dx dθ d (2 cosec θ) = dθ sec θ tan θ cos θ sin θ sin θ cos θ cos2 θ sin θ cos θ sin2 θ cos3 θ = − cot3 θ sin3 θ d2 y = − cot3 = − When θ = rad, dx (tan 1)3 = −0.2647, correct to significant figures Problem When determining the surface tension of a liquid, the radius of curvature, ρ, of part of the surface is given by: 1+ ρ= dy y = tan θ, hence = sec2 θ dθ (b) From equation (2), sin θ cos θ =− dx = sec θ tan θ dθ From equation (1), dy dy sec2 θ sec θ dθ = = = dx dx sec θ tan θ tan θ dθ 2 cos θ = = or cosec θ sin θ sin θ cos θ dy When θ = rad, = = 2.377, correct to dx sin significant figures 317 dy dx G d2 y dx Find the radius of curvature of the part of the surface having the parametric equations x = 3t , y = 6t at the point t = dx = 6t dt dy =6 y = 6t, hence dt x = 3t , hence dy dy = dt = = From equation (1), dx dx 6t t dt From equation (2), d2 y dx = d dt dy dx dx dt = d dt t 6t 1 t =− = 6t 6t − 318 DIFFERENTIAL CALCULUS dy dx 1+ Hence, radius of curvature, ρ = 1+ When t = 2, ρ = − (2)3 Use this in Problems and Determine the equation of the normal drawn 1 to the parabola x = t , y = t at t = [y = −2x + 3] 3 Find the equation of the normal drawn to the cycloid x = 2(θ − sin θ), y = 2(1 − cos θ) at π θ = rad [y = −x + π] d2 y dx 1+ t − 6t = 2 = (1.25)3 − 48 = − 48 (1.25)3 = −67.08 d2 y , correct to sigdx π nificant figures, at θ = rad for the cardioid x = 5(2θ − cos 2θ), y = 5(2 sin θ − sin 2θ) Determine the value of [0.02975] The radius of curvature, ρ, of part of a surface when determining the surface tension of a liquid is given by: Now try the following exercise Exercise 129 Further problems on differentiation of parametric equations A cycloid has parametric equations x = 2(θ − sin θ), y = 2(1 − cos θ) Evaluate, at θ = 0.62 rad, correct to significant d2 y dy (b) figures, (a) dx dx [(a) 3.122 (b) −14.43] The equation of the normal drawn to a curve at point (x1 , y1 ) is given by: (x − x1 ) y − y1 = − dy1 dx1 1+ ρ= dy dx 3/2 d2 y dx Find the radius of curvature (correct to significant figures) of the part of the surface having parametric equations (a) x = 3t, y = at the point t = t π (b) x = cos3 t, y = sin3 t at t = rad [(a) 13.14 (b) 5.196] Differential calculus 30 Differentiation of implicit functions 30.1 Implicit functions When an equation can be written in the form y = f (x) it is said to be an explicit function of x Examples of explicit functions include y = 2x − 3x + 4, 3ex and y = cos x Differentiating implicit functions It is possible to differentiate an implicit function by using the function of a function rule, which may be stated as du du dy = × dx dy dx Thus, to differentiate y3 with respect to x, the subdu stitution u = y3 is made, from which, = 3y2 dy d dy Hence, (y ) = (3y2 ) × , by the function of a dx dx function rule A simple rule for differentiating an implicit function is summarised as: d dy d [ f ( y)] = [ f ( y)] × dx dy dx (a) 2y4 (b) sin 3t (a) Let u = 2y4 , then, by the function of a function rule: du du dy d dy = × = (2y4 ) × dx dy dx dy dx y = 2x ln x In these examples y may be differentiated with respect to x by using standard derivatives, the product rule and the quotient rule of differentiation respectively Sometimes with equations involving, say, y and x, it is impossible to make y the subject of the formula The equation is then called an implicit function and examples of such functions include y3 + 2x = y2 − x and sin y = x + 2xy 30.2 Problem Differentiate the following functions with respect to x: (1) dy dx (b) Let u = sin 3t, then, by the function of a function rule: du du dt d dt = × = (sin 3t) × dx dt dx dt dx dt = cos 3t dx = 8y3 Problem Differentiate the following functions with respect to x: (a) ln 5y (b) e3θ−2 (a) Let u = ln 5y, then, by the function of a function rule: du du dy d dy = × = (4 ln 5y) × dx dy dx dy dx = dy y dx (b) Let u = e3θ−2 , then, by the function of a func5 tion rule: du du dθ d 3θ−2 dθ = × = e × dx dθ dx dθ dx = 3θ−2 dθ e dx G 320 DIFFERENTIAL CALCULUS Now try the following exercise For example, d d d (x y) = (x ) (y) + (y) (x ), dx dx dx by the product rule Exercise 130 Further problems on differentiating implicit functions In Problems and differentiate the given functions with respect to x √ (a) 3y5 (b) cos 4θ (c) k ⎤ ⎡ dθ dy ⎢ (a) 15y dx (b) −8 sin 4θ dx ⎥ ⎥ ⎢ ⎦ ⎣ dk (c) √ k dx (a) ln 3t (b) e2y+1 (c) tan 3y ⎡ dt dy ⎤ (b) e2y+1 (a) ⎢ 2t dx dx ⎥ ⎦ ⎣ dy (c) sec2 3y dx dy + y(2x), dx by using equation (1) = (x ) = x2 Problem Thus d d d (2x y2 ) = (2x ) (y2 ) + (y2 ) (2x ) dx dx dx dy + (y2 )(6x ) = (2x ) 2y dx dy + 6x y2 dx dy = 2x2 y 2x + 3y dx = 4x y Problem Find d 3y dx 2x In the quotient rule of differentiation let u = 3y and v = 2x d d (2x) (3y) − (3y) (2x) d 3y dx dx = Thus dx 2x (2x)2 (2x) = 30.3 Differentiating implicit functions containing products and quotients The product and quotient rules of differentiation must be applied when differentiating functions containing products and quotients of two variables d (2x y2 ) dx In the product rule of differentiation let u = 2x and v = y2 Differentiate the following with respect to y: √ (a) sin 2θ (b) x (c) t e ⎤ ⎡ √ dx dθ (b) (a) cos 2θ x ⎢ dy dy ⎥ ⎥ ⎢ ⎦ ⎣ −2 dt (c) t e dy Differentiate the following with respect to u: 2 (a) (b) sec 2θ (c) √ (3x + 1) y ⎤ ⎡ −6 dx (a) ⎥ ⎢ (3x + 1)2 du ⎥ ⎢ ⎢ dθ ⎥ ⎥ ⎢ (b) sec 2θ tan 2θ ⎢ du ⎥ ⎥ ⎢ −1 dy ⎦ ⎣ (c) y du Determine dy + 2xy dx = 6x dy − (3y)(2) dx 4x dy − 6y dy dx = x −y 4x dx 2x Problem Differentiate z = x + 3x cos 3y with respect to y DIFFERENTIATION OF IMPLICIT FUNCTIONS d d dz = (x ) + (3x cos 3y) dy dy dy dx dx = 2x + (3x)(−3 sin 3y) + ( cos 3y) dy dy dx dx = 2x − 9x sin 3y + cos 3y dy dy 321 dy An expression for the derivative in terms of dx x and y may be obtained by rearranging this latter equation Thus: (2y + 1) from which, dy = − 6x dx − 6x dy = dx 2y + Now try the following exercise Exercise 131 Further problems on differentiating implicit functions involving products and quotients d (3x y3 ) dx Determine Each term in turn is differentiated with respect to x: 3xy2 3x Find d dx 2y 5x Determine d du dy + 2y dx dy x −y 5x dx 3u 4v dv v−u 4v du dz √ Given z = y cos 3x find dx cos 3x dy √ − y sin 3x √ y dx Determine 30.4 dz given z = 2x ln y dy dx x 2x + ln y y dy Further implicit differentiation An implicit function such as 3x + y2 − 5x + y = 2, may be differentiated term by term with respect to x This gives: d d d d d (3x ) + (y2 ) − (5x) + (y) = (2) dx dx dx dx dx dy dy − + = 0, dx dx using equation (1) and standard derivatives i.e 6x + 2y Problem Given 2y2 − 5x − − 7y3 = 0, dy determine dx d d d d (2y2 ) − (5x ) − (2) − (7y3 ) dx dx dx dx d = (0) dx dy dy i.e 4y − 20x − − 21y2 =0 dx dx Rearranging gives: Hence (4y − 21y2 ) i.e Problem dy = 20x dx G dy 20x3 = dx (4y − 21y2 ) Determine the values of x = given that x + y2 = 25 dy when dx Differentiating each term in turn with respect to x gives: d d d (x ) + (y2 ) = (25) dx dx dx dy =0 i.e 2x + 2y dx 2x x dy =− =− Hence dx 2y y Since x + y2 = 25, when x = 4, y = (25 − 42 ) = ±3 Thus when x = and y = ±3, 4 dy =− =± dx ±3 322 DIFFERENTIAL CALCULUS x + y2 = 25 is the equation of a circle, centre at the origin and radius 5, as shown in Fig 30.1 At x = 4, the two gradients are shown + = 25 −5 dy dx and Gradient = − 43 y2 8x + 2y3 = (10y − 6xy2 ) 8x + 2y3 4x + y3 dy = = dx 10y − 6xy2 y(5 − 3xy) (b) When x = and y = 2, y x2 Rearranging gives: dy 4(1) + (2)3 12 = = = −6 dx 2[5 − (3)(1)(2)] −2 x −3 Gradient = 43 −5 Problem Find the gradients of the tangents drawn to the circle x + y2 − 2x − 2y = at x = Figure 30.1 Above, x + y2 = 25 was differentiated implicitly; actually, the equation could be transposed to y = (25 − x ) and differentiated using the function of a function rule This gives −1 (25 − x ) (−2x) dy = dx and when x = 4, obtained above =− (25 − x ) Problem dy (a) Find in terms of x and y given dx 4x + 2xy3 − 5y2 = dy (b) Evaluate when x = and y = dx (a) Differentiating each term in turn with respect to x gives: d d d d (4x ) + (2xy3 ) − (5y2 ) = (0) dx dx dx dx dy dx + (y3 )(2) dy =0 dx dy dy =0 8x + 6xy2 + 2y3 − 10y dx dx − 10y i.e d d d d d (x ) + (y2 ) − (2x) − (2y) = (3) dx dx dx dx dx x dy 4 =− as =± dx (25 − 42 ) i.e 8x + (2x) 3y2 dy dx Differentiating each term in turn with respect to x gives: The gradient of the tangent is given by 2x + 2y i.e dy dy −2−2 =0 dx dx dy = − 2x, dx − 2x 1−x dy = = from which dx 2y − y−1 The value of y when x = is determined from the original equation Hence (2y − 2) Hence (2)2 + y2 − 2(2) − 2y = i.e + y2 − − 2y = or y2 − 2y − = Factorising gives: (y + 1)(y − 3) = 0, from which y = −1 or y = When x = and y = −1, 1−x 1−2 −1 dy = = = = dx y−1 −1 − −2 When x = and y = 3, 1−2 −1 dy = = dx 3−1 Hence the gradients of the tangents are ± DIFFERENTIATION OF IMPLICIT FUNCTIONS 323 y x 2+ y 2− 2x −2y = Now try the following exercise Gradient = − 12 Exercise 132 Further problems on implicit differentiation dy In Problems and determine dx 2x + x + y2 + 4x − 3y + = − 2y r=√ 1 x −1 Gradient = 12 −2 Figure 30.2 The circle having√ the given equation has its centre at (1, 1) and radius (see Chapter 14) and is shown in Fig 30.2 with the two gradients of the tangents Problem 10 Pressure p and volume v of a gas are related by the law pvγ = k, where γ and k are constants Show that the rate of change of dp p dv pressure = −γ dt v dt Since pvγ = k, then p = k = kv−γ vγ dp dv dp = × dt dv dt by the function of a function rule dp d = (kv−γ ) dv dv = −γkv−γ−1 = −γk dv dp = γ+1 × dt v dt Since i.e −γk vγ+1 k = pv , dp −γ(pvγ ) dv −γpvγ dv = = dt vγ+1 dt vγ v1 dt γ dp p dv = −γ dt v dt − 6y2 2y3 − y + 3x − = dy Given x + y2 = evaluate when dx √ √ x = and y = − 25 In Problems to 7, determine x + 2x sin 4y = 3y2 + 2xy − 4x = 2x y + 3x = sin y 3y + 2x ln y = y4 + x dy dx −(x + sin 4y) 4x cos 4y 4x − y 3y + x x(4y + 9x) cos y − 2x − ln y + (2x/y) − 4y3 dy when If 3x + 2x y3 − y2 = evaluate dx x = and y = [5] Determine the gradients of the tangents drawn to the circle x + y2 = 16 at the point where x = Give the answer correct to significant figures [±0.5774] 10 Find the gradients of the tangents drawn to x y2 + = at the point where the ellipse x=2 [±1.5] 11 Determine the gradient of the curve [−6] 3xy + y2 = −2 at the point (1,−2) G 712 ESSENTIAL FORMULAE Integral Calculus y Standard integrals y dx y (x + a2 ) x n+1 +c n+1 (except where n = −1) ax n (x + a2 ) sec ax tan ax (x (x − a2 ) t = tan sec ax + c a x ln x + c tan ax ln ( sec ax) + c a cos2 x sin 2x x+ 2 +c sin2 x sin 2x x− 2 +c tan2 x tan x − x + c (a2 − x ) (a2 − x ) + x2 ) −cot x − x + c sin−1 x +c a (x + a2 ) +c a cosh−1 x + c or a x+ (x − a2 ) +c a a2 x x cosh−1 + c (x − a2 ) − 2 a θ substitution To determine ax e +c a x+ a2 x x sinh−1 + (x + a2 ) + c a ln eax cot2 x − a2 ) x + c or a sinh−1 ln a cos ax sin ax + c a sin ax − cos ax + c a sec2 ax tan ax + c a cosec ax − cot ax + c a cosec ax cot ax − cosec ax + c a (a2 y dx dθ let a cos θ + b sin θ + c sin θ = 2t (1 + t ) dθ = dt (1 + t ) cos θ = − t2 + t2 and Integration by parts If u and v are both functions of x then: u dv dx = uv − dx v du dx dx Reduction formulae x a2 x sin−1 + (a2 − x ) + c a x n ex dx = In = x n ex − nIn−1 x tan−1 + c a a x n cos x dx = In = x n sin x + nx n−1 cos x −n(n − 1)In−2 ESSENTIAL FORMULAE π x n cos x dx = In = −nπn−1 − n(n − 1)In−2 x n sin x dx = In = −x n cos x + nx n−1 sin x −n(n − 1)In−2 π/2 sinn x dx = mean value = b−a b y dx a R.m.s value: n−1 In−2 sinn x dx = In = − sinn−1 x cos x + n n n−1 cosn x dx = In = cosn−1 sin x + In−2 n n π/2 Mean value: cosn x dx = In = n−1 In−2 n b y2 dx a Volume of solid of revolution: b volume = tann−1 x tan x dx = In = − In−2 n−1 n b−a r.m.s value = πy2 dx about the x-axis a (ln x)n dx = In = x( ln x)n − nIn−1 Centroids With reference to Fig FA5: With reference to Fig FA4 b xy dx y x¯ = y ϭ f (x) a y¯ = and b b y2 dx a b y dx y dx a a A y xϭa xϭb y ϭ f (x) x Area A Figure FA4 C x y Area under a curve: b area A = y dx a Figure FA5 xϭa xϭb x 713 714 ESSENTIAL FORMULAE Second moment of area and radius of gyration Shape Rectangle length l breadth b Position of axis Second moment of area, I (1) Coinciding with b (2) Coinciding with l (3) Through centroid, parallel to b (4) Through centroid, parallel to l Triangle Perpendicular height h base b Circle radius r Semicircle radius r (1) Coinciding with b Radius of gyration, k bl 3 lb3 bl 12 √ b √ √ 12 lb3 12 b √ 12 bh3 12 h √ h √ 18 h √ (2) Through centroid, parallel to base bh3 36 (3) Through vertex, parallel to base bh3 πr r √ (2) Coinciding with diameter πr 4 (3) About a tangent 5πr 4 r √ r πr r (1) Through centre, perpendicular to plane (i.e polar axis) Coinciding with diameter G Theorem of Pappus With reference to Fig FA5, when the curve is rotated one revolution about the x-axis between the limits x = a and x = b, the volume V generated is given by: V = 2πA¯y B C Area A d Parallel axis theorem: If C is the centroid of area A in Fig FA6 then 2 2 AkBB = AkGG + Ad or kBB = kGG + d2 G Figure FA6 B ESSENTIAL FORMULAE Differential Equations Perpendicular axis theorem: If OX and OY lie in the plane of area A in Fig FA7, then = AkOZ AkOX + AkOY or kOZ = kOX + kOY First order differential equations Separation of variables If dy = f (x) dx then y = If dy = f (y) dx then If dy = f (x) · f (y) dx Z Area A O Y Figure FA7 Numerical integration Trapezoidal rule width of interval first + last ordinates + sum of remaining ordinates Mid-ordinate rule ydx ≈ width of interval f (x) dx dx = then dy f (y) dy = f (y) f (x) dx Homogeneous equations X ydx ≈ 715 sum of mid-ordinates dy If P = Q, where P and Q are functions of both dx x and y of the same degree throughout (i.e a homogeneous first order differential equation) then: dy dy Q (i) Rearrange P = Q into the form = dx dx P (ii) Make the substitution y = vx (where v is a function of x), from which, by the product rule, dy dv = v(1) + x dx dx dy (iii) Substitute for both y and in the equation dx dy Q = dx P (iv) Simplify, by cancelling, and then separate the dy variables and solve using the = f (x) · f (y) dx method y (v) Substitute v = to solve in terms of the original x variables Linear first order Simpson’s rule ydx ≈ width of interval first + last ordinate +4 +2 sum of even ordinates sum of remaining odd ordinates dy + Py = Q, where P and Q are functions of If dx x only (i.e a linear first order differential equation), then (i) determine the integrating factor, e P dx (ii) substitute the integrating factor (I.F.) into the equation y (I.F.) = (I.F.) Q dx (iii) determine the integral (I.F.)Q dx 716 ESSENTIAL FORMULAE (c) complex, say m = α ± jβ, then the general solution is Numerical solutions of first order differential equations y1 = y0 + h(y )0 y = eαx (A cos βx + B sin βx) yP1 = y0 + h(y )0 = y0 + h[(y )0 + f (x1 , yp1 )] (iv) given boundary conditions, constants A and B can be determined and the particular solution obtained Euler’s method: Euler-Cauchy method: and yC1 If a Runge-Kutta method: dy To solve the differential equation = f (x, y) given dx the initial condition y = y0 at x = x0 for a range of values of x = x0 (h)xn : Identify x0 , y0 and h, and values of x1 , x2 , x3 , Evaluate k1 = f (xn , yn ) starting with n = h h Evaluate k2 = f xn + , yn + k1 2 h h Evaluate k3 = f xn + , yn + k2 2 Evaluate k4 = f(xn + h, yn + hk3 ) Use the values determined from steps to to evaluate: h yn+1 = yn + {k1 + 2k2 + 2k3 + k4 } Repeat steps to for n = 1, 2, 3, d2 y dy +b + cy = f (x) then: dx dx2 (i) rewrite the differential equation as (aD2 + bD + c)y = (ii) substitute m for D and solve the auxiliary equation am2 + bm + c = (iii) obtain the complimentary function (C.F.), u, as per (iii) above (iv) to find the particular integral, v, first assume a particular integral which is suggested by f (x), but which contains undetermined coefficients (See Table 51.1, page 482 for guidance) (v) substitute the suggested particular integral into the original differential equation and equate relevant coefficients to find the constants introduced (vi) the general solution is given by y = u + v (vii) given boundary conditions, arbitrary constants in the C.F can be determined and the particular solution obtained Higher derivatives Second order differential equations d2 y dy If a + b + cy = dx dx constants) then: (where a, b and c are eax sin ax (i) rewrite the differential equation as (aD2 + bD + c)y = (ii) substitute m for D and solve the auxiliary equation am2 + bm + c = (iii) if the roots of the auxiliary equation are: cos ax xa sinh ax (a) real and different, say m = α and m = β then the general solution is y = Ae αx + Be βx (b) real and equal, say m = α twice, then the general solution is y = (Ax + B)eαx y(n) y cosh ax ln ax an eax nπ nπ n a cos ax + a! a−n x (a − n)! an {[1 + (−1)n ] sinh ax + [1 − (−1)n ] cosh ax} an sin ax + an {[1 − (−1)n ] sinh ax +[1 + (−1)n ] cosh ax} (n − 1)! (−1)n−1 xn ESSENTIAL FORMULAE is: Leibniz’s theorem To find the n’th derivative of a product y = uv: y = Ax v − y(n) = (uv)(n) = u(n) v + nu(n−1) v(1) + n(n − 1) (n−2) (2) u v 2! + n(n − 1)(n − 2) (n−3) (3) u v + ··· 3! (a) Leibniz-Maclaurin method (i) Differentiate the given equation n times, using the Leibniz theorem, (ii) rearrange the result to obtain the recurrence relation at x = 0, (iii) determine the values of the derivatives at x = 0, i.e find (y)0 and (y )0 , x2 22 (v + 1) + x4 24 × 2!(v + 1)(v + 2) − x6 + ··· 26 × 3!(v + 1)(v + 2)(v + 3) + Bx −v + Power series solutions of second order differential equations + x4 x2 + 22 (v − 1) 24 × 2!(v − 1)(v − 2) x6 + ··· 26 × 3!(v − 1)(v − 2)(v − 3) or, in terms of Bessel functions and gamma functions: y = AJv (x) + BJ−v (x) x =A x2 − (v + 1) (1!) (v + 2) v + (iv) substitute in the Maclaurin expansion for y = f (x), (v) simplify the result where possible and apply boundary condition (if given) +B (b) Frobenius method x (ii) differentiate the trial series to find y and y , + Jv (x) = and J−v (x) = d2 y dy + x + (x − v2 )y = dx dx ∞ x v x −v k=0 (−1)k x 2k (v + k + 1) 22k (k!) ∞ k=0 (−1)k x 2k 22k (k!) (k − v + 1) and in particular: Jn (x) = Bessel’s equation x4 − ··· 24 (2!) (3 − v) In general terms: (iii) substitute the results in the given differential equation, (iv) equate coefficients of corresponding powers of the variable on each side of the equation: this enables index c and coefficients a1 , a2 , a3 , from the trial solution, to be determined x4 − ··· 24 (2!) (v + 4) x2 − (1 − v) (1!) (2 − v) −v (i) Assume a trial solution of the form: y = x c {a0 + a1 x + a2 x + a3 x + · · · + a0 = 0, ar x r + · · · } The solution of x 717 x n x − n! (n + 1)! + x (2!)(n + 2)! 2 − ··· 718 ESSENTIAL FORMULAE J0 (x) = − x2 x4 + 22 (1!)2 24 (2!)2 Binomial probability distribution x6 − + ··· (3!) and J1 (x) = x3 x5 x − + 2 (1!)(2!) (2!)(3!) − x7 27 (3!)(4!) If n = number in sample, p = probability of the occurrence of an event and q = − p, then the probability of 0, 1, 2, 3, occurrences is given by: n(n − 1) n−2 q p , 2! n(n − 1)(n − 2) n−3 q p , 3! qn , + ··· (i.e successive terms of the (q + p)n expansion) Legendre’s equation The solution of (1−x ) d2 y dy +k(k +1)y = −2x dx dx Normal approximation to a binomial distribution: Mean = np is: y = a0 − Standard deviation σ = √ (npq) k(k + 1) x 2! k(k + 1)(k − 2)(k + 3) x − ··· + 4! + a1 x − + nqn−1 p, (k − 1)(k + 2) x 3! (k − 1)(k − 3)(k + 2)(k + 4) x − ··· 5! If λ is the expectation of the occurrence of an event then the probability of 0, 1, 2, 3, occurrences is given by: e−λ , λe−λ , λ2 e−λ , 2! λ3 e−λ , 3! Product-moment formula for the linear correlation coefficient Rodrigue’s formula Pn (x) = Poisson distribution d n (x − 1)n 2n n! dx n Coefficient of correlation r = xy x2 y2 Statistics and Probability Mean, median, mode and standard deviation If x = variate and f = frequency then: mean x¯ = fx f The median is the middle term of a ranked set of data The mode is the most commonly occurring value in a set of data where x = X − X and y = Y − Y and (X1 , Y1 ), (X2 , Y2 ), denote a random sample from a bivariate normal distribution and X and Y are the means of the X and Y values respectively Normal probability distribution Partial areas under the standardized normal curve — see Table 58.1 on page 561 Standard deviation σ= f (x − x¯ )2 f Student’s t distribution for a population Percentile values (tp ) for Student’s t distribution with ν degrees of freedom — see Table 61.2 on page 587 ESSENTIAL FORMULAE 719 Chi-square distribution Estimating the mean of a population (σ known) Percentile values (χp2 ) for the Chi-square distribution with ν degrees of freedom—see Table 63.1 on page 609 (o − e)2 χ2 = where o and e are the observed e and expected frequencies The confidence coefficient for a large sample size, (N ≥ 30) is zc where: Symbols: Population number of members Np , mean µ, standard deviation σ Sample number of members N, mean x, standard deviation s Sampling distributions mean of sampling distribution of means µx standard error of means σx standard error of the standard deviations σs Confidence level % Confidence coefficient zc 99 98 96 95 90 80 50 2.58 2.33 2.05 1.96 1.645 1.28 0.6745 The confidence limits of a population mean based on sample data are given by: Np − N zc σ x± √ Np − N for a finite population of size Np , and by zc σ x ± √ for an infinite population N Standard error of the means Standard error of the means of a sample distribution, i.e the standard deviation of the means of samples, is: σ σx = √ N Np − N Np − for a finite population and/or for sampling without replacement, and Estimating the mean of a population (σ unknown) The confidence limits of a population mean based on sample data are given by: µx ± zc σx Estimating the standard deviation of a population The confidence limits of the standard deviation of a population based on sample data are given by: s ± z c σs σ σx = √ N for an infinite population and/or for sampling with replacement The relationship between sample mean and population mean µx = µ for all possible samples of size N are drawn from a population of size Np Estimating the mean of a population based on a small sample size The confidence coefficient for a small sample size (N < 30) is tc which can be determined using Table 61.1, page 582 The confidence limits of a population mean based on sample data is given by: tc s x± √ (N − 1) 720 ESSENTIAL FORMULAE Laplace Transforms Function f (t) Fourier Series Laplace transforms L{f (t)} = 0∞ e−st f (t) dt 1 s k k s s−a eat sin at a s2 +a2 cos at s s2 +a2 s2 t t n (n = positve integer) n! sn+1 cosh at s s2 −a2 sinh at a s2 −a2 e−at t n n! (s+a)n+1 e−at sin ωt ω (s+a)2 +ω2 e−at cos ωt If f (x) is a periodic function of period 2π then its Fourier series is given by: ∞ f (x) = a0 + n=1 where, for the range −π to +π: a0 = 2π an = π bn = π s+a (s+a)2 −ω2 e−at sinh ωt ω (s+a)2 −ω2 π −π f (x) dx π −π f (x) cos nx dx (n = 1, 2, 3, ) f (x) sin nx dx (n = 1, 2, 3, ) π −π If f (x) is a periodic function of period L then its Fourier series is given by: ∞ f (x) = a0 + + bn sin 2πnx L an cos where for the range − a0 = L an = L bn = L L L to + : 2 L/2 −L/2 f (x) dx L/2 −L/2 f (x) cos 2πnx L dx (n = 1, 2, 3, ) f (x) sin 2πnx L dx (n = 1, 2, 3, ) L/2 −L/2 Complex or exponential Fourier series The Laplace transforms of derivatives ∞ f (x) = First derivative dy L = sL{y} − y(0) dx where y(0) is the value of y at x = Second derivative dy L dx cn e j 2πnx L n=−∞ where cn = L For even symmetry, L − L2 f (x)e−j 2πnx L L 2 f (x) cos L For odd symmetry, cn = = s L{y} − sy(0) − y (0) dy at x = where y (0) is the value of dx 2πnx L n=1 s+a (s+a)2 +ω2 e−at cosh ωt (an cos nx + bn sin nx) cn = −j L L 2πnx L f (x) sin dx 2πnx L dx dx Index Adjoint of matrix, 274 Algebra, Algebraic method of successive approximations, 80 substitution, integration, 391 Amplitude, 154, 157 And-function, 94 And-gate, 106 Angle between two vectors, 238 of any magnitude, 148 of depression, 119 of elevation, 119 Angular velocity, 142 Applications of complex numbers, 257 differentiation, 298 rates of change, 298 small changes, 311 tangents and normals, 310 turning points, 302 velocity and acceleration, 299 integration, 374 areas, 374 centroids, 378 mean value, 376 r.m.s value, 376 second moment of area, 382 volumes, 377 Arc, 137 length, 138 Area of triangle, 125 irregular figures, 216 of sector, 138 under curve, 374 Argand diagram, 250 Argument, 254 Arithmetic mean, 538 progression, 51 Astroid, 314 Asymptotes, 203 Auxiliary equation, 475 Average, 538 value of waveform, 219 Bessel’s correction, 598 equation, 504 Bessel functions, 504, 506, 508 Binary numbers, 86 Binomial distribution, 553, 591, 594 expression, 58 series/theorem, 58, 59 practical problems, 64 Bisection method, 76 Boolean algebra, 94 laws and rules of, 99 Boundary conditions, 444, 512 Brackets, Cardioid, 314 Cartesian complex numbers, 249 co-ordinates, 133 Catenary, 43 Centre of area, 379 gravity, 378 mass, 379 Centripetal acceleration, 144 force, 144 Centroids, 378 Chain rule, 295 Change of limits, 393 Chi-square values, 607, 609 Chord, 137 Circle, 137, 192 equation of, 140, 192 Circumference, 137 Class interval, 532 Coefficient of correlation, 567, 568 Cofactor, 273 Combinational logic networks, 107 Combination of periodic functions, 232 Common difference, 51 logarithms, 24 ratio, 54 Comparing two sample means, 602 Complementary function, 481 Completing the square, 16 Complex numbers, 249 applications of, 257 Cartesian form, 249 coefficients, 691 conjugate, 251 equations, 253 exponential form, 264 form of Fourier series, 690 polar form, 254 powers of, 261 roots of, 262 Complex wave, 160 considerations, 686 Compound angles, 176 Computer numbering systems, 86 Conditional probability, 545 722 INDEX Confidence coefficients, 582 intervals, 581, 582 levels, 582 limits, 582 Continuous data, 527 function, 199, 657 Contour map, 357 Conversion of a sin ωt + b cos ωt into R sin(ωt + α), 178 Correlation, linear, 567 Cosecant, 116 Cosh, 41 series, 48 Cosh θ substitution, 405 Coshec, 41 Cosine, 116 curves, 152 rule, 124 wave production, 151 Cotangent, 116 Coth, 41 Couple, 102 Cramer’s rule, 283 Critical regions, 593 values, 593 Cross product, 241 Cubic equations, 191 Cumulative frequency distribution, 533, 536 Curve sketching, 209 Cycloid, 314 Deciles, 543 Definite integrals, 371 Degree of differential equation, 444 Degrees of freedom, 586 De Moivre’s theorem, 261 De Morgan’s laws, 101 Denary number, 86 Dependent event, 545 Depression, angle of, 119 Derivatives, 288 Determinant, 267, 271, 273 to solve simultaneous equations, 279 Determination of law, 38 Diameter, 137 Differential coefficient, 288 Differential equations, 444 dy d2 x a + b + cy = type, 475 dy dx d2 x dy a + b + cy = f (x) type, 481 dy dx dy = f (x) type, 444 dx dy = f (y) type, 446 dx dy = f (x) · f (y) type, 448 dx dy + Py = Q type, 455 dx dy = Q type, 451 dx degree of, 444 first order, separation of variables, 444 homogeneous first order, 451 linear first order, 455 partial, 512 power series method, 491 numerical methods, 460 simultaneous, using Laplace transforms, 650 using Laplace transforms, 645 Differentiation, 287, 288 applications, 298 from first principles, 288 function of a function, 295 implicit, 319 inverse hyperbolic function, 338 trigonometric function, 332 logarithmic, 324 methods of, 287 of common functions, 288 of hyperbolic functions, 330 of parametric equations, 314 partial, 343 first order, 343 second order, 346 product, 292 quotient, 293 successive, 296 Direction cosines, 240 Discontinuous function, 199 Discrete data, 527, 541 Distribution-free tests, 613 Dividend, Divisor, D-operator form, 475 Dot product, 238 Double angles, 182 P Elastic string, 516 Elevation, angle of, 119 Ellipse, 192, 314 Equations, Bessel’s, 504 complex, 253 heat conduction, 515, 520 hyperbolic, 47 indicial, 26, 498 Laplace, 514, 515, 522 Legendre’s, 509 normal, 310 of circle, 140, 192 quadratic, simple, simultaneous, solving by iterative methods, 76 tangents, 310 transmission, 515 trigonometric, 166, 167 wave, 515, 516 INDEX Euler–Cauchy method, 465 Euler’s formula, 699 Euler’s method, 460 Even function, 43, 199, 669, 686, 695 Expectation, 545 Exponential form of complex number, 264 Fourier series, 690 Exponential function, 28, 193 graphs of, 31, 193 power series, 29 Extrapolation, 572 Factorization, 2, 16 Factor theorem, Family of curves, 443 Final value theorem, 636 First moment of area, 382 Formulae, 3, 705 Fourier coefficients, 658 Fourier series, 160, 657 cosine, 669 exponential form, 690 half range, 672, 680 non-periodic over range 2π, 663 over any range, 676 periodic of period 2π, 657 sine, 669 Frequency, 157, 527 curve, 559 distribution, 532, 533, 535, 559 domain, 698 polygon, 533, 535 relative, 527 spectrum, 698 Frobenius method, 498 Functional notation, 288 Function of a function, 295, 319 Functions of two variables, 355 Fundamental, 658 Gamma function, 506 Gaussian elimination, 284 General solution of a differential equation, 444, 476 Geometric progression, 54 Gradient of a curve, 287 Graphs of exponential functions, 31 hyperbolic functions, 43 inverse functions, 333 logarithmic functions, 27 standard functions, 191 trigonometric functions, 148 Grouped data, 532, 539, 542 Growth and decay laws, 35 Half range Fourier series, 672, 680 Half-wave rectifier, 163 Harmonic analysis, 160, 683 Harmonic synthesis, 160 Heat conduction equation, 515, 520 Hexadecimal number, 90 Higher order differentials, 491 Histogram, 532, 535, 539 of probabilities, 555, 557 Homogeneous, 451, 475 Homogeneous first order differential equations, 451 Horizontal bar chart, 528 Hyperbola, 193, 314 rectangular, 193, 314 Hyperbolic functions, 41, 173 differentiation of, 330 graphs of, 43 inverse, 332 solving equations, 47 Hyperbolic identities, 44, 174 logarithms, 24, 33 Hypotenuse, 115 Hypotheses, 590 Identities, hyperbolic, 44, 174 Identities, trigonometric, 166 Imaginary part, 249 Implicit differentiation, 319 Implicit function, 319 Independent event, 545 Indices, laws of, Indicial equations, 26, 498 Industrial inspection, 554 Inequalities, simple, 12 involving a modulus, 13 involving quotients, 14 involving square functions, 15 quadratic, 16 Initial conditions, 512 Initial value theorem, 636 Integrating factor, 455 Integration, 367 algebraic substitution, 391 applications of, 374 areas, 374 centroids, 378 mean value, 376 r.m.s value, 376 second moment of area, 382 volumes, 377 by partial fractions, 408 by parts, 418 change of limits, 393 cosh θ substitution, 405 definite, 371 numerical, 71, 433 reduction formulae, 424 sin θ substitution, 401 sinh θ substitution, 403 standard, 367 tan θ substitution, 403 t = tan (θ/2) substitution, 413 trigonometric substitutions, 397 Interpolation, 572 Interval estimate, 581 723 724 INDEX Inverse functions, 201, 332 hyperbolic, 332 differentiation of, 338 trigonometric, 202, 332 differentiation of, 332 Inverse Laplace transforms, 638 using partial fractions, 640 Inverse matrix, 272, 274 Invert-gate, 106 Iterative methods, 76 Karnaugh maps, 102 Lagging angle, 154, 157 Lamina, 378 Laplace’s equation, 514, 515, 522 Laplace transforms, 627, 632 common notations, 627 definition, 627 derivatives, 634 for differential equations, 645 for simultaneous differential equations, 650 inverse, 638 using partial fractions, 640 linearity property, 627 of elementary functions, 627, 632 Laws of Boolean algebra, 99 growth and decay, 35 indices, logarithms, 24, 324 probability, 545 Leading angle, 154, 158 Least-squares regression lines, 571 Leibniz notation, 288 theorem, 493 Leibniz–Maclaurin method, 495 Legendre polynomials, 509, 510 Legendre’s equation, 509 Level of significance, 592 L’Hopital’s rule, 73 Limiting values, 72, 288 Linear correlation, 567 first order differential equation, 455 regression, 571 second order differential equation, 475 velocity, 142 Logarithmic differentiation, 324 forms of inverse hyperbolic functions, 337 scale, 38 Logarithms, 24, 324 graphs of, 27, 193 laws of, 24, 324 Logic circuits, 106 universal, 110 Log-linear graph paper, 38 Log-log graph paper, 38 Lower class boundary value, 532 Maclaurin’s series/theorem, 67 numerical integration, 71 Mann–Whitney test, 620 Matrices, 267 to solve simultaneous equations, 277 Matrix, 267 adjoint, 274 determinant of, 267, 271, 273 inverse, 272, 274 reciprocal, 272, 274 unit, 271 Maximum point, 302, 355 practical problems, 306 Mean value, 376, 538 of waveform, 219 Measures of central tendency, 538 Median, 538 Mid-ordinate rule, 216, 435 Minimum point, 302, 355 practical problems, 306 Mode, 538 Modulus, 13, 239, 254 Moment of a force, 244 Nand-gate, 107, 110 Napierian logarithms, 24, 33 Natural logarithms, 24, 33 Newton–Raphson method, 83 Non-homogeneous differential equation, 475 Non-parametric tests, 607, 614 Nor-gate, 107, 110 Norm, 239 Normal, 310 approximation to binomial distribution, 591, 595 curve, 559 distribution, 559 equations, 571 probability paper, 563 standard variate, 559 Nose-to-tail method, 226 Not-function, 94 Not-gate, 106 Numerical integration, 71, 433 methods for first order differential equations, 460 Numerical method of harmonic analysis, 160, 683 Octal numbers, 88 Odd function, 43, 199, 669, 686, 695 Ogive, 533, 536 One-sided test, 593 Order of precedence, Or-function, 94 Or-gate, 106 Osborne’s rule, 44 Pappus theorem, 380 Parabola, 191, 314 Parallel axis theorem, 383 Parallelogram method, 226 Parameter, 314 Parametric equations, 314 Partial differential equations, 512, 513 INDEX Partial differentiation, 343 equations, 512 Partial integration, 512 Partial fractions, 18 inverse Laplace transforms, 640 integration, using, 408 linear factors, 18 quadratic factors, 22 repeated linear factors, 21 Particular solution of differential equation, 444, 476 Particular integral, 481 Pascal’s triangle, 58 Percentage component bar chart, 528 Percentile, 543 Period, 153, 657 Periodic function, 153, 199, 657 combination of, 232 Periodic time, 157 Perpendicular axis theorem, 384 Phasor, 157, 225, 232, 699 Pictogram, 528 Pie diagram, 528 Planimeter, 216 Point of inflexion, 302 Point estimate, 581 Poisson distribution, 556, 595 Polar co-ordinates, 133 curves, 194 form, 254 Poles, 642 Pole-zero diagram, 643 Polynomial division, Polynomial, Legendre’s, 510 Population, 527 Power series for ex , 29 cosh x and sinh x, 48 Power series methods of solving differential equations, 491 by Frobenius’s method, 498 by Leibniz–Maclaurin method, 495 Power waveforms, 185 Powers of complex numbers, 261 Precedence, Probability, 545 laws of, 545 paper, 563 Product rule of differentiation, 292 Product-moment formula, 567 Pythagoras, theorem of, 115 Quadrant, 137 Quadratic equations, graphs, 191 inequalities, 16 Quartiles, 543 Quotient rule of differentiation, 293 Radian, 138, 158 Radius, 137 of curvature, 317 of gyration, 383 Radix, 86 Rates of change, 298, 350 Reciprocal matrix, 272, 274 ratios, 116 Rectangular co-ordinates, 136 Rectangular hyperbola, 193, 314 Recurrence formula, 495 relation, 495, 505 Reduction formulae, 424 of exponential laws to linear form, 38 Regression, coefficients, 571 linear, 571 Relation between trigonometric and hyperbolic functions, 173 Relative frequency, 527 velocity, 231 Reliability, 581 Remainder theorem, 10 Resolution of vectors, 227 Right-angled triangles, 118 R.m.s values, 376 Rodrigue’s formula, 511 Roots of complex numbers, 262 Runge–Kutta method, 469 Saddle point, 355, 356 Sample, 527, 577 Sampling distributions, 577 statistics, 581 Scalar multiplication, 268 Scalar product, 237, 238 application of, 241 Scalar quantity, 225 Scatter diagram, 567, 574 Secant, 116 Sech, 41 Second moment of area, 382 Second order differential equations, 475, 481 Sector, 137 area of, 138 Segment, 137 Semicircle, 137 Semi-interquartile range, 543 Separation of variables, 444, 515 Series, binomial, 58, 59 exponential, 29 Fourier, 657 Maclaurin’s, 67 sinh and cosh, 48 Set, 527 Significance testing, 590, 597 tests, 597 Sign test, 614 Simple equations, Simpson’s rule, 217, 437 Simultaneous differential equations by Laplace transforms, 650 725 726 INDEX Simultaneous equations, by Cramers rule, 283 by determinants, 279 by Gaussian elimination, 284 by matrices, 277 Sine, 116 curves, 152 rule, 124 wave, 220 wave production, 151 Sin θ substitution, 401 Sinh, 41 series, 48 Sinh θ substitution, 403 Sinusoidal form, A sin(ωt ± α), 157 Small changes, 311, 352 Solution of any triangle, 124 right-angled triangles, 118 Space diagram, 231 Spectrum of waveform, 698 Standard curves, 191 derivatives, 289 deviation, 541 error of the means, 578 integration, 367 Stationary points, 302, 357 Statistical tables: Chi-square, 609 Mann–Whitney, 622, 623 normal curve, 561 sign test, 614 Student’s t, 587 Wilcoxon signed test, 617 Straight line, 191 Student’s t distribution, 586 Sum to infinity, 54 Successive differentiation, 296 Switching circuits, 94 Symmetry relationships, 695 Tables, statistical: Chi-square, 609 Mann–Whitney, 622, 623 normal curve, 561 sign test, 614 Student’s t, 587 Wilcoxon signed test, 617 Tally diagram, 532, 535 Tangent, 116, 137, 310 Tangential velocity, 244 Tanh, 41 Tan θ substitution, 403 Tan 2θ substitution, 413 Taylor’s series, 460 Testing for a normal distribution, 563 Theorems: binomial, 58, 59 factor, Maclaurin’s, 67 Pappus, 380 parallel axis, 383 perpendicular axis, 384 Pythagoras, 115 remainder, 10 Total differential, 349 Transfer function, 642 Transformations, 194 Transmission equation, 515 Transposition of formulae, Trapezoidal rule, 216, 433, 683 Trial solution, 516 Triangle, area of, 125 Trigonometric ratios, 116 evaluation of, 121 functions, 173, 191 and hyperbolic substitutions, integration, 397, 398, 403 equations, 166, 167 identities, 166 inverse function, 202, 332 waveforms, 148 Trigonometry, 115 practical situations, 128 Truth tables, 94 t = tan(θ/2) substitution, 413 Two-state device, 94 Two-tailed tests, 593 Turning points, 302 Type I and II errors, 590 Ungrouped data, 528 Unit matrix, 271 Unit triad, 237 Universal logic gates, 110 Upper class boundary value, 532 Vector addition, 225 nose-to-tail method, 226 parallelogram method, 226 Vector equation of a line, 245 Vector products, 241 applications of, 244 Vector quantities, 225 Vector subtraction, 229 Velocity and acceleration, 299 Vertical bar chart, 528 Volumes of irregular solids, 218 of solids of revolution, 377 Wallis’s formula, 430 Wave equation, 515, 516 Waveform analyser, 160 Wilcoxon signed-rank test, 616 Work done, 241 Zeros (and poles), 643 ... 2x + 2y i.e dy dy 2 2 =0 dx dx dy = − 2x, dx − 2x 1−x dy = = from which dx 2y − y−1 The value of y when x = is determined from the original equation Hence (2y − 2) Hence (2) 2 + y2 − 2( 2) − 2y... + tan 2 − dθ 2( θ − 2) y = ⎡ 3e2θ sec 2 dy + tan 2 − = √ dθ 2( θ − 2) (θ − 2) x ln 2x with Problem Differentiate y = x e sin x respect to x (x + 1)(2x + 1)3 (x − 3 )2 (x + 2) 4 (x + 1)(2x + 1)3... π π /2 −1 A y = tanϪ1 x π D y = cosϪ1 x π /2 π /2 B +1 x −π /2 y y 3π /2 −1 C +1 x −π /2 −π −π −3π /2 333 x −π /2 −3π /2 (a) (c) (b) y y 3π /2 π 3π /2 y = secϪ1 x y= cosecϪ1 π x π π /2 π /2 −1 +1 −π /2 x −1