The Chemistry Maths Book, Second Edition

90 Chapter 3Transcendental functions Section 3.3 12.Find the principal values of (i) (ii)sin − 1 (1) (iii) (iv)cos − 1 (−1) 13.T ...

3.10 Exercises 91 28.Find the cartesian coordinates of the points whose polar coordinates are (i)r 1 = 1 3, θ 1 = 12 π 2 3, (ii) ...

92 Chapter 3Transcendental functions wheref 1 = 1 γpis the fugacity and γis the fugacity coefficient. Express pas an explicit fu ...

4 Differentiation 4.1 Concepts In the physical sciences we are interested in the value of a physical quantity and how it is rela ...

94 Chapter 4Differentiation If the pressurepof the gas is changed by an amount∆pat constantTandn,the change in the volume is Fig ...

4.2 The process of differentiation 95 More generally, if the variable changes by ∆x, from p 1 = 1 xto q 1 = 1 x 1 + 1 ∆x, the co ...

96 Chapter 4Differentiation As Q moves through Q′towards P, the gradient of the line PQ approaches the gradient of the tangent a ...

4.3 Continuity 97 An alternative symbol for the derivative of the functionf(x)isf′(x): (4.9) Another symbol is Df, by which is i ...

98 Chapter 4Differentiation Point a.The function has a finite discontinuityatx 1 = 1 a. For example, the function is discontinuo ...

4.4 Limits 99 This is an example of a removable discontinuity; we have and the discontinuity can be removed by redefining the fu ...

100 Chapter 4Differentiation EXAMPLE 4.5 From the properties of the logarithm, 0 Exercises 16, 17 4.5 Differentiation from first ...

4.5 Differentiation from first principles 101 EXAMPLE 4.6Find from first principles for. (1) (2) (3) 0 Exercise 20 EXAMPLE 4.7Di ...

102 Chapter 4Differentiation 3 Many of the rules of differentiation appeared in Leibniz’s 1684 paper on the differential calculu ...

4.6 Differentiation by rule 103 EXAMPLES 4.9Differentiating powers (i) y 1 = 1 x 5 (ii)f(x) 1 = 1 x − 122 (iii)f(x) 1 = 1 x 0.3 ...

104 Chapter 4Differentiation Linear combination of functions A linear combination of the functions u, v, and wof xhas the form y ...

4.6 Differentiation by rule 105 EXAMPLE 4.12Product rule The function y 1 = 1 (2x 1 + 13 x 2 ) 1 sin 1 x is easily differentiate ...

106 Chapter 4Differentiation can be differentiated by first expanding the cube and then differentiating term by term: y 1 = 1 (2 ...

4.6 Differentiation by rule 107 EXAMPLES 4.15The chain rule (i) y 1 = 1 sin 12 x 1 = 1 sin 1 u, whereu 1 = 12 x (ii)y 1 = 1 cos( ...

108 Chapter 4Differentiation Example 4.15(i) demonstrates the important special case of u(x) 1 = 1 ax, for which . Then For exam ...

4.6 Differentiation by rule 109 EXAMPLE 4.17Inverse trig functions If thenx 1 = 1 a 1 sin 1 yand If yhas its principal value the ...