The Chemistry Maths Book, Second Edition

50 Chapter 2Algebraic functions Algebraic functions Polynomials are the simplest examples of algebraic functions. More generally ...

2.6 Rational functions 51 (2.28) is defined for all values of xwith the exception of the roots of the polynomial Q(x)in the deno ...

52 Chapter 2Algebraic functions EXAMPLE 2.26Dividex 3 1 − 17 x 2 1 + 116 x 1 − 110 byx 1 − 11. The cubic in this example is (num ...

2.7 Partial fractions 53 To derive this result, write It is required therefore that x 1 + 121 = 1 A(x 1 + 1 4) 1 + 1 B(x 1 − 1 3 ...

54 Chapter 2Algebraic functions EXAMPLE 2.30A repeated linear factor in the denominator and it follows thatA 1 = 13 andB 1 = 1 − ...

2.8 Solution of simultaneous equations 55 or if the same quadratic factor occurs mtimes, (2.35) For example (see Example 2.22c), ...

56 Chapter 2Algebraic functions defines a second function y 1 = 1 g(x). The two equations have simultaneoussolutions for those v ...

2.8 Solution of simultaneous equations 57 EXAMPLE 2.33Solve (1) x 1 + 1 y 1 = 13 (2) 2x 1 + 12 y 1 = 16 In this case, doubling e ...

58 Chapter 2Algebraic functions This has rootsx 1 1 = 1 − 4 andx 2 1 = 14 , with corresponding value of y,y 1 1 = 14 andy 2 1 = ...

Simplify if possible: Section 2.4 Find xas a function of y: 23.y 1 = 1 x 1 − 12 24. 25. 26. Fo ...

60 Chapter 2Algebraic functions 42.Explain howKandΛ 0 m in Kohlrausch’s law (Exercise 33), can be obtained graphically from the ...

2.9 Exercises 61 Section 2.8 Solve the simultaneous equations: 66.x 1 + 1 y 1 = 1 3, x 1 − 1 y 1 = 11 67. 3 x 1 − 12 y 1 = 1 1, ...

3 Transcendental functions 3.1 Concepts The mathematical description of physical phenomena often involves functions other than t ...

3.2 Trigonometric functions 63 triangles, 1 and is important in structural and architectural design, astronomy, and navigation. ...

64 Chapter 3Transcendental functions Several derived functions are also defined, the most important being the secant(sec), cosec ...

3.2 Trigonometric functions 65 EXAMPLE 3.2For the triangle in Figure 3.3, 3 2 1 + 14 2 1 = 15 2 and 0 Exercises 2 Units of angle ...

66 Chapter 3Transcendental functions (angles are often quoted as multiples of π). The length of an arc of a circle iss 1 = 1 rθ ...

3.2 Trigonometric functions 67 The trigonometric functions are defined for all points on the circle as (3.10) In the first quadr ...

68 Chapter 3Transcendental functions Special values The values for the angles on the boundaries of the four quadrants, and for a ...

3.2 Trigonometric functions 69 Negative angles Each point on the circle can be reached by either anti-clockwise rotation or by c ...