Higher Engineering Mathematics

22 NUMBER AND ALGEBRA Hence 5 x^2 − 2 x− 19 (x+3)(x−1)^2 ≡ 2 (x+3) + 3 (x−1) − 4 (x−1)^2 Problem 7. Resolve 3 x^2 + 16 x+ 15 (x+ ...

PARTIAL FRACTIONS 23 A Identity (1) may be expanded as: 7 x^2 + 5 x+ 13 ≡Ax^2 +Ax+Bx+B+Cx^2 + 2 C Equating the coefficients ofx^ ...

Number and Algebra 4 Logarithms and exponential functions 4.1 Introduction to logarithms With the use of calculators firmly esta ...

LOGARITHMS AND EXPONENTIAL FUNCTIONS 25 A (b) Letx=log 10 10 then 10x=10 from the defin- ition of a logarithm, i.e. 10x= 101 , f ...

26 NUMBER AND ALGEBRA log (x−1)+log (x+1)=log (x−1)(x+1), from the first law of logarithms =log (x^2 −1) 2 log (x+2)=log (x+2)^2 ...

LOGARITHMS AND EXPONENTIAL FUNCTIONS 27 A Problem 8. Solve the equation 2x=3, correct to 4 significant figures. Taking logarithm ...

28 NUMBER AND ALGEBRA Figure 4.2 In general, with a logarithm to any basea, it is noted that: (i)loga 1 = 0 Let loga=x, thenax=1 ...

LOGARITHMS AND EXPONENTIAL FUNCTIONS 29 A (b) 5 ( e^0.^25 −e−^0.^25 e^0.^25 +e−^0.^25 ) = 5 ( 1. 28402541 ...− 0. 77880078 ... 1 ...

30 NUMBER AND ALGEBRA for ex. Thus e^0.^05 = 1 + 0. 05 + (0.05)^2 2! + (0.05)^3 3! + (0.05)^4 4! + (0.05)^5 5! +··· = 1 + 0. 05 ...

LOGARITHMS AND EXPONENTIAL FUNCTIONS 31 A Now try the following exercise. Exercise 19 Further problems on the power series for e ...

32 NUMBER AND ALGEBRA From the graph,whenx=2.2,y=3.87andwhen y=1.6,x=−0.74. Problem 17. Plot a graph ofy=^13 e−^2 xover the rang ...

LOGARITHMS AND EXPONENTIAL FUNCTIONS 33 A In a chemical reaction the amount of start- ing materialCcm^3 left aftertminutes is g ...

34 NUMBER AND ALGEBRA (a) ln e^2.^5 lg 10^0.^5 = 2. 5 0. 5 = 5 (b) 4e^2.^23 lg 2. 23 ln 2. 23 = 4(9. 29986607 ...)(0. 34830486 . ...

LOGARITHMS AND EXPONENTIAL FUNCTIONS 35 A The work done in an isothermal expansion of a gas from pressurep 1 top 2 is given by: ...

36 NUMBER AND ALGEBRA Problem 25. In an experiment involving Newton’s law of cooling, the temperatureθ(◦C) is given by θ=θ 0 e−k ...

LOGARITHMS AND EXPONENTIAL FUNCTIONS 37 A (a) Transposing the formula to makeθ 1 the subject gives: θ 1 = θ 2 (1−e −t T) = 50 1 ...

38 NUMBER AND ALGEBRA 4.10 Reduction of exponential laws to linear form Frequently, the relationship between two variables, sayx ...

LOGARITHMS AND EXPONENTIAL FUNCTIONS 39 A Gradient of straight line, k= AB BC = ln 100−ln 10 3. 12 −(− 1 .08) = 2. 3026 4. 20 = ...

40 NUMBER AND ALGEBRA When the voltage is 30.0 volts, 30. 0 =2090 e −t (^0) , hence e −t (^0) = 0 2090 and e t (^0) = 20 ...

A Number and Algebra 5 Hyperbolic functions 5.1 Introduction to hyperbolic functions Functions which are associated with the geo ...