Understanding Engineering Mathematics

3.3.6 Inequalities ➤➤ 89 97 ➤ Find the ranges of values ofxfor which the following are satisfied: (i) 2x− 3 > 2 (ii) 2 x x− 3 ...

3.3.10 Infinite series ➤➤ 89 105 ➤ A.Identify which are geometric sequences, and sum them to infinity. (i) 1, 2 , 3 , 4 ,... (ii ...

3.The following summation is used in the multiplication ofmatrices(see Chapter 13): ∑n k= 1 aikbkj whereaik,bkj, are elements of ...

Answers to reinforcement exercises 3.3.1 Definition of a function A. (i) (a) − 2 (b) 0 (c) 6 (d) −^23 (ii) (a) 2 (b) −1(c)26(d)− ...

(ii) y − 6 − 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 6 x 3 2 1 − 1 − 2 − 3 − 4 y =^3 x (iii) y − 4 − 3 − 2 − 10 1 2 3 4 x 10 15 20 5 y = 2 ...

B. y − 3 − 2 − 1 0 1 2 3 4 x 15 25 20 10 5 − 5 − 10 y = 2 x^2 − 3 x − 2 > 0 for x > 2 and x < −^12 3.3.3 Formulae (i) O ...

3.3.4 Odd and even functions (i) odd (ii) even (iii) odd (iv) neither (v) even (vi) even (vii) odd (viii) neither 3.3.5 Composit ...

3.3.9 Finite series (i) a=1, r= 1 2 S 6 = 2 ( 1 − 1 26 ) = 63 32 (ii) a= 0 .1, r= 0. 1 S 6 = 1 9 ( 1 − 1 106 ) = 0. 1111111 (iii ...

(iii) 1+(− 2 )( 4 x)+ (− 2 )(− 3 )( 4 x)^2 2! + (− 2 )(− 3 )(− 4 )( 4 x)^3 3! = 1 − 8 x+ 48 x^2 − 256 x^3 (iv) 1+ ( 1 2 ) (−x)+ ...

4 Exponential and Logarithm Functions The exponential function is one of the most important in engineering. It describes behavio ...

4.1 Review 4.1.1 y=an,n=an integer ➤120 136➤➤ (i) Plot the values of 2nforn=−4,−3,−2,−1, 0, 1, 2, 3, 4 using rectangular Cartesi ...

B.Express in simplest form (i) e^2 A−e^2 B eA+eB (ii) (eA−e−A)(eA+e−A) (iii) e^2 A+ 1 e^2 A+e−^2 A+ 2 (iv) (eA+e−A)^2 −(eA−e−A)^ ...

behaviour of the exponential function by looking at the behaviour of the power function for different values of the index, as sh ...

m,nare ‘constant’ and to think in terms of apower functionxn, etc. where the base,x, is variable andnis given. However, it is ju ...

We need to say something about the definition ofaxfor different values ofx, building on the work in Section 1.2.7. Ifx=nis a pos ...

(ii) a^3 xax a^2 x =a^3 x+x−^2 x=a^2 x (iii) (ax)^3 a−^2 x (a^4 )x = a^3 xa−^2 x a^4 x =a−^3 x (iv) ax 2 a−^2 x a(x−^1 )^2 =ax ( ...

One rigorous method of defining the exponential functionexis by means of alimit.This is quite an advanced concept that we only a ...

This is all very well – but how do we actuallycalculatethis? Asngets larger and larger it gets more and more difficult. In parti ...

All we need here is to note that such things as 1/n, 2 /n,....‘tend to’ zero asn‘tends to’ infinity, i.e. gets infinitely large. ...

Solution to review question 4.1.3 A. We m a y d e fi n eeby the limit e= lim n→∞ ( 1 + 1 n )n but it is more easily evaluated fr ...