Understanding Engineering Mathematics

4.2.4 Manipulation of the exponential function ➤ 119 136➤ As noted earlier, any exponential function, includingex, satisfies the ...

= eA(eA+e−A) (eA+e−A)^2 = eA eA+e−A = e^2 A e^2 A+ 1 (iv)(eA+e−A)^2 −(eA−e−A)^2 =(eA)^2 + 2 eAe−A +(e−A)^2 −((eA)^2 − 2 eAe−A+(e ...

The two results x=elnxandx=lnex expressing the fact that the exponential and the log are inverse functions of each other are ext ...

logxα=αlogx Putx=as,sos=logx,thenxα=(as)α=asα=aαsand soαs= logxα=αlogx The last result holds for any real numberα, positive or ...

y (^0) x (0,1) (1,0) y = ax y^ = log a^ x y^ = x Figure 4.4The exponential and logarithm functions. Note that as observed above, ...

B.If log 2 x=6 then from the change of base formula logax= logbx logba we have log 8 x= log 2 x log 28 = log 2 x 3 = 6 3 = 2 Alt ...

Logs can also be used to simplify graphical representation of certain functions. Thus, given any function of the form y=kxα We c ...

4.3 Reinforcement 4.3.1 y=an,n=an integer ➤➤ 119 120 ➤ A.Plot the values of 3nforn=−2,−1, 0, 1, 2, using Cartesian axes withnon ...

4.3.5 Logarithms to general base ➤➤ 120 130 ➤ A.Findxif (i) 8=log 2 x (ii) 3=log 2 x (iii) 4=lnx (iv) 6=log 3 x (v) 4=log 3 x (v ...

4.3.7 Some applications of logarithms ➤➤ 120 134 ➤ A.Solve the following equations, giving your answers to 3 decimal places. (i) ...

currentIis approximately exponential, i.e I Ioexp ( eV kT ) In the general case obtain an expression forV in terms ofI. 3.The sp ...

4.3.2 The general exponential functionax (i) 1 (ii) 12 (iii) 1 (iv) − 1 x √ x^2 + 1 (v) e−^2 x(x+^1 ) (vi) a^2 (vii) a−^3 4.3.3 ...

B. (i) ln 3+2lnx+lny (ii) 3+2log 2 x+3log 2 y (iii) A−B (iv) x+alogay (v) 3+2log 2 ax+4log 2 ay (vi) 2 lnx+2lny+2lnz C. (i) 9 x ...

5 Geometry of Lines, Triangles and Circles One can hardly get more practical than surveying and designing buildings and other st ...

Motivation You may need the material of this chapter for: the study of structures – from molecular and crystalline to ‘big’ eng ...

C.For (i) and (ii) determine the lettered anglesa,b,c,d,e. Equal arrows denote parallel lines. 60 ° b c a (i) 85 ° a e d b c (ii ...

5.1.5 Similar triangles ➤152 162➤➤ The trianglesABCandDEFare similar. B A C E D F Figure 5.5 IfBC= 5 .0cm,AB= 4 .0cm,AC= 3 .0cma ...

5.1.8 Pythagoras’ theorem ➤154 163➤➤ The largest sides of a right angled triangle have lengths 5 and 6 units. What is the length ...

5.2 Revision 5.2.1 Division of a line in a given ratio ➤ 143 160➤ Points and lines are regarded asundefined primitive concepts(t ...

SoAP=90 cm andPB=60 cm. APB x x − 30 30 5.2.2 Intersecting and parallel lines and angular measurement ➤ 143 160➤ Given two inter ...