Understanding Engineering Mathematics

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 Cartesian

axes withnon the horizontal axis and 2non the vertical axis.

(ii) Repeat (i) with

( 1

2

)n

= 2 −nand compare the results.

(iii) Sketch the graph of (a)y= 2 x,(b)y= 2 −x,(c)y= 3 x,(d)y= 3 −xon rectangular

x-,y-axes.

4.1.2 The general exponential functionax ➤121 136➤➤

A.Use the laws of indices to show that iff(x)=ax,whereais a positive constant, then

f(x)·f(y)=f(x+y)

What isf(x−y)?

B.Simplify the following (a>0)

(i) axa−x (ii)

a^3 xax

a^2 x

(iii)

(ax)^3 a−^2 x

(

a^4

)x (iv)

ax

2

a−^2 x

a(x−^1 )^2

(v)

2 x 16 −^3 x

42 x 8 −x+^1

4.1.3 The natural exponential functionex ➤124 136➤➤

A.Define the base of natural logarithms,e. Write down the value ofeto 3 decimal places.

Can you write down the exact value ofe?

B. Given thatex=2, evaluate

(i) e^2 x (ii) e−x

(iii) e^3 x (iv) e^4 x− 4 e^2 x

C.Plot the graphs ofy=ex,e−x,e^2 x,e−^3 xon the same axes.

4.1.4 Manipulation of the exponential function ➤129 136➤➤

A.Express each of the following as a single exponential

(i) eAeB (ii) eA/e−B (iii) e^2 B(e^3 B)^2

(iv)

eAe^2 Be−C

e^2 AeBeC

(v) e−Be−CeB (vi) (eA)^3 e−^2 A