Further algebra
7.5.1 Approximations when multiplying or dividing quantities
Suppose we have (1 + a)n (1 + b)m where both a and b are small compared with 1. We can then use the binomial theorem to make the approximation:
(1 + a)n (1 + b)m (1 + an)(1 + bm)
Since (1 + an)(1 + bm) = 1 + an + bm + abnm and we can neglect abnm as being small because it involves the products of two numbers which are smaller than 1, we can further approximate this to:
(1 + a)n(1 + b)m (1 + an)(1 + bm) 1 + an + bm
Suppose we have (1 + a)n(1 + b)m where both a and b are small compared with 1. We can then use the binomial theorem to make the approximation:
(1 + a)n(l + b)-m (1 + an)(1 - bm) 1 + an - bm Example
Determine the approximate percentage change in the volume of a cylinder when its diameter d increases by 2% and its length h decreases by 3%.
The volume V of a cylinder is ¼πd2h. The new value of the diameter will be d + 2% of d = d + 2d/100 = d(1 + 0.02). The new value of h will be h - 3% of h = h - 3h/100 = h(1 - 0.03). The new volume is thus:
new volume = ¼π[d(1 + 0.02)2h(1 - 0.03)]
= ¼πd2h(1 + 0.02)2)(1 - 0.03)1
¼πd2h(1 + 2 x 0.02)(1 - 0.03)
¼πd2h(1 + 0.04 - 0.03)
¼πd2h x1.01
The volume thus increases by a factor of about 1.01. The percentage change in volume is thus about 1%.
Example
Simplify the expression (1 - 2x)2/(l + x)4 if powers of x greater than the first can be neglected.
We can write the expression as (1 - 2x)2(1 + x)-4 and use the binomial expression to expand each binomial term, neglecting powers of x greater than the first. Thus:
(1 - 2x)2(1 + x)-4 [1 + 2(-2x)][l + (-4)x]
( 1 - 4 x ) ( 1 - 4 x ) 1 - 8 x
Revision
16 Kinetic energy is ½mv2. Determine the percentage change in kinetic energy when the mass m increases by 2% and the velocity is reduced by 4%.
17 Simplify the expression (1 + x)1/2/(l - 2x)2 if powers of x greater than the first can be neglected.
Problems 1 State the rules determining the terms in the following sequences: (a) 2, 3, 4, 5, ..., (b) -3, 6, -12, 24, ..., (c) 5, 6, 7, 8, ..., (d) 1, 4, 16, 64, ..., (e) -1, 1, -1, 1, ...
2 Find the 10th term of the following arithmetic sequences: (a) 2, 5, 8, ..., (b) l, 6, 11, ..., (c) 5, 9, 13, ...
3 State the first five terms of the arithmetic sequence when: (a) the first term is 2 and the common difference 3, (b) the first term is - 4 and the common difference is 2.
4 Find the 6th term of the following geometric sequences: (a) 6, 18, 54, ..., (b) l, 3, 9, ..., (c) l, - ½ , ¼, ...
5 State the first five terms of the geometric sequence when: (a) the first term is 24 and the common ratio is ½, (b) the first term is 10 and the common ratio is 0.1.
6 A machine is bought for £5000. If its value depreciates at the rate of
£300 per year, what is its value after 4 years?
7 A ball dropped onto level ground rebounds each time to 0.7 of the height from which it fell. If it starts from a height of 200 cm, list the heights of the first four bounces.
8 A machine tool is to have five rotational speeds with the slowest being 20 rev/min and the fastest 200 rev/min. If the speeds are to be in a geometrical sequence, determine the common ratio.
9 Determine the sum of the first 20 terms of the following arithmetic series: (a) 3 + 6 + 9 + 12 + ..., (b) 2 + 5 + 8 + 11 + ..., (c) 1.2 + 2.2 + 3.2 + 4.2 + ...
10 Determine the number of terms required in the series 11 + 15 + 19 + 24 + ... to give a total of 341.
11 An arithmetic series has a first term of 27 and a common difference of - 2 . How many terms will be needed to give a sum of 192?
12 An arithmetic series has a first term of 3 and a common difference of 4.
How many terms will be needed to give a sum of 2775?
13 Determine the sums of the following geometric series: (a) 128 + 64 + 32 + ... for the first 8 terms, (b) 2 + 8 + 32 + ... for the first 4 terms, (c) 8.00 + 8.15 + 8.30 + ... for the first 30 terms.
14 Determine the sums to infinity of the following series: (a) 9 - 6 + 4 - ..., (b) 12 + 6 + 3 + ..., (c) 5 + 0.5 + 0.05 + ...
15 A ball falls vertically onto a horizontal surface and, on each bounce, rebounds to 4/5 of the height from which it fell. If the ball starts at a height of 3.5 m, determine the total distance it travels before coming to rest.
16 £100 is invested and earns compound interest of 2% per year. What will be the amount in the account after 8 years?
17 Determine, using the binomial series, the expansion of the following as series and state where there are any restrictions on the validity of the result:
(a) (3 + x)5, (b) (2 + x)7, (c) (1 + 2x)4 (d) (2 - x)6, (e) (y - 2x)3, (f) (x + 2y)4, (g) (1 + x)-1/2, (h) (2 + x)1/2, (i) (1 - 2x)-1/2, (j) (1 + 3x)1/3, (k) (1 + x)-2, (l) (3 - 2x)-1.
18 The periodic time T of a simple pendulum is related to its length L and the acceleration due to gravity g by T = 2π√(L/g). What will be the percentage change in the periodic time if the length increases by 2%?
19 The maximum bending stress σ at the surface of a cantilever when subject to bending is related to its length L, breadth b, depth d and the load F applied at the free end by σ = 6FL/(bd2). Determine the
percentage change in σ if (a) F increases by 1% and the dimensions remain unchanged, (b) b is reduced by 1% and d is reduced by 1%, the other terms remaining unchanged.
20 Evaluate, using the binomial theorem, the following to five decimal places (a) 1.0024, (b) 1/0.98, (c) 0.987.
21 Simplify the following expressions if powers of x greater than the first can be neglected:
(a) (1 + x)3/(1 - x)1/2, (b) (1 + 2x)1/2/(1 - x), (c) √[(1 - x)/(1 + x)].
8.1 Introduction There are many situations in engineering where we are concerned with the rate at which some quantity grows or the rate at which it decays. Examples are:
1 The variation with time of the temperature of a cooling object.
2 The variation with time of the charge on a capacitor when it is being charged and when it is being discharged.
3 The variation with time of the current in a circuit containing inductance when the current is first switched on and then when it is switched off.
4 The decay with time of the radioactivity of a radioactive isotope.
5 The growth of bacteria or indeed any uncontrolled population.
This chapter is about equations we can use to describe such growth or decay of quantities.
8.2 Exponentials We can describe growth and decay processes by an equation of the form:
y = at
where a is some constant, and y the value of the quantity at a time t. Such equations are called exponentials with a being called the base. An exponential is not a polynomial. The powers of a polynomial are constants, e.g. the 2 in x2, whereas the power of an exponential is the variable. An exponential could be 2t, 3t, 4t etc.
8.2.1 Growth
To illustrate the use of y = at for a growing quantity, consider how the value of y changes with time when we have a = 2 and so the equation:
y = 2t
At t = 0 we have y = 2° = 1, at t = 1 we have y = 21 = 2, at t = 2 we have y
= 22 = 4, at t = 3 we have y = 23 = 8, and so on. Thus we have the data:
y = 2t 1 2 4 8 16 32
t 0 1 2 3 4 5
This data describes a quantity that increases with time, starting at t = 0 with the value 1 and rapidly increasing as t increases. Figure 8.1 shows this data
y 30
20
10
0 1 2 3 4 5
Figure 8.1 y = 2t
t
plotted as a graph. Note that the graph describes a function which doubles for every successive interval of t = 1.
8.2.2 Decay
For a quantity that decreases with time, consider how the value of y changes with time when we have a = 2 and the equation
y = 2-t
At t = 0 we have y = 2° = 1, at t = 1 we have y = 2-1 = 0.5, at t = 2 we have y = 2-2 = 0.25, at y = 3 we have y = 2-3 = 0.125, and so on. Thus we have the data:
y = 2-t 1 0.5 0.25 0.125 0.0625 0.03125
t 0 1 2 3 4 5
This data describes a quantity that decreases with time, starting with a value 1 at t = 0 and rapidly decreasing in size as t increases. Figure 8.2 shows this data plotted as a graph. Note that the graph describes a function which halves for every successive interval of t = 1.