Maths for A Level Physics - updated edition

In Data Exercise 4.1, you should have found the values given in Table 4.1.Most of the values of (1 + p ) 1 p in the table were found using a calculator, and you should check them ! ➊ The graph of (1 + p ) 1 p against p is Figure 4.1. It appears to cross the vertical axis ( p = 0) at just over 2.7. The exact value is the important ‘natural number’, known as e . Why didn’t we evaluate e directly, by including p = 0 in the table above? You may have found out when doing Data Exercise 4.1! Instead we have to sneak up on p = 0. Table 4.2 shows the results when we use smaller and smaller positive values for p . Table 4.2 p 0.001 0.0001 0.00001 0.000001 (1 + p ) 1 p 2.71692 2.71814 2.71827 2.71828 Note the smaller and smaller changes in (1 + p ) 1 p as we decrease p . We say that is homing in on a limiting value or limit , which is the natural number e . We write: lim p →0 (1 + p ) 1 p = e Like π , the number e is irrational: it can’t be expressed as a ratio of two whole numbers. Its value is 2.718 to four signiϐicant ϐigures, but more will be given by your calculator, using the function e  x (with x = 1). Another way of calculating e is to use the series: 1 +1 + 1 2! + 1 3! + 1 4! + … It converges on, i.e. it homes in on, e . Try it. 7 terms gives 2.718. But how do we know that this series will converge to e ? We’ll use the binomial expansion from Chapter 3. (1 + x ) n = 1 + nx + n ( n – 1) 2!  x 2 + …, which converges as long as | x | < 1. Putting x = p and n = 1 p : (1 + p ) 1 p = 1 + 1 p p + 1 p ( 1 p – 1 ) 2!  p 2 + 1 p ( 1 p – 1 )( 1 p – 2 ) 3! p 3 + …. . The ϐirst two terms are 1 + 1. The 3rd term is, multiplying it out: 1 – p 2! , which tends to 1 2! as p →0. Similarly, the 4th term tends to 1 3! , etc. So we can write e = 1 + 1 + 1 2! + 1 3! + 1 4! + … . We’ve made a lot of fuss about e , because you will meet it again – often. ➊ Some of the values in Table 4.1 don’t need a calculator, e.g. when p = 1 2 , a bit of manipulation gives y = 4. Data Exercise 4.1 Use your calculator or a spreadsheet to plot a graph of y against p for the function y = (1 + p ) 1 p for − 1 2 ≤ p ≤ 1, in other words, for p between − 1 2 and +1. Suggested initial values of p are: − 1 2 , − 1 4 , 1 4 , 1 2 and 1. Don’t try p = 0; well you can try! y 4 3 2 p Ϋ ¹ Μ Ϋ ¹ Ξ ¹ Ξ ¹ Μ ³ Ξ 1 Fig 4.1 Table 4.1 p (1 + p ) 1 p −0.50 4.00 −0.25 3.16 0.25 2.44 0.50 2.25 1.00 2.00 QUICKFIRE 4.10 Use a calculator to find (a) (1 + 0.1) 10 and (b) (1 + 0.01) 100 . To find e using your calculator use the e x button [on the author’s calculator this is accessed using SHIFT + ln] with x = 1. 4 Indices and Logarithms 37