Maths for A Level Physics - updated edition

4.3.2 The exponential function, exp( x ) The function exp( x ) is deϐined by: exp( x ) = lim p →0 (1 + xp ) 1 p . This looks very familiar. Using the same process as in 4.3.1 you should be able to show that exp( x ) = 1 + x + x 2 2! + x 3 3! + x 4 4! + … (1) This is the same as e with the addition of the powers of x . How does it relate to e ? Clearly e = exp(1), but what about exp(2), exp(2.5), etc.? How do they relate to e ? We get a big clue if we consider the product exp( a ) × exp( b ). Using equation (1): exp( a ) × exp( b ) = ( 1 + a + a 2 2! + a 3 3! + … )( 1 + b + b 2 2! + b 3 3! + … ) There are a lot of terms to multiply out here (an inϐinite number!) but let’s make a start and gather together all the terms of order 1, all the terms of order 2, etc. exp( a ) × exp( b ) = 1 + ( a + b ) + ( a 2 2! + ab + b 2 2! ) + ( a 3 3! + a 2 b 2! + ab 2 2! + b 3 3! ) + … = 1 + ( a + b ) + 1 2!  ( a 2 + 2 ab + b 2 ) + 1 3!  ( a 3 + 3 a 2 b + 3 ab 2 + b 3 ) + … = 1 + ( a + b ) + ( a + b ) 2 2! + ( a + b ) 3 3! + … = exp( a + b ) So we see that exp(1) = e 1 and exp( a ) × exp( b ) = exp( a + b ), which means that exp( a )= e a . Calculator warning: The EXP button on your calculator does not give the exp( x ) function. The EXP button is short for “×10 x ”, so the key strokes 5 EXP 3 enter the number 5 × 10 3 . The reason is that the EXP here is short for exponent , which is the power (in this case) of 10. 4.3.3 e x – the growth function Figure 4.2 shows graphs of the functions 1.5 x , 2 x , e x and 4 x . Their common characteristics, as with all other functions a x , with a > 1, are: ▪ they all pass through (0, 1) ▪ they all tend to 0 as x → − ∞ ▪ their gradients all increase with x . ▪ their gradients are all proportional to the value of the function. The last bullet point is not obvious. We need to do some work on it. It is proved in a formal way in Chapter 13, but an example will illustrate it. In fact the gradient of the function e x at a point is equal to the value of e x at that point. Because of this, e x [aka exp( x )] is often referred to as the growth function . See Example C. y 8 6 4 2 0 x Ϋ 1 0 1 2 3 y = 4 x y = 2 x y = e x y = 1.5 x Fig 4.2 Data Exercise 4.2 Use a spreadsheet and the function exp( x ) = 1 + x + x 2 2! + x 3 3! + x 4 4! + …, to work out the value of exp(1.5) to 4 significant figures and compare your answer with using the e x button on your calculator. On many calculators, the e x function is accessed using inv ln or shift ln , so the key strokes for e 1.5 are: 1 . 5 inv ln or 1 . 5. shift ln See footnote: ➋ QUICKFIRE 4.11 Use your calculator to determine (a) e 2 , (b) e −2 and (c) √ e . QUICKFIRE 4.12 Write the numbers in QF 4.11 in the style exp( x ). ➋ Successive partial sums of the infinite series are: 1, 2.5, 3.625, 4.1875, 4.3984, 4.4617, 4.4775, 4.4809, 4.4816. These compare with the e 1.5 result of 4.482 (4 s.f.). Thus the answer is correct to 4 s.f. after the 8th order term. Mathematics for Physics 38