Module 7 - Functions (self Study)

SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING Self Study Course

MODULE 7

FUNCTIONS

Module Topics 1. Functions and inverse functions 2. Trigonometric and inverse trigonometric functions 3. Exponential and logarithmic functions 4. Hyperbolic and inverse hyperbolic functions 5. Differentiation of inverse trigonometric and hyperbolic functions

A:

Work Scheme based on JAMES (THIRD EDITION)

1. This module concerns parts of a long chapter in J. on functions. Read section 2.1, the introduction to chapter 2, on p.58. 2. Study section 2.2.1 to the bottom of p.59 and then work through Example 2.1. The most important property to note for functional relationships y = f (x) is that for each value of the independent variable x there is one, but only one, value of the dependent variable y. Complete this section by studying the paragraph near the top of p.61. 3. Turn on to p.64 and study carefully section 2.2.3 on inverse functions, including Examples 2.4 to 2.6. Figure 2.10 shows that the inverse function y = f −1 (x), if it exists, is found by reflecting the graph y = f (x) in the line y = x. The discussion in Example 2.6 on finding the inverse function is particularly important. 4. Study section 2.2.4 on composite functions stopping just under Figure 2.13. 5. Study section 2.2.5, including Examples 2.9 and 2.10. Knowledge of odd, even and periodic functions is particularly useful in discussing Fourier series, which most of you will study in more detail either this year or next year. ***Do Exercises 6, 11(a),(c) on p.75*** 6. Omit sections 2.3, 2.4 and 2.5 on linear, polynomial and rational functions and move on to p.111. Read quickly through the text in sections 2.6.1 to 2.6.4, much of which has been covered in previous modules. Study carefully section 2.6.5 on defining inverse trigonometric functions. Work through Examples 2.45, and then the much harder Example 2.46. ***Try Exercises 54(a),(c) (both hard) on p.129*** 7. Read the introduction to section 2.7 on p.130 and then study section 2.7.1 on exponential functions. Properties 2.33 are consistent with the indicial laws revised in module 1. Note the comments below properties 2.33 on the use of the exp notation in certain situations. –1–

8. Study section 2.7.2 on the logarithmic function and study Example 2.50. The graph of ln x is on the Formula sheet and in the Data Book, and it is easily seen that y = ln x is the inverse function corresponding to y = ex . As stated in J. it is customary to use the notation log to denote a logarithm to base 10 and ln to denote the corresponding function to base e. Properties 2.35 and 2.36(a),(b),(c) are especially useful. 9. Study section 2.7.3 on hyperbolic functions. The hyperbolic cosine and hyperbolic sine are defined on p.134 and these definitions appear on the Formula sheet and in the Data Book. The definitions of the other hyperbolic functions, tanh x, sech x, cosech x and coth x, are defined from cosh x and sinh x in exactly the same way as the trigonometric functions tan x, sec x, cosec x and cot x are defined in terms of cos x and sin x. Graphs of cosh x, sinh x and tanh x are shown on the Formula sheet and in the Data Book. The definitions of cosh x and sinh x lead to equation 2.38, which is an important relation connecting these two hyperbolic functions. Equation 2.38 also appears on the Formula sheet (but NOT in the Data Book). In a similar way the relations 2.39 can be determined, although these need not be remembered. Study Example 2.51. Osborn’s rule on p.136 indicates how a trigonometric identity is related to the corresponding hyperbolic one. In general we will not need to use this result although it does indicate that the hyperbolic identity cosh2 x − sinh2 x = 1 is related to the trigonometric expression cos2 x + sin2 x = 1. Study Examples 2.52 and 2.53. 10. Study section 2.7.4 on inverse hyperbolic functions. Graphs of these functions are shown in Figure 2.78. Note, in particular, the restriction on the range of cosh −1 x which is necessary to obtain a function. You are not expected to memorise equations 2.40 to 2.42. Work through Example 2.54. ***Do Exercises 59, 60, 61, 64(a),(e) on p.140*** 11. To complete this module you need to consider the differentiation of the inverse functions introduced above. Turn to p.510 and study the part of sectionp8.3.7 below equation 8.17. Study Example 8.16 parts (f ) (the denominator in the solution should read (1 − 36x2 ) ), (g) (quotient rule in the solution in J. should be replaced by product rule) and (h). Then study Example 8.17(b). 12. Turn to p.516 and study how the differentials of the hyperbolic functions can be derived. Only the differentials of sinh x and cosh x are listed on the Formula sheet - you should be able to derive the other results given by equations 8.23(c)–(f ) if required. Note that equations 8.24(a),(b) for the differentials of the inverse hyperbolic functions sinh−1 x and cosh−1 x are quoted on the Formula sheet and in the Data Book. Study Example 8.19. ***Do Exercises 26(a),(c),(f ), 29(c),(d), 30(a),(b) on p.518***

B:

Work Scheme based on STROUD (FIFTH EDITION)

S. contains most of the algebraic material in this module but omits some of the initial definitions and the graphical interpretations. Hence, to start this module you are advised to work through the items 1-5, 7 and 8 in the above work scheme based on J. Turn to S. p.475 and work through frames 1-38 of Programme 3 on hyperbolic functions. Note that the references to series in the early frames can be omitted, they will be discussed later in the year, and the results in logarithmic form in frame 31 need not be memorised. In order that cosh−1 x is a function each value of x must lead to only 1 value of cosh−1 x. Thus the plus sign must be taken as the answer in frame 30. –2–

The differentiation of hyperbolic functions is considered in Programme 7 (in the frames omitted when you were studying module 4). Work through frame 2 on p.601, question 5 in frame 13 on p.605 and frames 17 and 18 on pp.607 and 608. Finally, Programme 9 starting on p.643 contains the material on differentiating inverse trigonometric and hyperbolic functions. Work through frames 1-17 of this programme. The crucial point not emphasised in S. is that for a function y = f (x) to have an inverse each value of y must correspond to exactly one value of x.

Specimen Test 7 x2 + 1 for an appropriate domain. The function f (x) is defined by the equation f (x) = √ x2 − 1

1.

(i) Write down the domain of the function. (ii) State whether the function over its domain is even or odd or neither even nor odd. 2. (i) Draw the graph of cos−1 x, showing carefully the domain and range of the function.
(ii) Find the value of cos−1 − 1/2 , and illustrate it on your graph.

3.

  1 (x > 3), use the elementary properties of ln and exp y = exp ln x − ln(x − 3) 2 to simplify the right-hand expression as much as you can.

4.

Given that sinh x = −5/12 find

Given that

(i)

cosh x,

(ii)

tanh x,

(iii)

sech x.

5. (i) Define sinh y in terms of exponentials. (ii) If x = sinh y use the expression from (i) to obtain a quadratic equation for ey involving x. √   (iii) Solve this quadratic equation for ey and hence prove that sinh−1 x = ln x + x2 + 1 . Differentiate the following functions with respect to x:

6. (i)

tan−1 (4x),

(ii)

x3 sinh x. –3–