Module 3 - Differentiation 1 (self Study)

SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING Self Study Course

MODULE 3

DIFFERENTIATION I

Module Topics 1. Basic rules of differentiation 2. Differentiation of standard functions 3. Newton-Raphson procedure for finding roots 4. Partial differentiation The previous two modules were self-contained but this module and most of the remaining ones are written around a book. The main text is Modern Engineering Mathematics by James, which was chosen by a team of both mathematicians and engineers as the best for most Part 1 students. As mentioned in Module 0, some of you will find Engineering Mathematics by Stroud to be easier to work through. The latter is a programmed learning text which is found particularly useful by those with a less mathematical background. Although it covers a little less of the syllabus than James it contains sufficient mathematics for your needs. Work schemes based on the above books are included in each module. You need to obtain a copy of one of the books, since it is essential to have access to it when working through the module. There are a few copies of each in the short loan section of the library if you are working through the module on site (and you have left your heavy copy at home!), and in the Workshop cupboard. The material in this module will be mostly revision, but the work on partial derivatives is likely to be new to most of you. Calculus is extremely useful in many branches of engineering and so it is important that you are very proficient at differentiation and integration. Do work through the module, therefore, and attempt the Exercises and Specimen Test, the solutions to which start on p.5. Note that the solutions are written in an expanded form which includes the intermediate variables. The better you are at differentiation the more likely it is you will omit many of these intermediate steps.

A:

Work Scheme based on JAMES (THIRD EDITION)

1. Read the introduction to chapter 8 on p.479. Note that the major aim of the mathematics units this year is to develop your understanding of a number of useful mathematical methods. Symbolic algebra packages are of great assistance in solving complicated problems but they will not form part of your first year mathematics course. 2. Read section 8.2.1 on rates of change. Remind yourself of the definition of a derivative in section 8.2.2 and then look through Example 8.1. Note, however, that deriving results from first principles is much less important than being able to use the standard results. Read section 8.2.3 and work through Example 8.2. 3. Study section 8.2.4. Roughly speaking a function is differentiable provided there are no sharp changes in the value of the function or its derivative. –1–

4. Read section 8.2.5 and then look through section 8.2.6. The latter shows some of the many applications of calculus in modelling engineering problems. In the mathematics this year we concentrate on the methods, the applications you will see in your engineering units. 5. Read the beginning of section 8.2.7 on p.492 and the top of page p.493. Finding maxima and minima will be discussed in more detail in module 7. ***Do Exercise 1(c) on p.494*** 6. Read the introduction to section 8.3 on p.496 and then study section 8.3.1. Rules 1-5 on pp.496 and 497 are extremely important and must be remembered, although the special case of rule 5 for a linear function is not normally memorised. Rule 6, which applies to inverse functions (to be discussed in detail in dy dx = 1. The verifications of the six rules, on pp.497-499, are not required. module 7), can be rewritten dx dy 7. Read section 8.3.2 and, of course, you must be absolutely sure of equation 8.10. Study Example 8.9. 8. Study the introduction to section 8.3.3 and Examples 8.10 and 8.11. 9. Study section 8.3.4 on differentiating rational functions and then work through all three parts of Example 8.13. 10. The chain rule, introduced in section 8.3.5, is very important and must be studied carefully. Study Examples 8.14 and 8.15. Intermediate steps can often be omitted when you are familiar with the process, but you should always leave in sufficient working to enable you to get the correct answer! ***Do Exercises 17(d),(h), 18(e),(f ), 22(a),(e), 23(c),(e),(h) on p.507*** 11. The basic rules of differentiation have been illustrated so far in this module using polynomial functions. In the next few sections the differentials of trigonometric, exponential and logarithmic functions are considered. Functions will be considered in more detail in module 7 but before studying section 8.3.7 you may wish to look at section 2.6.2 on circular functions. Study the first part of section 8.3.7. The differentials of sin x and cos x are derived from first principles, and the corresponding results for tan x, cot x, sec x and cosec x are then found using the rule for differentiating a quotient. Stop at equation 8.17, since the inverse trigonometric functions are not investigated in this module. The differentials for tan x, cot x, sec x and cosec x can be found on the Formula sheet, but it is often useful to have them at your fingertips and lecturers of other units on your course might expect you to know them. The differentials for sin x and cos x are so fundamental that they must be memorised, with the minus sign in the correct place. If you are in a department which uses the Data Book you must get familiar with what results are stated in the Book, where the results appear and the notation which is used. In a number of places the notation in James (and these modules) and the Data Book are different. Study Examples 8.16(a)-(e). 12. Study section 8.3.8 on p.513 and then study Example 8.17, part (a) only. 13. Turn back to p.130 in chapter 2 and look briefly at the introduction to section 2.7 on exponential and logarithmic functions, and then read sections 2.7.1 (omitting Example 2.49) and 2.7.2. Moving on to section 8.3.9 on p.514 the important results for the differentials of ex and ln x are stated by equations 8.21 and 8.22 and these should be known (they do not appear on the Formula sheet). Study Example 8.18. –2–

***Do Exercises 25(c),(f ), 27(b),(e), 28(b),(c) on p.518*** 14. Higher derivatives are mentioned for the first time on p.523. Study the first paragraph of section 8.3.13, and then study Example 8.25. ***Do Exercises 41(a), 43 on p.528*** 15. A very useful iterative method for solving equations of the form f (x) = 0 is the Newton-Raphson procedure, and many of you will have met this technique previously. Turn to p.619 and study section 9.5.8. It is easily shown from Figure 9.20 that the formula (9.19), which appears on the Formula sheet (and which appears in the Data Book in a simpler form as Newton’s Method), should lead to a more accurate root of f (x) = 0. The comments below (9.19) concern the order of convergence of the process, and when the method fails, and can be omitted. Work through Examples 9.21 and 9.22. Note that any equation g(x) = C, where C is a constant, must be expressed in the form g(x) − C = 0, before applying the Newton-Raphson formula to the function g(x) − C (= f (x)). When you are told an approximate root of f (x) = 0 the use of the Newton-Raphson method is straightforward. If no initial approximation is prescribed then you must consider the signs of f (x) for different values of x to ascertain a starting value for the iteration. ***Do Example A: Given that the equation x3 = 7 has the approximate root x = 2 use the NewtonRaphson procedure to obtain three further approximations (giving answers correct to 4 decimal places). ***Do Exercise 39 on p.622*** 16. Up to now all the work in this module has involved a function depending on 1 variable, say f (x). In practice, most engineering problems involve quantities which depend on 2 or more variables, e.g. the temperature in a machine generally depends on time t and the 3 spatial coordinates x, y and z, i.e. f = f (x, y, z, t). For these more complicated functions differentiation is more involved and it is necessary to define partial derivatives. At this stage only the basic definition is introduced, with some simple examples. Further details will be considered in module 19 later in the year. Study the introductory comments to section 9.7 on p.627, and then study section 9.7.1 stopping at the top of p.629. Get familiar with the ‘curly dee’ notation, and use it, since it is important in your working to distinguish between ordinary and partial derivatives. It should be emphasised that the partial derivative of a function is the differential with respect to a given variable with all the other independent variables kept ∂f constant. That is, for the function f (x, y) the derivative is differentiation with respect to x, with y ∂x constant. 17. Study Examples 9.26 and 9.27. Note that part (b) of Example 9.27 concerns partial differentiation of a composite function f (u), where u = u(x, y). In these situations the chain rule can be used so, for example, df ∂u ∂f = . ∂x du ∂x Note the use of ordinary and partial derivatives in the above. The function f depends on the single variable u, and hence differentiation of f with respect to u gives an ordinary derivative. On the other hand, u depends on 2 variables x and y and hence differentiation of u with respect to x leads to a partial derivative (with the other variable y kept constant). ***Do Exercise 52 on p.634*** –3–

B:

Work Scheme based on STROUD (FIFTH EDITION)

Work first through Programme F.10 on Differentiation, starting on p.299 in the Foundations part of S.. Then turn to Programme 7 on p.599 of S. Frame 1 contains a list of differentials. You should memorise results 1, 2, 5, 7 and 8 which do not appear on the Formula sheet, but other lecturers will probably expect you to know many of the remaining ones in the list. Frame 2 should be omitted, hyperbolic functions are considered later in the course. You can also omit questions 6, 9, 12, 16 and 18 in frame 3. Study frames 5-16, leaving out question 10 in frame 10 and question 5 in frame 13. The Newton-Raphson procedure for finding roots is not in S. so you should work through section 15 of the work scheme based on J, (copies of James are in the library). Finally, work through frames 1-14 of Programme 10 on Partial Differentiation beginning on p.667 of S.

Specimen Test 3 1. (i) Write down the definition of

df at the point x. dx

(ii) Find the derivative of f (x) = 2x2 from first principles.

2.

Differentiate the following functions with respect to x: (i)

(vi)

1

6x 4 , x √ , x2 + 1

(ii)

2 , x3 (vii)

(iii)

(x2 + x + 1)3 ,

ln(2 + sin x),

(viii)

(iv)

5 + cos(x4 ),

(v)

e−2x sin(2x),

sin2 (x3 + 1).

3.

An approximate solution of the equation x2 = 5 is x = 2. Use the Newton-Raphson procedure to obtain the next two approximations.

4.

Find (i)

∂f ∂f and when f (x, y) is ∂x ∂y

x2 y 2 + xy − x2 + y 2 − 5x,

(ii)

ln(xy) .

–4–