Module 20 - Further Calculus 2 (self Study)

SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING Self Study Course

MODULE 20

FURTHER CALCULUS II

Module Topics 1. Sequences and series 2. Rolle’s theorem and the mean value theorems 3. Taylor’s and Maclaurin’s theorems 4. L’Hˆopital’s rule

A:

Work Scheme based on JAMES (THIRD EDITION)

1. When obtaining a mathematical solution to an engineering problem it is possible to proceed in an obvious way but sometimes obtain the wrong answer, even though the initial model and all the algebraic manipulations are correct. In these situations the errors usually arise because some mathematical procedure has been assumed to be true, whereas a much closer inspection would have revealed a flaw in the argument. Analysis is the area of mathematics in which the rigour of a process is investigated. Very little on this topic is covered in your engineering course but a few ideas are introduced in this module. You should know that many mathematical results are correct only if certain conditions are satisfied. These conditions will normally be valid in an engineering process, but they will not always be true and in these cases a new solution procedure must be found. First some features of sequences and series will be investigated. Start by reading section 7.1, the introduction to chapter 7, on p.416. 2. Read the paragraph on notation at the beginning of p.416. The notation {fk }nk=0 is not universally adopted, but in this course we shall follow J. and use it. Work through Examples 7.1, 7.2 and 7.3. Note that Example 7.2 is not really a problem, it simply states the sequence of answers for the ducting of an increasing number of cables. The elements in this sequence do not follow from neighbouring elements in any easily prescribed way. Complete section 7.2.1 by studying the concluding paragraph, below the horizontal line on p.418. 3. Read the introductory sentence to section 7.3 on p.422 and then study section 7.3.1 on arithmetical sequences and series. This is the simplest finite sequence and series, and one you will almost certainly have seen before. You should remember formula (7.2). Work through Examples 7.7 and 7.8. 4. Study section 7.3.2 on geometric sequences and series, both common types. You need to know result (7.4). Work through Example 7.9. ***Do Exercise 11 on p.426*** –1–

5. Read the introductory paragraph to section 7.5 on p.438, and then move on to section 7.5.1 on convergent sequences. You need to understand the general ideas of convergent sequences, but you will not be expected to reproduce the full mathematical detail. Study p.438, although the highlighted mathematical statement near the bottom of the page is not necessary for this module. Study Figure 7.8 and the bottom eight lines on p.438, noting that a sequence {an }∞ n=0 is convergent to the limit a when for any small positive value of  you prescribe there always exists a place in the sequence (where n = N ) beyond which every term in the sequence lies between a −  and a + . The smaller the value of  then the further along the sequence you need to go to find N . If a sequence does not converge then we say it diverges. 6. Study the properties of convergent sequences at the beginning of section 7.5.2. Some rules are stated, but not proved. These rules are often very useful in answering questions. Work through Example 7.17. ***Do Exercise A: Find the limits of the sequences {xn }∞ n=0 defined by (a)

xn =

n+1 , n2 + 1

(b)

xn =

3n2 + 2n + 1 . 6n2 + 5n + 2

7. Read the introduction to section 7.6 on infinite series starting at the bottom of p.444. The manipulations in this paragraph show how easy it is to obtain fallacious results. Study section 7.6.1 including Example 7.20, parts (a) and (c) only. The result (7.13) for the sum of an infinite geometric series is particularly useful, and must be remembered. 8. Little time is available to discuss the tests used to establish the convergence or divergence of an infinite series. Read the introductory paragraph to section 7.6.2 and then move to p.447 and study part (b) on D’Alembert’s ratio test, including Example 7.22(a). Complete the section by studying the material below the horizontal line on p.448. The highlighted result is very useful and is used in Example 7.23, which you should work through. ***Do Exercise 37 on p.450*** 9. We now change topic and consider a number of theorems. Start by reading the introduction to section 9.4 on p.599. 10. Move on to section 9.4.1 and study theorem 9.1, Rolle’s theorem, and the first paragraph below the statement of the theorem, together with figure 9.14. The theorem essentially says that for all reasonable (non-constant) functions satisfying f (a) = f (b), there exists at least one maximum or minimum between a and b, at x = c say, where f 0 (c) = 0. When the function is constant then all points c between a and b satisfy f 0 (c) = 0. Rolle’s theorem only requires f (x) to be differentiable on (a, b), the open interval, not including the endpoints x = a and x = b. The theorem holds, therefore, for the semi-circle x2 + y 2 = 1, y ≥ 0, which has infinite slope at the end-points. You do need to be a bit careful, however. Suppose firstly that f (x) satisfies f (a) = f (b) but the function does not have a derivative at every point between a and b. Figure 1 below shows such a function, and clearly there is no point between a and b at which the tangent to the graph is parallel to the x-axis. Consider next a function f (x) which satisfies f (a) = f (b) but is not continuous at all points in a < x < b. A function with one point of discontinuity in this interval is displayed in Figure 2 below, and it is easily seen from the graph that again there is no point between a and b at which the tangent is parallel to the x-axis. –2–

N.B. Although Rolle’s theorem states the existence of a point x = c, it is very important to observe that it does not tell you how to find the value of c. ...... ... ...... .... ... ..... ... . . ..... ...... .... ...... ... . . . . ....... ... . ......... . . . . ......... ........... .. .... ... . . . . .... .... .... ... . . .. . ... . .............................................................................................................................................................. .

.. ... ... ... . . . ... .... .... ...... .......... ... . ........ .. ........ . .......... .. ... ........... . . . . . . . .. . . . . . . .... . .... .... .... ... . . .. . ... . ........................................................................................................................................................... .

Figure 1

Figure 2

a

b

x

a

b

x

11. Continue reading section 9.1.4, starting with the last three lines on p.599. Look at theorems 9.2 and 9.3 on the first mean value theorems, stopping four lines below the stated end of theorem 9.3. Z b f (x) dx, can always be equated to Theorem 9.2 states that the area under the graph of a function f (x), a

the area of a suitable rectangle, (b − a)f (c), for some suitably chosen c satisfying a < c < b. Figure 9.15(a) shows there can be more than one value for c. Theorem 9.3 can be obtained by applying theorem 9.2 to the function f 0 (x). The result is illustrated in Figure 9.15(b) and discussed in the paragraph below the statement of Theorem 9.3. Once again you should observe that the theorem states the existence of at least one point x = c, whilst the figure shows that more than one solution is certainly possible. Theorem 9.3 is really a rotated version of Rolle’s theorem. [In fact, another way of proving 9.3 is to apply Rolle’s theorem to the function g(x) = f (x) − f (a) − f (b) − f (a) (x − a) . ] b−a 12. Read the introduction to section 9.5 on p.606, and then study the introductory paragraph to section 9.5.1 on finding polynomial approximations. Work through Example 9.12. 13. Turn to p.608 and study from below the horizontal line to the end of section 9.5.1. Equation (9.11) states the rather surprising result that however many terms you choose to have in your polynomial approximation to f (x), the difference between f (x) and the polynomial can always be expressed in a comparatively simple form, Lagrange’s form of the remainder. It should be emphasised that the precise value for θ is not known, although it must satisfy 0 < θ < 1, but knowledge of this range and the functional form of the remainder means that the maximum magnitude of the remainder can often be calculated. This is an important application of Taylor’s theorem. 14. Before studying section 9.5.2 let us look briefly at power series. Turn back to p.451 and read the introduction to section 7.7. Infinite series have been considered above, but when the elements have the form an xn , where the an are independent of x, then we have a power series. Study section 7.7.1 on convergence, and how the radius of convergence is determined in part(a) of Example 7.24. A list of special power series, with their domains of validity, is given in Figure 7.13 on p.455. The most important of these series appear on the Formula sheet and a smaller number can be found in the Data Book. 15. Return to p.609 and study section 9.5.2 on Taylor and Maclaurin series up to equation (9.15) (You did look briefly at this topic in Module 8). Equations (9.12) and (9.13) are equivalent, one is found from the other by a change of variable. Note that equation (9.13) appears on the Formula sheet and in the Data Book (in a slightly different form). –3–

The Maclaurin series expansion of f (x) is obtained from the Taylor series expansion by choosing a = 0. In a similar way, putting a = 0 into equations (9.12) or (9.13) leads to Maclaurin’s theorem f (x) = f (0) +

x2 x3 xn (n) x 0 f (0) + f 00 (0) + f 000 (0) + . . . f (0) + Rn (x) , 1! 2! 3! n!

where Rn (x) =

xn+1 (n+1) f (θx) , (n + 1)!

with 0 < θ < 1 .

Note that in Taylor’s and Maclaurin’s theorems J. defines Rn as above, containing the term xn+1 , with the preceding term in the series containing xn . It is more usual to write Rn+1 for the term displayed above but to avoid confusion with J. we shall use his notation throughout this module. Work through Example 9.14, although this is not an easy one. The two expressions for f 0 (x) are equal because  π π 1 1 π = sin x cos + cos x sin = √ sin x + √ cos x . sin x + 4 4 4 2 2 To obtain f 00 (x) you should differentiate the result for f 0 (x) as a product, and then rewrite the answer to yield the stated form. A pattern for the higher derivatives then emerges. The general expression for Rn can be written down. Since | sin ω| ≤ 1 and |eθx | < ex , J. is able to show that Rn → 0 as n → ∞ for all x. Thus Maclaurin’s series expansion for ex sin x exists, and takes the displayed form. 16. Now study the following Example. Example 1 Use Maclaurin’s theorem, with two terms and a remainder, to show that the error in writing sin x as x is less than 0.005 if 0 < x < 0.1. With f (x) = sin x, it follows that f 0 (x) = cos x, f 00 (x) = − sin x. Maclaurin’s theorem, with two terms and a remainder, implies x2 00 f (θx) 0 < θ < 1 2! x2 sin x = sin 0 + x cos 0 + (− sin(θx)), 0 < θ < 1 2! x2 sin(θx) , 0<θ<1 where R1 = − = x + R1 , 2 f (x) = f (0) + x f 0 (0) +

i.e.

Now

2 x sin(θx) 1 2 ≤ x , |R1 | = − 2 2

since |sin(θx)| ≤ 1,

and in the given range 0 < x < 0.1 it then follows that |R1 | <

1 (0.1)2 = 0.005. 2

Thus, replacing sin x by x has an error of maximum magnitude 0.005 (provided 0 < x < 0.1). 1 Use Maclaurin’s theorem to show that if 0 < x < π/2 then cos x = 1− x2 +R3 (x) , 2 x2 x4 . Obtain an expression for the maximum error if the approximation cos x = 1− where 0 < R3 (x) < 4! 2 π is used to calculate cos . 10 ***Do Exercise B:

Use your calculator to evaluate cos x and 1 − x2 /2 when x = π/10, and comment briefly on your results. –4–

***Do Exercise C: Apply Maclaurin’s theorem with two terms and a remainder to the function f (x) = (1 + x)1/2 to show that 1 (1 + x)1/2 = 1 + x + R1 (x). 2 Write down Lagrange’s form of the remainder R1 (x) and deduce that the error in writing (1.02)1/2 as 1.01 is at most 0.00005. ***Do Exercise D:

Find the Maclaurin’s series expansion of ex .

17. To complete this module we consider L’Hˆ opital’s rule. Turn to section 9.5.3 on p.614. It is often necessary to find limits of the form f (x) . lim x→a g(x) A number of situations can arise. Suppose lim f (x) = A and lim g(x) = B, and that B is non-zero, then x→a

lim

x→a

x→a

A f (x) = . g(x) B

When B = 0, the quotient can have a limit only when A = 0, but simple division then leads to the 0 indeterminate form . How do we proceed? 0 Study section 9.5.3, and then work through Example 9.18. Note that in the worked Examples the numerator and denominator are both differentiated once at each step, until one of the derivatives becomes non-zero at x = a. 18. L’Hˆopital’s rule can also be used when both numerator and denominator tend to infinity. Consider the following Example. Example 2.

Find lim

x→∞

x . ex

Clearly both x and ex approach infinity as x → ∞. The quotient is indeterminate so differentiate both numerator and denominator. Then x 1 lim = lim x , = 0, x→∞ ex x→∞ e since ex gets larger and larger as x → ∞. ***Do Exercises 32(a),(b),(c),(e) on p.616

B:

Work Scheme based on STROUD (FIFTH EDITION)

The material on sequences and series can be found in programme 13 of S., starting on p.747. Study frames 1–18, then 24–38 and finally 43–49. S. contains some frames on the remaining topics in this module. Turn to programme 14 in S., starting on p.775, and study frames 1–30, which mainly concern Maclaurin’s series. Next read frame 31, and then study frames 37–45 on L’Hˆ opital’s rule. Complete programme 14 by studying frames 46–49 on Taylor’s series. After studying the above you should finish the module by going through sections 9–18 of the above Work scheme based on JAMES (THIRD EDITION), since the latter contains some topics not covered in S. Note in particular the additional material on Rolle’s, Taylor’s and Maclaurin’s theorems, and the discussion of the remainder term. –5–

Specimen Test 20 1.

The second and fourth terms of an arithmetical sequence are 2 and 18 respectively. (i) Find the first term and the common difference. (ii) What is the tenth term in the sequence?

2.

A geometric sequence has first term 3 and common ratio

2 3.

(i) Find the third term in the sequence. (ii) Determine the sum of the first eight terms in the sequence.

3.

For each of the following sequences, defined for n = 1, 2, . . ., state whether or not the sequence converges and, if it does, find the limit: (i) an =

1 − 2n2 1 + n + n2

(ii) an = 1 + 4n − n2

1 2 3 4 + + + + ... 3 4 5 6

is convergent.

4.

Determine whether the series

5.

State Maclaurin’s theorem, putting in the first three terms, the term involving xn and the Lagrange form of the remainder.  lim

 cos x − 1 . x2

6.

Use l’Hˆopital’s rule to evaluate

7.

Apply Maclaurin’s theorem with two terms and a remainder to the function f (x) = (1 + x)2/3 to show that 2 (1 + x)2/3 = 1 + x + R1 (x) . 3 Write down Lagrange’s form of the remainder R1 .

x→0

Deduce that if 0 < x < 0.3 then |R1 (x)| < 0.01 .

–6–