Module 8 - Differentiation 2 (self Study)

SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING Self Study Course

MODULE 8

DIFFERENTIATION II

Module Topics 1. Maxima, minima and points of inflection 2. Curve sketching 3. Parametric differentiation, implicit differentiation of functions of one variable and logarithmic differentiation 4. Taylor series, Maclaurin series

A:

Work Scheme based on JAMES (THIRD EDITION)

1. Study section 8.2.7 on p.492 and the top of p.493. 2. Read the introduction to chapter 9 on p.586. Study section 9.2.1 up to and including Example 9.2. The latter section considers the familiar topic of calculating critical or stationary points, and then determining their nature. The methods based on the change in signs of the first derivative and the value of the second derivative must both be known, since each is important in certain situations. Examples 9.3 to 9.5 which complete the section can be read, but need not be studied. They indicate the wide range of problems for which the calculation of optimal values is important but unfortunately we do not have sufficient time to include these applications within this year’s mathematics units. 3. Before sketching curves it is necessary to turn back to p.108 on asymptotes. Study the whole of section 2.5.3, including Examples 2.34 and 2.35. 4. Your graphical calculators can display graphs of functions. However, you do need to be able to use the above techniques to obtain the major properties of the graphs by calculation. To sketch a graph of the function f (x), therefore, it is necessary to: (i) look for any useful general property of f , such as eveness, oddness or periodicity; (ii) find any stationary points of f and determine whether they are maxima, minima or points of inflection; (iii) determine any asymptotes and the behaviour of f on both sides of these lines; (iv) find the values of f (x) as x → ±∞ ; (v) determine any points of intersection of the graph with the coordinate axes. N.B. In graph questions in examinations marks will be awarded for each of these tasks so even though your calculator will be of assistance in the plot it is important that you carry out the necessary algebraic steps listed above. Given a polynomial f (x), note that in determining where y = f (x) intersects the x-axis, in other words finding the roots of f (x) = 0, it is often useful to use the factor theorem. This states that if x = a satisfies –1–

f (a) = 0 then (x − a) is a factor of f (x), i.e. f (x) = (x − a)g(x), where g(x) is a polynomial of lower order. For example, consider the particular case f (x) = x3 −6x2 +11x−6. It is easily seen that f (1) = 0, so (x−1) is a factor of f (x), or x3 − 6x2 + 11x − 6 = (x − 1)g(x). The function g(x) is clearly a quadratic and the coefficient of x2 and the constant can then be found by inspection: x3 − 6x2 + 11x − 6 = (x − 1)(x2 + bx + 6). The remaining constant b is then determined by multiplying out, and must take the value −5. Hence, it follows that x3 − 6x2 + 11x − 6 = (x − 1)(x2 − 5x + 6) = (x − 1)(x − 2)(x − 3). ***Do Exercises 39(a),(b) on p.111*** ***Do Exercises 1(a), 2(c) on p.593*** 5. Turn back to p.519. Study the paragraph on parametric differentiation and then work through Example 8.20. 6. Equations of the form g(x, y) = 0 often arise in applications. In simple situations the equations can dy be solved for y to give y = f (x), an explicit expression for y in terms of x, and calculation of is then dx usually straightforward. The above simplification is not always possible. For instance, the equation y 4 + sin(xy) = 0 cannot be written in the explicit form y = f (x), but remains as an equation in which x and y are linked implicitly. dy The calculation of is still necessary for such equations, however, and you must study on p.520 the dx method of implicit differentiation, which uses the chain rule. Work through Example 8.21. 7. Study p.521 on logarithmic differentiation. This method can simplify the calculation of derivatives, as shown in Example 8.22. Study Examples 8.23 and 8.24. ***Do Exercises 35, 37, 38(but for the equation x3 + y 3 − xy − x = 0), 39(a) on p.523*** 8. Study section 8.3.13 on the determination of second derivatives using parametric or implicit differentiation d2 x calculated on starting on p.523. Study Example 8.26. The general expressions, such as the one for dy 2 p.526, should not be memorised but you could be asked to perform the calculation in particular situations. ***Do Exercises (harder) 46, 52(a),(c) on pp.528 and 529*** 9. Most standard functions of x can be expressed in terms of powers x through the appropriate Maclaurin series or Taylor series. This topic is introduced briefly here (since it is used in some of your first year engineering units) but will be considered in greater depth in Module 20, or next year. Turn to p.609 and study section 9.5.2. The general result with series and remainder is stated by equation (9.13), but the remainder often goes to zero. In these situations we obtain equation (9.14) (the Taylor series expansion of f (x) about x = a) or more importantly equation (9.15) (the Maclaurin series expansion of f (x) about x = 0). Work through the first half of Example 9.14, omitting the sentence beginning “It remains to show...” and the remainder of that page. Look at the expansions in Figure 9.18 and read the paragraph below this figure.

***Do Exercise A

Use the Maclaurin series expansion to confirm that ex = 1 + –2–

x2 x3 x + + + .... 1! 2! 3!

***Do Exercise B Obtain the first four non-zero terms of the Maclaurin series expansion of (1 + x) sin x (a) by calculations using the Maclaurin series expansion with f (x) = (1 + x) sin x; (b) the known Maclaurin series expansion for sin x and multiplication.

B:

Work Scheme based on STROUD (FIFTH EDITION)

The material in this module can be found in S., but in different places in the book. First work through frames 18-35 of Programme 9. There are different ways of determining points of inflection and S. does not use the same approach as J. Next turn to Programme 12 on p.709, on curves and curve fitting. Read through frames 1-17 which contain revision material on straight lines, polynomials, conics, exponential, logarithmic and hyperbolic functions, and then study frames 18-39 on asymptotes and curve sketching. Note that the method in S. of obtaining asymptotes by substituting y = mx + c is different from that discussed in J. For the next topic go to p.608 and work through frames 19-36 in programme 7. Go to p.776 and work through frames 1-18 of Programme 14, and then finally work through frames 46-49 of the same programme. This material on Maclaurin series and Taylor series will be expanded in Module 20.

Specimen Test 8 1. (i) Write down the condition for the point x = x0 to be a stationary point of the function f (x). (ii) Write down the extra conditions to be satisfied if the stationary point x0 is to be a point of inflection. 2.

Find the stationary points (if any) of the following functions, and determine their nature: (i)

3.

f (x) = x − x2 ,

(ii)

f (x) = x + ex ,

Sketch the graph of the function

(iii)

y = x3 − 3x2 + 2

f (x) = x3 − 3x + 3 showing clearly your working.

[You may find it useful to note that y = 0 when x = 1. ]

4.

d2 y dy and in terms A curve is defined parametrically by the equations x = t2 and y = t3 . Find dx dx2 of t.

5.

Find the equation of the tangent to the curve x3 + y 2 + xy − 3 = 0 at the point (1, 1).

6.

Use the Maclaurin series expansion to confirm that sin x = x − –3–

x5 x3 + + ... 3! 5!