By Phil Dyke (auth.)

ISBN-10: 0333660927

ISBN-13: 9780333660928

ISBN-10: 1349140767

ISBN-13: 9781349140763

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Although such a visualisation is useful, the beauty of the Lagrange multiplier technique is that it is not necessary. Jx2 + i + z2 , the distance of an arbitrary point (x, y, z) from the origin. Note also that the values of x, y and z that minimise r will also minimise r 2 so we can work with the alto+ i + z2 • This is a commonly used and very useful trick that saves gether cleaner expression considerable algebra. Thus we form r where we have used two multipliers A1 and Az. There is no restriction on the number of multipliers in theory, however in practice too many constraints can lead to the absence of a true extremum.

If a and I both increase by 5 per cent, p decreases by 10 per cent and J1 decreases by 30 per cent, find the percentage change in V. L _ dl J1 a I using the above formulae for the partial derivatives. 4 Interestingly, if we take the formula V = 7rpa and take logarithms we obtain 8j11 In V = d Differentiating and since -(In V) dV + In In( ; ) p + 4ln a - In J1 - In I 1 dV ~ V V p = -so d(ln V) = -, d(ln p) = - etc. re-derives the formula for d;. Sometimes the phrase 'logarithmic differentiation' is used for this process.

If x = e"secv and y = e"tanv show that = volume. Show that where f(x, y) is a differentiable function of two variables. 14. By writing u = x + y, v = xy transform the equation into one involving derivatives of solve it. 15. If u = tan _,x + tan _,y and with respect to u and v. Hence Find the rates of change of V with respect to b, a and h. 5b. 17. The area of a triangle ABC with sides a, b, c is given by ~ absinC. 2 if schooldays are too distaut a memory. x+y v = -1- - determine whether - xy A there is a functional relationship between u and v by finding the J b.

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