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By Phil Dyke (auth.)

ISBN-10: 0333660927

ISBN-13: 9780333660928

ISBN-10: 1349140767

ISBN-13: 9781349140763

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Extra info for Advanced Calculus

Example text

Rn is the remainder term. ' /;j oxi f(a 1 = + h 1, a 2 + h 2 , -... ar ' oxpxj ••• , an + hn), the h; (i = 1, 2, ... , and fand all its derivatives on the right-hand side are evalu- ated at X; = a;. The first (single) summation is a generat linear expression since all of the /;'s are simply numbers, the second (double) summation is, similarly, a generat quadratic form and so on. Each 'term' is a higher-order homogeneous function. Little use of Taylor's Theorem in more than two dimensions will be made in this text.

10. Find the maximum value of the function of three variables xyz(1 - x - y - z). 0, (e) f(x, y) = ex cosy. 5. 6, or otherwise. 6. Jx2 + y2 has a relative maximum at the origin (0, 0), but that none of its partial derivatives exist there. What kind of point is it? 7. A functionf(x, y) is defined by f(x, y) = sinxsiny. Investigate all the maxima, minima and saddle points of this function. 8. Investigate the extreme values of the function f(x, y) x 4 - 2(x - y) 2 + l. In particular, find out in some detail what happens at the origin (0, 0).

The curve we need to consider is the intersection of this with the plane y = bx, which will resemble a hyperbola. 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.

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Advanced Calculus by Phil Dyke (auth.)


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