Mathematical Methods for Physics and Engineering - Matematica.NET

2.3 EXERCISES2.10 The function y(x) is defined by y(x) =(1+x m ) n .(a) Use the chain rule to show that the first derivative of y is nmx m−1 (1 + x m ) n−1 .(b) The binomial expansion (see section 1.5) of (1 + z) n is(1 + z) n =1+nz +n(n − 1)z 2 + ···+2!n(n − 1) ···(n − r +1)z r + ··· .r!Keeping only the terms of zeroth and first order in dx, apply this result twiceto derive result (a) from first principles.(c) Expand y in a series of powers of x before differentiating term by term.Show that the result is the series obtained by expanding the answer givenfor dy/dx in (a).2.11 Show by differentiation and substitution that the differential equation4x 2 d2 y dy− 4xdx2 dx +(4x2 +3)y =0has a solution of the form y(x) =x n sin x, and find the value of n.2.12 Find the positions and natures of the stationary points of the following functions:(a) x 3 − 3x +3;(b)x 3 − 3x 2 +3x; (c)x 3 +3x +3;(d) sin ax with a ≠0;(e)x 5 + x 3 ;(f)x 5 − x 3 .2.13 Show that the lowest value taken by the function 3x 4 +4x 3 − 12x 2 +6is−26.2.14 By finding their stationary points and examining their general forms, determinethe range of values that each of the following functions y(x) can take. In eachcase make a sketch-graph incorporating the features you have identified.(a) y(x) =(x − 1)/(x 2 +2x +6).(b) y(x) =1/(4+3x − x 2 ).(c) y(x) =(8sinx)/(15 + 8 tan 2 x).2.15 Show that y(x) =xa 2x exp x 2 has no stationary points other than x =0,ifexp(− √ 2)