Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PRELIMINARY CALCULUS

O

C

P

Q

ρ

ρ

r

r+∆r

c

p

p+∆p

Figure 2.13 The coordinate system described in exercise 2.20.

2.20 A two-dimensional coordinate system useful for orbit problems is the tangential-
polar coordinate system (figure 2.13). In this system a curve is defined byr,the
distance from a fixed pointOto a general pointPof the curve, andp,the
perpendicular distance fromOto the tangent to the curve atP. By proceeding
as indicated below, show that the radius of curvature,ρ,atPcanbewrittenin
the formρ=r dr/dp.
Consider two neighbouring points,PandQ, on the curve. The normals to the
curve through those points meet atC, with (in the limitQ→P)CP=CQ=ρ.
Apply the cosine rule to trianglesOP CandOQCto obtain two expressions for
c^2 ,oneintermsofrandpand the other in terms ofr+∆randp+∆p.By
equating them and lettingQ→Pdeduce the stated result.
2.21 Use Leibnitz’ theorem to find

(a) the second derivative of cosxsin 2x,

(b) the third derivative of sinxlnx,

(c) the fourth derivative of (2x^3 +3x^2 +x+2)exp2x.

2.22 Ify=exp(−x^2 ), show thatdy/dx=− 2 xyand hence, by applying Leibnitz’
theorem, prove that forn≥ 1

y(n+1)+2xy(n)+2ny(n−1)=0.

2.23 Use the properties of functions at their turning points to do the following:

(a) By considering its properties nearx= 1, show thatf(x)=5x^4 − 11 x^3 +

26 x^2 − 44 x+ 24 takes negative values for some range ofx.

(b) Show thatf(x)=tanx−xcannot be negative for 0≤x<π/2, and deduce

thatg(x)=x−^1 sinxdecreases monotonically in the same range.

2.24 Determine what can be learned from applying Rolle’s theorem to the following
functionsf(x): (a)ex;(b)x^2 +6x;(c)2x^2 +3x+1; (d) 2x^2 +3x+2; (e)
2 x^3 − 21 x^2 +60x+k.(f)Ifk=−45 in (e), show thatx=3isonerootof
f(x) = 0, find the other roots, and verify that the conclusions from (e) are
satisfied.
2.25 By applying Rolle’s theorem toxnsinnx,wherenis an arbitrary positive integer,
show that tannx+x=0hasasolutionα 1 with 0<α 1 <π/n. Apply the
theorem a second time to obtain the nonsensical result that there is a realα 2 in
0 <α 2 <π/n, such that cos^2 (nα 2 )=−n. Explain why this incorrect result arises.