Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PRELIMINARY CALCULUS

Find the volume of a cone enclosed by the surface formed by rotating about thex-axis

the liney=2xbetweenx=0andx=h.

Using (2.46), the volume is given by

V=

∫h

0

π(2x)^2 dx=

∫h

0

4 πx^2 dx

=

[ 4

3 πx

3 ]h

0 =

4

3 π(h

(^3) −0) = 4
3 πh
(^3) .
As before, it is also possible to form a volume of revolution by rotating a curve
about they-axis. In this case the volume enclosed betweeny=aandy=bis
V=
∫b
a
πx^2 dy. (2.47)
2.3 Exercises
2.1 Obtain the following derivatives from first principles:
(a) the first derivative of 3x+4;
(b) the first, second and third derivatives ofx^2 +x;
(c) the first derivative of sinx.
2.2 Find from first principles the first derivative of (x+3)^2 and compare your answer
with that obtained using the chain rule.
2.3 Find the first derivatives of
(a) x^2 expx,(b)2sinxcosx,(c)sin2x,(d)xsinax,
(e) (expax)(sinax)tan−^1 ax,(f)ln(xa+x−a),
(g) ln(ax+a−x), (h)xx.
2.4 Find the first derivatives of
(a) x/(a+x)^2 ,(b)x/(1−x)^1 /^2 ,(c)tanx,assinx/cosx,
(d) (3x^2 +2x+1)/(8x^2 − 4 x+2).
2.5 Use result (2.12) to find the first derivatives of
(a) (2x+3)−^3 ,(b)sec^2 x,(c)cosech^33 x,(d)1/lnx,(e)1/[sin−^1 (x/a)].
2.6 Show that the functiony(x)=exp(−|x|) defined by
y(x)=







expx forx< 0 ,

1forx=0,

exp(−x)forx> 0 ,

isnotdifferentiable atx= 0. Consider the limiting process for both ∆x>0and
∆x<0.
2.7 Finddy/dxifx=(t−2)/(t+2) andy=2t/(t+1) for−∞<t<∞. Show that
it is always non-negative, and make use of this result in sketching the curve ofy
as a function ofx.
2.8 If 2y+siny+5=x^4 +4x^3 +2π, show thatdy/dx=16whenx=1.
2.9 Find the second derivative ofy(x)=cos[(π/2)−ax]. Now seta=1andverify
that the result is the same as that obtained by first settinga= 1 and simplifying
y(x) before differentiating.