Higher Engineering Mathematics, Sixth Edition

Some applications of differentiation 311

7. The power developed in a resistorRby a
   battery of emfEand internal resistanceris
   given byP=

E^2 R

(R+r)^2

. DifferentiatePwith
respect toRand show that the power is a
maximum whenR=r.
8. Find the height and radius of a closed cylin-
der of volume 125cm^3 which has the least
surface area. [
height= 5 .42cm;
radius= 2 .71cm

]

9. Resistance to motion,F, of a moving vehi-
   cle, is given byF=^5 x+ 100 x. Determine the
   minimum value of resistance. [44.72]


10. An electrical voltageEis given by
    E=(15sin50πt+40cos50πt)volts,
    wheretis the time in seconds. Determine the
    maximum value of voltage.
    [42.72volts]


11. The fuel economyEof a car, in miles per
    gallon, is given by:

E= 21 + 2. 10 × 10 −^2 v^2

− 3. 80 × 10 −^6 v^4

wherevis the speed of the car in miles per

hour.

Determine, correct to 3 significant figures,

the most economical fuel consumption, and

the speed at which it is achieved.

[50.0 miles/gallon, 52.6 miles/hour]

28.5 Tangents and normals

Tangents

The equation of the tangent to a curvey=f(x)at the
point (x 1 ,y 1 ) is given by:

y−y 1 =m(x−x 1 )

wherem=

dy

dx

=gradient of the curve at (x 1 ,y 1 ).

Problem 21. Find the equation of the tangent to

the curvey=x^2 −x−2 at the point (1,−2).

Gradient,m

=

dy

dx

= 2 x− 1

At the point (1,−2),x=1andm= 2 ( 1 )− 1 =1.

Hence the equation of the tangent is:

y−y 1 =m(x−x 1 )

i.e. y−(− 2 )= 1 (x− 1 )

i.e. y+ 2 =x− 1

or y=x− 3

The graph ofy=x^2 −x−2 is shown in Fig. 28.12. The

lineABis the tangent to the curve at the pointC,i.e.(1,

−2), and the equation of this line isy=x−3.

Normals

The normal at any point on a curve is the line which

passes through the point and is at right angles to the

tangent. Hence, in Fig. 28.12, the lineCDis the normal.

It may be shown that if two lines are at right angles

then the product of their gradients is−1. Thus ifmis the

gradient of the tangent, then the gradient of the normal

is−

1

m

Hence the equation of the normal at the point (x 1 ,y 1 )is

given by:

y−y 1 =−

1

m

(x−x 1 )

2

1

0 1 23

 1

 2  1

 2

 3

yx^2 x 2

y

x

B

C

A D

Figure 28.12