3.9Differentials

Let y = f(x) represent a function that is differentiable inan open interval containing x. The differential of x (denoted by dx) is any nonzero real number. The differentialof y (denoted by dy) is given by

dy = f’(x) dx

x

y

dx

dyx Δ

ΔΔ 0lim

→=

x xx Δ+

xdx Δ=

dy

yΔ

)()( xfxxfy −+= ΔΔ

Comparing dyandyΔLet y = x2. Find dy when x = 1 and dx = 0.01. Comparethis value to when x = 1 and = 0.01.yΔ xΔ

dy = f’(x) dx dy = 2x dx

dy = 2(1)(.01) = .02

)()( xfxxfy −+= ΔΔ = f(1.01) – f(1)

= 1.012 – 12 = .0201

(1, 1)

01.== xdx Δ

0201.0=yΔdy = 0.02

Estimation of error.

The radius of a ball bearing is measured to be .7 inch. If themeasurement is correct to within .01 inch, estimate thepropagated error in the Volume of the ball bearing.

r = .7

r = .7 and 01.01. ≤≤− rΔ

3

3

4rV π=

drrdVV 24πΔ ==

( ) ( )01.07.04 2 ±= π

06158.0±≈ propagated error

relative error is

( ) %29.4100 ≈V

dV( )0429.

7.3406158.

3±=±

π Percentage error

Finding differentials

Function Derivative Differential

y = x2 xdx

dy2= dy = 2x dx

y = 2sin x xdx

dycos2= dy = 2cos x dx

y = x cos x )1(cos)sin( xxxdx

dy+−=

( )dxxxxdy cossin +−=

y = sin 3x xdx

dy3cos3= dxxdy 3cos3=

xxf =)(101Find

)()( xfxxfy −+= ΔΔ

yxfxxf ΔΔ +=+ )()(

dyxfxxf +=Δ+ )()( dyy ≈Δand dy = f’(x) dx

dxxfxfxxf )(')()( +=+∴ Δ

Let x = 100and 1==dxxΔ

xxf

2

1)(' =

( )11002

1100 + 05.10

20

110 =+=

05.10101 ≈∴

=+ )1100(f