082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

18.3 Second-order Taylor polynomials 515
The required second-order Taylor polynomial, p 2
(x), isthus given by
p 2
(x) =y(2) +y ′ (2)(x −2) +y ′′ (x −2)2
(2)
2
(x −2)2
=6+16(x−2)+14
2
=6+16x−32+7(x 2 −4x+4)
=7x 2 −12x+2
(b) Using (a)we can see that
and so
Hence
(c) We have
p 2
(x) =7x 2 −12x +2,p ′ 2 (x) = 14x −12,p′′ 2
(x) =14
p 2
(2) =6,p ′ 2 (2) = 16,p′′ 2
(2) =14
y(2) = p 2
(2),y ′ (2) = p ′ 2 (2),y′′ (2) = p ′′ 2 (2)
y(2.1) = (2.1) 3 + (2.1) 2 −6 = 7.671
p 2
(2.1) = 7(2.1) 2 −12(2.1) +2 = 7.67
Clearly there is a very close agreement between values of y(x) and p 2
(x) near to
x=2.
Engineeringapplication18.4
Quadraticapproximationtoadiodecharacteristic
In Engineering application 10.5 we derived a linear approximation to a diode characteristic
suitable for small signal variations around an operating point. Sometimes
it is not possible to use a linear approximation because the variations are too large
to maintain sufficient accuracy. Even so, an approximate model may be desirable.
In general, a higher order Taylor polynomial will give a more accurate model than
that of a lower order polynomial. We will consider a quadratic model for a diode
characteristic. TheV −I characteristic of typical diode at room temperature can be
modelledby the equation
I =I(V) =I s
(e 40V −1)
Given anoperating point,V a
, the second-order Taylor polynomialis
p 2
(V)=I(V a
)+I ′ (V a
)(V −V a
)+I ′′ (V a
) (V −V a )2
2
Now
I ′ (V) = 40I s
e 40V I ′′ (V) = 1600I s
e 40V ➔