Upload
trankhanh
View
243
Download
0
Embed Size (px)
Citation preview
317
CHAPTER 11CHAPTER 11
Linear Tangent Approximations and Euler’s Method
Before the arrival of calculators, a method for estimating values by extrapolation was
sometimes effected by the use of the fact that for small changes in x , dy y
dx x
!"!
.
Graphically, this meant that on the graph below
provided h was small, then points Q and R were virtually the same point. This
meant that their y co-ordinates were approximately equal.
i.e. ( )MQ MR MT TR! = +
This means that ( ) ( ) ( )'f a h f a f a h+ ! + .
This approximation is called a linear approximation or a linear tangent
approximation. This is often written: ( ) ( ) ( )'f x x f x f x x+ ! " + ! .
318
Example
Question: Find, without the use of a calculator, an approximate value for 4.01 .
Answer: Let ( )f x x= , let 4a = and 0.01h = .
Note that ( )1
'2
f xx
=
Then ( )4.01 4.01f=
( )4 0.1f= +
( )1
4 0.012 4
! +
2 0.0025= +
2.0025=
! 4.01 2.0025! (This is an excellent approximation)
Example
Evaluate, approximately, the value of 10 55x x x+ + when 1.01x = .
Let ( ) 10 55f x x x x= + + .
Then ( ) 9 4' 10 25 1f x x x= + +
( ) ( ) ( )( )' 1.01 1 ' 1 0.01f f f! +
( )7 36 0.01= +
7.36=
In fact, ( )1.01 7.36967f = (approx.)
319
Example
The relation 2 32 8x y xy+ = defines y as a function of x near to ( 2,1 ).
Call this function ( )y f x= . Use the linear tangent approximation to find an
approximate value for ( )1.92f .
( ) ( ) ( )( )1.92 2 ' 2 0.08f f f! + "
To find ( )' 2f we need to find dydx
when 2x = and 1y = .
Differentiate with respect to x .
2 3 22 2 3 0
dy dyy x x y x y
dx dx
! "+ + + =# $
% &
At ( 2,1 )
4 4 2 1 6 0dy dy
dx dx
! "+ + + =# $
% &
i.e. 3
8
dy
dx= !
Substituting in yields:
f 1.92( ) ! 1+ "3
8
#$%
&'("0.08( )
1.03=
*
*
320
Worksheet 1
1. Without the use of a calculator find the approximate value of
a) ( )1
38.02 b) sin31° (note 1 0.01745° = radians)
c) ( )1.5
4.1 d) 3 0.126
2. Find an approximate value for 3 23 2 1x x x! + ! when 1.998x = without the aid
of a calculator.
3. The surface area of a sphere is 24 r! . If the radius of the sphere is increased
from 10 cm to 10.1 cm, what is the approximate increase in area?
4. One side of a rectangle is three times another side. If the perimeter increases
by 2% what is the approximate percentage increase in area?
5. A new spherical ball bearing has a 3 cm radius. What is the approximate value
of the metal lost when the radius wears down to 2.98 cm?
6. Find the percentage error in the volume of a cube if an error of 1% is made in
measuring the edge of the cube.
7. ( 1,1 ) is a point on the graph of 2 22x y y x+ = . Find a reasonable
approximation for the y co-ordinate of a point near ( 1,1 ) whose
x co-ordinate is 1.01.
8. The equation 4 41x y xy+ + = defines y implicitly in terms of x near the point
( -1,1 ). Use the tangent line approximation at the point ( -1,1 ) to estimate the
value of y when 0.9x = ! .
321
9. The local linear approximation of a function f will always be greater than the
function’s value if, for all x in the interval containing the point of tangency,
(A) ' 0f < (B) ' 0f > (C) " 0f > (D) " 0f < (E) ' " 0f f= =
Answers to Worksheet 1
1. a) 2.0017 b) 0.5151 c) 8.3 d) 0.5013
2. -1.004
3. 8!
4. 4.04%
5. 0.72! cubic cm
6. 3.03%
7. 0.99
8. 0.9
9. D
322
Euler’s Method
Euler’s Method involves the use of the linear tangent approximation more than once
and is essentially the same. It is somewhat more accurate.
For example to find 4.02 where ( )f x x= and ( )1
'
2
f xx
= , 4a = and 0.02h = ,
we have ( ) ( ) ( )( )4.02 4.02 4 ' 4 0.02f f f= ! +
( )1
2 0.02 2.0052 4
= + =
If we use Euler’s Method where we do the approximation twice we have
0.01x! = repeated.
i.e. ( ) ( ) ( )( )4.01 4.01 4 ' 4 0.01f f f= ! +
2 0.0025= +
Then ( ) ( ) ( )( )4.02 4.02 4.01 ' 4.01 0.01f f f= ! +
( )1
2.0025 0.012 4.01
= +
( )1
2.0025 0.014.005
= +
2.004997=
In fact 4.02 2.0049938= (approx.)
323
Example
Given ( )1
'f xx
= and ( )1 0f = find ( )1.5f using Euler’s Method with two
iterations of 0.25x! = .
( ) ( ) ( )( )1.25 1 ' 1 0.25f f f! +
( )1
0 0.251
= +
0.25=
( ) ( ) ( )1
1.5 1.25 0.251.25
f f= +
( )4
0.25 0.255
= + = 0.45
Graphically Euler’s Method can be viewed and follows:
To evaluate ( )1.5f : we want the y co-ordinate of P
Euler’s method yields the y co-ordinate of Q
One linear approximation yields the y co-ordinate of R
324
Worksheet 2
1. 0.5dy
xydx
= . Use Euler’s method to find y when 2x = given that 2y = when
0x = . Use steps each of size 1.
2. The solution of the differential equation 2dy x
dx y= ! contains the point ( 3,-2 ).
Find using approximation methods the value of y when 2.7x = with
0.3x! = " .
3. 2
dy x y
dx y
!= and 2y = ! when 3x = .
An estimate for the value of y when 3.2x = using a linear tangent
approximation is:
(A) -2 (B) -2.15 (C) -2.2 (D) -2.25 (E) -2.30
325
4.
In the figure above PC is tangent to the graph of ( )y f x= at point P. Points
A and B are on PC and points D and E are on the graph of ( )y f x= .
Which statement is true?
I. Euler’s method uses the y co-ordinate of point A to
approximate the y co-ordinate of point E.
II. Euler’s method uses the x co-ordinate of point B to
approximate the root of the function ( )f x at point D.
III. Euler’s method uses the x co-ordinate of C to approximate a
root of the function ( )f x at point D.
(A) I only (B) II only (C) III only (D) I and II (E) I and III
Answers to Worksheet 2
1. 3 2. -3.35 3. D 4. A