Download f x( ) = 0

Name: __________________
Class:
Date: _____________
1 The graph of a function f ( x ) is given below. Suppose that Newton's method is used to approximate the roots r and s of the equation f ( x ) = 0 .
The 1 and 2 lines are the tangent lines corresponding to initial approximations for finding these roots. In the approximation of the value of r , what is
x ?
2
x
2
________
2 The graph of a function is given. For which of the following initial approximations does Newton's method fail? Round your answer to the nearest
integer.
x
1
=
________
3 Use Newton's method with the specified initial approximation x
1
to find x , the third approximation to the root of the given equation. (Give your
3
answer to four decimal places.)
x
3
x
x
2
6 = 0, x
1
= 3
________
3
4 Use Newton's method with the specified initial approximation x
answer to four decimal places.)
x
4
x
3
20 = 0 , x
PAGE 1
________
1
= 7
1
to find x , the third approximation to the root of the given equation. (Give your
3
Name: __________________
Class:
Date: _____________
5 Use Newton's method to approximate the given number, correct to eight decimal places.
9
6
20 __________
Use Newton's method to approximate the root of x
4
+ x 10 = 0 in the interval
1, 2 , correct to six decimal places. Use x
1
= 1.5 as the
initial approximation.
x ________
7 Use Newton's method to approximate the least positive root of the equation 2sinx = x correct to six decimal places.Use an initial approximation
x = 2.84 .
1
x ________
8 Use Newton's method to find all the roots of the equation correct to eight decimal places.
5
3x
x 4x
4
16x
3
24x
2
155x 20 = 0
_________________________
9 Use Newton's method to find all the roots of the equation, correct to four decimal places.
x
4
78x
2
10
595 +
x
x 2
= x
+ 8
2
x
2
+ 18
1
+ 8
_______________
10 The following algorithm used by the ancient Babylonians to compute
x
n+1
=
1
2
x
n
+
a
x
n
( You can derive it by applying Newton's method to the equation x
11
a = 0 .)
____________
The following algorithm enables a computer to find reciprocals without actually dividing: x
Newton's method to the equation
1 x
1
1.6005
y
1
=
a = 0 .) Use this algorithm to compute
n+1
1
= 2x
n
2
a x . (You can derive it by apply
n
, correct to six decimal places.
1.6005
__________
12 Find the initial approximation y
13
2
145 , correct to six decimal places.
Use this algorithm to compute
145 a :
1
for which Newton's method succeeds when applied to the equation 6
3
y = 0.
________
A grain silo consists of a cylindrical main section, with height 27 ft, and a hemispherical roof. In order to achieve a total volume of 25000 ft
(including the part inside the roof section), what would the radius of the silo have to be? Find the result and round to two decimal places.
________ ft
PAGE 2
3
ANSWER KEY
Name: __________________
Class:
Date: _____________
1.
( answer 3 ) ( answer<3.7 )
5.
1.39495079
9.
2.
3.
3,10
2.2396
6.
7.
10. 12.041595
11. 0.624805
4.
3.9827
8.
1.697472
1.895494
4, 2.53518376, 0.13148291
PAGE 1
9.2195, 9.2195
12. 0
13.
14.7