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Nonlinear Equations
Your nonlinearity confuses me
ax
5
bx
4
cx
3 tanh(
x
)
dx
2
x
ex
f
0
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“The problem of not knowing what we missed is that we believe we haven't missed anything” –
Stephen Chew on Multitasking
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Example – General Engineering You are working for ‘DOWN THE TOILET COMPANY’ that makes floats for ABC commodes. The floating ball has a specific gravity of 0.6 and has a radius of 5.5 cm. You are asked to find the depth to which the ball is submerged when floating in water.
x
3 0 .
165
x
2 3 .
993 10 4 0
Figure:
Diagram of the floating ball
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For the trunnion-hub problem discussed on first day of class where we were seeking contraction of 0.015”, did the trunnion shrink enough when dipped in dry-ice/alcohol mixture?
1. Yes 2. No
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1.
2.
Example – Mechanical Engineering
Since the answer was a resounding NO, a logical question to ask would be: If the temperature of -108 o F is not enough for the contraction, what is?
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Finding The Temperature of the Fluid
D
T
a
D
= 80
T c T a
o (
T
F
T
c = ???
o F )
dT D
= 12.363" ∆
D = -
0.015" 5 4.5
4 3.5
3 2.5
7 6.5
6 5.5
2 -350 -300 -250 -200 -150 T -100 -50 0 50 100 6 .
033 0 .
009696
T
0 .
015 5 .
992 10 8
T c
2 7 .
457 10 5
T c
6 .
349 10 3
f
(
T c
) 5 .
992 10 8
T c
2 7 .
457 10 5
T c
8 .
651 10 3 0
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Finding The Temperature of the Fluid
D
D T a
T c
(
T
)
dT T
a = 80 o F
T
c = ???
o F
D
= 12.363" ∆
D = -
0.015" 1 .
228 10 5
T
2 6 .
195 10 3
T
6 .
015 0 .
015 5 .
059 10 9
T c
3 3 .
829 10 6
T c
2 7 .
435 10 5
T c
6 .
166 10 3
f
(
T c
) 5 .
059 10 9
T c
3 3 .
829 10 6
T c
2 7 .
435 10 5
T c
8 .
834 10 3 0
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Nonlinear Equations
(Background) http://nm.mathforcollege.com
How many roots can a nonlinear equation have?
3 2 1 0 -1 -2 -3 -20 -10 sin(x)=2 has no roots 0 10 20
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How many roots can a nonlinear equation have?
sin(x)=0.75 has infinite roots 3 2 1 0 -1 -2 -3 -20 -10 0 10 20
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How many roots can a nonlinear equation have?
sin(x)=x has one root 3 2 1 0 -1 -2 -3 -20 -10 0 10 20
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How many roots can a nonlinear equation have?
3 0 -1 2 1 -2 -3 -20 sin(x)=x/2 has finite number of roots -10 0 10 20
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The value of
x
that satisfies
f
(
x
)=0 is called the A.
B.
C.
D.
root of equation
f
(
x
)
=0
root of function
f
(
x
) zero of equation
f
(
x
)
=0
none of the above
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2.
3.
4.
A quadratic equation has ______ root(s) A. one B. two C. three D. cannot be determined
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2.
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4.
For a certain cubic equation, at least one of the roots is known to be a complex root. The total number of complex roots the cubic equation has is
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A. one B. two C. three D. cannot be determined
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2.
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Equation such as
tan
(
x
)
=x
has __ root(s) 1.
2.
3.
4.
zero one two infinite
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2.
3.
4.
A polynomial of order
n
has
1. n
-1 2.
n
3.
4.
n n
+1 +2 zeros
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The velocity of a body is given by
v
(
t
)
=
5
e
-
t
+4 , where
t
is in seconds and
v
is in m/s. The velocity of the body is
6
m/s at
t
=___.
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A. 0.1823 s B. 0.3979 s C. 0.9162 s D. 1.609 s
A.
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B.
C.
D.
END http://nm.mathforcollege.com
Newton Raphson Method
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This is what you have been saying about your TI-30Xa A. I don't care what people say The rush is worth the price I pay I get so high when you're with me But crash and crave you when you are away B. Give me back now my TI89 Before I start to drink and whine TI30Xa calculators you make me cry Incarnation of of Jason will you ever die C. TI30Xa – you make me forget the high maintenance TI89.
D. I never thought I will fall in love again!
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C.
D.
Newton-Raphson method of finding roots of nonlinear equations falls under the category of __________ method.
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A. bracketing B. open C. random D. graphical
1.
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3.
4.
The next iterative value of the root of the equation
x
2 =4 using Newton-Raphson method, if the initial guess is 3 is
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A.
B.
C.
D.
1.500
2.066
2.166
3.000
1.
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3.
4.
The root of equation
f
(x)=0 is found by using Newton-Raphson method. The initial estimate of the root is
x
o =3,
f
(3)=5. The angle the tangent to the function
f
(x) makes at
x
=3 is 57 o . The next estimate of the root,
x
1 most nearly is A. -3.2470
B.
C.
D.
-0.2470
3.2470
6.2470
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The Newton-Raphson method formula for finding the square root of a real number
R
the equation
x
2
-R
=0 is,
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from
25%
A.
x i
1
x i
2 B.
C.
D.
x i
1 3
x i
2
x i
1 1 2
x i
R x i
x i
1 1 2 3
x i
R x i
1.
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3.
4.
END http://nm.mathforcollege.com
Bisection Method
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Bisection method of finding roots of nonlinear equations falls under the category of a (an) method.
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A. open B. bracketing C. random D. graphical
1.
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3.
4.
If for a real continuous function
f(x)
,
f
(
a
)
f
(
b
)<0, then in the range [
a,b
] for
f
(
x
)=0, there is (are)
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A.
B.
C.
D.
one root undeterminable number of roots no root at least one root
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2.
3.
4.
The velocity of a body is given by
v
(
t
)
=5e -t +4
, where
t
is in seconds and
v
is in m/s. We want to find the time when the velocity of the body is 6 m/s. The equation form needed for bisection and Newton-Raphson methods is
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A.
B.
C.
D.
f (t)= 5e -t +4=0 f (t)= 5e -t +4=6 f (t)= 5e -t =2 f (t)= 5e -t -2=0
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To find the root of an equation
f
(
x
)
=0
, a student started using the bisection method with a valid bracket of [20,40]. The smallest range for the absolute true error at the end of the 2 nd iteration is
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A. 0 ≤ |E t |≤2.5
B. 0 ≤ |E t | ≤ 5 C. 0 ≤ |E t | ≤ 10 D. 0 ≤ |E t | ≤ 20
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2.
3.
4.
For an equation like
x 2 =
0, a root exists at
x=
0. The bisection method cannot be adopted to solve this equation in spite of the root existing at
x
=0 because the function
f
(
x
)
=x
2
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A.
B.
C.
D.
is a polynomial has repeated zeros at
x
=0 is always non-negative slope is zero at
x
=0
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END http://numericalmethods.eng.usf.edu
How and Why?
Study Groups Help?
Studying Alone Studying with Peers
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I walk like a pimp – Jeremy Reed You know it's hard out here for a pimp, When he tryin to get this money for the rent, For the Cadillacs and gas money spent
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END http://nm.mathforcollege.com
Final Grade vs. First Test Grade
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A New Book on How Brain Works The Compass of Pleasure: How Our Brains Make Fatty Foods, Orgasm, Exercise, Marijuana, Generosity, Vodka, Learning, and Gambling Feel So Good
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