nonlinear equations your nonlinearity confuses me
DESCRIPTION
Nonlinear Equations Your nonlinearity confuses me. “The problem of not knowing what we missed is that we believe we haven't missed anything” – Stephen Chew on Multitasking. http://numericalmethods.eng.usf.edu Numerical Methods for the STEM undergraduate. - PowerPoint PPT PresentationTRANSCRIPT
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Nonlinear EquationsYour nonlinearity confuses me
02345 fexdxcxbxax
“The problem of not knowing what we missed is that we believe we haven't missed anything” – Stephen Chew on Multitasking
xx )tanh(
http://numericalmethods.eng.usf.eduNumerical Methods for the STEM undergraduate
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Identify the picture of your EML3041 instructor
A. B. C. D.
25% 25%25%25%A.
B.
C.
D.
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3
Example – General EngineeringYou 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.
Figure Diagram of the floating ball
010993.3165.0 423 xx
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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. 2.
50%50%1. Yes2. No
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5
Example – Mechanical Engineering
Since the answer was a resounding NO, a logical question to ask would be:
If the temperature of -108oF is not enough
for the contraction, what is?
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6
Finding The Temperature of the Fluid
dTTDDc
a
T
T
)(
Ta = 80oFTc = ???oFD = 12.363"∆D = -0.015" -350 -300 -250 -200 -150 -100 -50 0 50 100
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
T
3528 10349.610457.710992.5015.0 cc TT
TT 009696.0033.6
010651.810457.710992.5)( 3528 ccc TTTf
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7
Finding The Temperature of the Fluid
dTTDDc
a
T
T
)(
Ta = 80oFTc = ???oFD = 12.363"∆D = -0.015"
352639 10166.610435.710829.310059.5015.0 ccc TTT
015.610195.610228.1 325 TT
010834.810435.710829.310059.5)( 352639 cccc TTTTf
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Nonlinear Equations(Background)
http://numericalmethods.eng.usf.eduNumerical Methods for the STEM undergraduate
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How many roots can a nonlinear equation have?
-20 -10 0 10 20-3
-2
-1
0
1
2
3sin(x)=2 has no roots
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How many roots can a nonlinear equation have?
-20 -10 0 10 20-3
-2
-1
0
1
2
3sin(x)=0.75 has infinite roots
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How many roots can a nonlinear equation have?
-20 -10 0 10 20-3
-2
-1
0
1
2
3sin(x)=x has one root
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How many roots can a nonlinear equation have?
-20 -10 0 10 20-3
-2
-1
0
1
2
3sin(x)=x/2 has finite number of roots
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The value of x that satisfies f (x)=0 is called the
1. 2. 3. 4.
0% 0%0%0%
1. root of equation f (x)=0 2. root of function f (x) 3. zero of equation f (x)=0 4. none of the above
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A quadratic equation has ______ root(s)
1. 2. 3. 4.
0% 0%0%0%
1. one2. two3. three4. cannot be determined
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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
1. 2. 3. 4.
0% 0%0%0%
1. one2. two3. three4. cannot be
determined
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Equation such as tan (x)=x has __ root(s)
1. 2. 3. 4.
0% 0%0%0%
1. zero2. one3. two4. infinite
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A polynomial of order n has zeros
1. 2. 3. 4.
0% 0%0%0%
1. n -12. n3. n +14. n +2
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The velocity of a body is given by v (t)=5e-t+4, where t is in seconds and v is in m/s. The velocity of the body is 6 m/s at t =
1. 2. 3. 4.
0% 0%0%0%
1. 0.1823 s2. 0.3979 s3. 0.9162 s4. 1.609 s
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END
http://numericalmethods.eng.usf.eduNumerical Methods for the STEM undergraduate
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Newton Raphson Method
http://numericalmethods.eng.usf.eduNumerical Methods for the STEM undergraduate
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This is what you have been saying about your TI-30Xa
A. B. C. D.
25% 25%25%25%A. I don't care what people say The rush is worth the priceI pay I get so high when you're with meBut crash and crave you when you are away
B. Give me back now my TI89Before I start to drink and whine TI30Xa calculators you make me cryIncarnation 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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Newton-Raphson method of finding roots of nonlinear equations falls under the category of __________ method.
1. 2. 3. 4.
0% 0%0%0%
1. bracketing2. open3. random4. graphical
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The next iterative value of the root of the equation x 2=4 using Newton-Raphson method, if the initial guess is 3 is
1. 2. 3. 4.
0% 0%0%0%
1. 1.5002. 2.0663. 2.1664. 3.000
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The root of equation f (x)=0 is found by using Newton-Raphson method. The initial estimate of the root is xo=3, f (3)=5. The angle the tangent to the function f (x) makes at x=3 is 57o. The next estimate of the root, x1 most nearly is
1. 2. 3. 4.
0% 0%0%0%
1. -3.24702. -0.24703. 3.24704. 6.2470
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The Newton-Raphson method formula for finding the square root of a real number R from the equation x2-R=0 is,
1. 2. 3. 4.
25% 25%25%25%
21i
i
xx
2
31
ii
xx
iii x
Rxx
2
11
iii x
Rxx 3
2
11
1.
2.
3.
4.
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END
http://numericalmethods.eng.usf.eduNumerical Methods for the STEM undergraduate
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Bisection Method
http://numericalmethods.eng.usf.eduNumerical Methods for the STEM undergraduate
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Bisection method of finding roots of nonlinear equations falls under the category of a (an) method.
1. 2. 3. 4.
0% 0%0%0%
1. open2. bracketing3. random4. graphical
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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)
1. 2. 3. 4.
0% 0%0%0%
1. one root2. undeterminable number of roots3. no root4. at least one root
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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
1. 2. 3. 4.
25% 25%25%25%1. f (t)= 5e-t+4=02. f (t)= 5e-t+4=63. f (t)= 5e-t=24. 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 2nd iteration is
1. 2. 3. 4.
25% 25%25%25%1. 0 ≤ |Et|≤2.5
2. 0 ≤ |Et| ≤ 5
3. 0 ≤ |Et| ≤ 10
4. 0 ≤ |Et| ≤ 20
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For an equation like x2=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
1. 2. 3. 4.
0% 0%0%0%
1. is a polynomial2. has repeated zeros at
x=03. is always non-negative4. slope is zero at x=0
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END
http://numericalmethods.eng.usf.eduNumerical Methods for the STEM undergraduate
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How and Why?
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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://numericalmethods.eng.usf.eduNumerical Methods for the STEM undergraduate
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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