Nonlinear Equations Your nonlinearity confuses me 0 2 3 4 5 f ex dx cx bx ax “The problem of not knowing what we missed is that we believe we haven't missed anything” – Stephen Chew on Multitasking x x ) tanh( http://numericalmethods.eng.usf.edu Numerical Methods for the STEM undergraduate
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 Presentation
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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
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
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
5
Example – Mechanical Engineering
Since the answer was a resounding NO, a logical question to ask would be:
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!
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
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
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
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.
END
http://numericalmethods.eng.usf.eduNumerical Methods for the STEM undergraduate
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
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
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
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
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
END
http://numericalmethods.eng.usf.eduNumerical Methods for the STEM undergraduate