1. 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 http://numericalmethods.eng.usf.edu Numerical Methods for the STEM undergraduate

2. 2 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. Figure Diagram of the floating ball

3. 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 30

4. 4 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?

5. 5 Finding The Temperature of the Fluid T a = 80 o F T c = ??? o F D = 12.363" ∆D = -0.015"

6. 6 Finding The Temperature of the Fluid T a = 80 o F T c = ??? o F D = 12.363" ∆D = -0.015"

7. Nonlinear Equations (Background) http://numericalmethods.eng.usf.edu Numerical Methods for the STEM undergraduate http://numericalmethods.eng.usf.edu

8. How many roots can a nonlinear equation have?

12. The value of x that satisfies f (x)=0 is called the 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

13. A quadratic equation has ______ root(s) 1.one 2. two 3. three 4. cannot be determined

14. 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.one 2. two 3. three 4.cannot be determined

15. Equation such as tan (x)=x has __ root(s) 1.zero 2. one 3. two 4. infinite

16. A polynomial of order n has zeros 1.n -1 2. n 3. n +1 4. n +2

17. END http://numericalmethods.eng.usf.edu Numerical Methods for the STEM undergraduate http://numericalmethods.eng.usf.edu

18. Bisection Method http://numericalmethods.eng.usf.edu Numerical Methods for the STEM undergraduate http://numericalmethods.eng.usf.edu

19. Bisection method of finding roots of nonlinear equations falls under the category of a (an) method. 1.open 2. bracketing 3. random 4. graphical

20. 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.one root 2. undeterminable number of roots 3. no root 4. at least one root

21. 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.f (t)= 5e -t +4=0 2.f (t)= 5e -t +4=6 3.f (t)= 5e -t =2 4.f (t)= 5e -t -2=0

22. 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 1.0 ≤ |E t |≤2.5 2.0 ≤ |E t | ≤ 5 3.0 ≤ |E t | ≤ 10 4.0 ≤ |E t | ≤ 20

23. 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 1. is a polynomial 2. has repeated zeros at x=0 3. is always non-negative 4. slope is zero at x=0

24. END http://numericalmethods.eng.usf.edu Numerical Methods for the STEM undergraduate http://numericalmethods.eng.usf.edu

25. Newton Raphson Method http://numericalmethods.eng.usf.edu Numerical Methods for the STEM undergraduate http://numericalmethods.eng.usf.edu

26. Newton-Raphson method of finding roots of nonlinear equations falls under the category of __________ method. 1.bracketing 2. open 3. random 4. graphical

27. 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. 1.500 2. 2.066 3. 2.166 4. 3.000

28. 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 1.-3.2470 2. -0.2470 3. 3.2470 4. 6.2470

29. The Newton-Raphson method formula for finding the square root of a real number R from the equation x 2 -R=0 is, 1. 2. 3. 4.

30. END http://numericalmethods.eng.usf.edu Numerical Methods for the STEM undergraduate http://numericalmethods.eng.usf.edu