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 Numerical Methods for the STEM undergraduate

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

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 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 Finding The Temperature of the Fluid T a = 80 o F T c = ??? o F D = " ∆D = "

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

Nonlinear Equations (Background) Numerical Methods for the STEM undergraduate

How many roots can a nonlinear equation have?

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

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

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

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

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

END Numerical Methods for the STEM undergraduate

Bisection Method Numerical Methods for the STEM undergraduate

Bisection method of finding roots of nonlinear equations falls under the category of a (an) method. 1.open 2. bracketing 3. random 4. 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.one root 2. undeterminable number of roots 3. no root 4. 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.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

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 |≤ ≤ |E t | ≤ ≤ |E t | ≤ ≤ |E t | ≤ 20

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

END Numerical Methods for the STEM undergraduate

Newton Raphson Method Numerical Methods for the STEM undergraduate

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

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

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

END Numerical Methods for the STEM undergraduate