Reference no: EM132402940
Question 1 - Consider the solution of the following nonlinear equation using Newton's method.
cos(1.1 x) = ½ x4 - 46.6976.
After two iterations, using ten digit floating point arithmetic, the estimated solution is: x2 = 3.1028670886.
Using Newton's method find the next estimate of the solution and use this to determine an absolute estimate of the error in x2.
Maintain at least TEN significant digits throughout all your calculations.
When entering your results you may round your answers to eight decimal digits.
Question 2 - Apply three iterations of Newton's method to find an approximate solution of the equation
0.6x4 = 4 - x
if your initial estimate is x0 = 2.03
Maintain at least eight digits throughout all your calculations.
When entering your final result you MAY round your estimate to five decimal digit accuracy. For example 1.67353.
YOU DO NOT HAVE TO ROUND YOUR FINAL ESTIMATE.
Question 3 - Starting with interval [1, 1.5] how many iterations of the bisection method are required to find an estimate correct to 7 decimal places? Find the minimum number required. Your answer should be a positive integer.
Question 4 - When finding the root of f(x) = 0 on the interval [a, b] using the bisection method we know that f(a) > 0 and f(b) < 0. If c is the midpoint of the interval [a, b] and f(c) > 0 then in the bisection method we would? Which statement is correct?
A. The root is between c and b, so we put a = c and go to the next iteration.
B. The root is between a and c, so we put b = c and go to the next iteration.
C. The root is between c and b, so we put b = c and go to the next iteration.
D. The root is between a and c, so we put a = c and go to the next iteration.
E. None of the above
Question 5 - Consider the equation 8cosx = 1 - 2e-x/7.
Perform 4 iterations of the bisection method to find an approximate solution in the interval [1.6, 1.75]. That is, do four bisections. Enter your answer in decimal form. Your answer should be exact.
Question 6 - Find the line of best fit in a least squares sense for the following data values (all points have equal weight):
|
Data Values
|
|
Xi
|
0.96
|
1.1
|
1.16
|
1.24
|
1.48
|
1.56
|
|
Yi
|
3.14
|
2.8
|
2.39
|
2.04
|
2.39
|
2.35
|
The line of best fit will be in the form: y = α + βx.
Enter the values of α and β.
When entering your final answers you may round your estimate to five decimal digit accuracy.