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A homework assignment from a geosystems engineering course, pge310, focusing on finding the zeros of nonlinear functions using analytical techniques and programming concepts. Students are required to find the exact and approximate roots of given functions using graphical methods, limits, and numerical methods such as newton's method and the bisection method. The assignment also includes tasks related to the natural frequencies of a uniform beam and numerical integration.
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February 28, 2001 Due on March 7, 2001
PGE310 (Unique No. 17135) Spring Semester 2001 Formulation and Solution of Geosystems Engineering Problems
Instructor: Carlos Torres-Verdín
The objective of this homework assignment is to test, reinforce, and exercise both analytical techniques and programming concepts used in class to describe practical ways in which one can find the zeros of a single-valued real nonlinear function. The last problem is intended as an introduction to the subject of numerical integration.
Homework assignments will be submitted to the instructor at the beginning of the lecture when they are due. Late homework will not be counted toward the final course’s homework grade; this includes homework submitted at the end of the lecture rather than at the beginning.
Submit your homework assignments in clean and letter-size paper. You are encouraged to use standard engineering sheets whenever possible. Write down the assignment number, your name, and date of the assignment on every sheet of paper you submit and staple them together. Clearly identify the question for which you are providing an answer and organize your answers in the same sequential order of the questions provided to you. Show all of your work in order to receive full credit for a given answer. Most importantly, develop your answers in an organized, clear, and tidy manner. Neatness will most definitely count toward assigning a grade to your computer project.
Graded homework projects will be returned to you promptly. Moreover, homework solutions will be posted on our course’s web site for your perusal. Should you disagree with your homework grade, please contact the course’s TA as quickly as possible for a clarification. Consult the homework solutions as much as possible for clues on solution strategies and ways of thinking. Exam questions will be designed in the same way homework assignments are. Therefore, it is to your advantage to understand and properly solve all of the homework assignments.
1. Give a graphical demonstration that the function tan( x )= x has infinitely many roots. Determine one root precisely and another approximately by using a graph. 2. Let p be a positive number. What is the value of
x = p + p + p + K? Note that this can be interpreted as meaning x = lim x n , where x 1 (^) = p , x (^) 2 = p + p , and so on. Observe that xn (^) + 1 = p + x n.
3. Let p > 1. What is the value of
p
p
x
1
Use the ideas of the preceding problem to solve this one.
4. Find the largest staring point for which Newton’s method will converge using f ( x )= tan−^1 ( x ). 5. Adapt your codes of Computer Project No. 3 to find the zero of each one of the following functions, using the bisection method: (a) x^3 − 3 x + 1 on the interval [ 0 , 1 ].
(b) x^3 − 2 sin( x )on the interval (^) ú û
ù êë
é , 2 2