Let's plot the function f(x)=e^ on the interval [0,3] using Excel and Matlab. The simplest approximation...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
Let's plot the function f(x)=e^ on the interval [0,3] using Excel and Matlab. The simplest approximation to a function is the 0th order polynomial (a constant). Assume f(x) is constant on the interval [a,b] and has the value f(a) there. One can construct a rectangle of a height f(a) to illustrate a constant function on the interval [a,b]. Problem 3b (10 points). above Plot the rectangle of a height f(0) on the graph of Problem on the interval [0,3] along with the original function. What is the area of your rectangle? The area of the rectangle is an estimate of the desired integral. From your plot, how accurate do you think your estimate is (compared to the area under the original function)? The approximation to the integral of the original function you have just developed is called the "Rectangular Rule" A more accurate approximation is a 1st order (linear) polynomial of the form f(x) = a+bx. This polynomial has two unknown coefficients (a and b). These coefficients can be determined using the function values at two points by solving the system of two linear equations with two unknowns. Typically, the points at the ends of the interval are used. In our case these are the points (0,f(0)) and (3.f(3)). (In graphical terms, to define a linear polynomial means to draw a straight line through these two points). Problem 3c (10 points). Build the straight line between the two end points of the interval [0,3]. The x-axis, the vertical lines x = 0 and x = 3, and the constructed line form a trapezoid. Plot the trapezoid on the graph. The area of the trapezoid is a more accurate approximation to the integral of f(x) on [0,3]. What is the approximate value of the integral? How accurate do you think it is? The approximation you have just developed is called the "Trapezoidal Rule". Moving up the scale of polynomials the next order of approximation utilizes the 2nd order (quadratic) polynomial of the form f(x) = a + bx + cx. This polynomial has 3 unknown coefficients. To calculate the value of these coefficients we need the values of the function at three points. Typically, the ends of the interval and the middle of the interval are used. In our case we will use the points (0,f(0)), (1.5,f(1.5)), and (3,f(3)). Problem 3d (10 points). Find the coefficients a,b,c for the quadratic polynomial that passes through the three specified points: (0,f(0)), (1.5,f(1.5)), and (3,f(3)). You will have to solve the 3x3 system of equations to accomplish that (3 equations with 3 unknowns). Plot the quadratic polynomial along with the graph of the original function. The area under the quadratic polynomial is an approximation to the integral of f(x). Integrate your quadratic polynomial to obtain the numerical value of the integral. (Recall the rule for integrating polynomials. If you do not remember how to integrate polynomials - let me know and I will help.) How accurate do you think the obtained approximation is? The case you have just developed is called "Simpson's Rule". SYSE 5350, Spring 2018, Exercise 1, Part II 2 Error Analysis: Without going through the numerical analysis derivations the following error results are stated. Consider your function on the interval (a,b). The error of the Rectangular Rule is O[(ba)2]. This notation means that the error is proportional to the square of the interval length. The Trapezoidal Rule has an error of O[(b-a)], while Simpson's Rule error is O[(b- a)5]. How can one increase the accuracy of such approximations? The only control we have over increasing the accuracy (reducing the error) of the numerical integration estimate when using the above-mentioned methods is by shortening the length of the interval: If (ba) is small ( <1) then raising the quantity to a power further reduces the term and hence reduces the error. Simpson's Composite Rule: The general expression for the integral of a function, f(x) using the basic Simpson's Rule is: b-a 6 {(a)+4(a+b) + (b)} Problem 3e (10 points). Verify that this expression yields the same resulting value for the integral of our exponential function as Problem 3d over the interval (0,3). Let's plot the function f(x)=e^ on the interval [0,3] using Excel and Matlab. The simplest approximation to a function is the 0th order polynomial (a constant). Assume f(x) is constant on the interval [a,b] and has the value f(a) there. One can construct a rectangle of a height f(a) to illustrate a constant function on the interval [a,b]. Problem 3b (10 points). above Plot the rectangle of a height f(0) on the graph of Problem on the interval [0,3] along with the original function. What is the area of your rectangle? The area of the rectangle is an estimate of the desired integral. From your plot, how accurate do you think your estimate is (compared to the area under the original function)? The approximation to the integral of the original function you have just developed is called the "Rectangular Rule" A more accurate approximation is a 1st order (linear) polynomial of the form f(x) = a+bx. This polynomial has two unknown coefficients (a and b). These coefficients can be determined using the function values at two points by solving the system of two linear equations with two unknowns. Typically, the points at the ends of the interval are used. In our case these are the points (0,f(0)) and (3.f(3)). (In graphical terms, to define a linear polynomial means to draw a straight line through these two points). Problem 3c (10 points). Build the straight line between the two end points of the interval [0,3]. The x-axis, the vertical lines x = 0 and x = 3, and the constructed line form a trapezoid. Plot the trapezoid on the graph. The area of the trapezoid is a more accurate approximation to the integral of f(x) on [0,3]. What is the approximate value of the integral? How accurate do you think it is? The approximation you have just developed is called the "Trapezoidal Rule". Moving up the scale of polynomials the next order of approximation utilizes the 2nd order (quadratic) polynomial of the form f(x) = a + bx + cx. This polynomial has 3 unknown coefficients. To calculate the value of these coefficients we need the values of the function at three points. Typically, the ends of the interval and the middle of the interval are used. In our case we will use the points (0,f(0)), (1.5,f(1.5)), and (3,f(3)). Problem 3d (10 points). Find the coefficients a,b,c for the quadratic polynomial that passes through the three specified points: (0,f(0)), (1.5,f(1.5)), and (3,f(3)). You will have to solve the 3x3 system of equations to accomplish that (3 equations with 3 unknowns). Plot the quadratic polynomial along with the graph of the original function. The area under the quadratic polynomial is an approximation to the integral of f(x). Integrate your quadratic polynomial to obtain the numerical value of the integral. (Recall the rule for integrating polynomials. If you do not remember how to integrate polynomials - let me know and I will help.) How accurate do you think the obtained approximation is? The case you have just developed is called "Simpson's Rule". SYSE 5350, Spring 2018, Exercise 1, Part II 2 Error Analysis: Without going through the numerical analysis derivations the following error results are stated. Consider your function on the interval (a,b). The error of the Rectangular Rule is O[(ba)2]. This notation means that the error is proportional to the square of the interval length. The Trapezoidal Rule has an error of O[(b-a)], while Simpson's Rule error is O[(b- a)5]. How can one increase the accuracy of such approximations? The only control we have over increasing the accuracy (reducing the error) of the numerical integration estimate when using the above-mentioned methods is by shortening the length of the interval: If (ba) is small ( <1) then raising the quantity to a power further reduces the term and hence reduces the error. Simpson's Composite Rule: The general expression for the integral of a function, f(x) using the basic Simpson's Rule is: b-a 6 {(a)+4(a+b) + (b)} Problem 3e (10 points). Verify that this expression yields the same resulting value for the integral of our exponential function as Problem 3d over the interval (0,3).
Expert Answer:
Related Book For
Computer Architecture A Quantitative Approach
ISBN: 9780128119051
6th Edition
Authors: John L. Hennessy, David A. Patterson
Posted Date:
Students also viewed these mathematics questions
-
CANMNMM January of this year. (a) Each item will be held in a record. Describe all the data structures that must refer to these records to implement the required functionality. Describe all the...
-
This question concerns lexical grammars. (a) Tree Adjoining Grammars contain two types of elementary tree. (i) What are these trees called? [1 mark] (ii) If one were building a grammar for English...
-
Microkernel operating systems aim to address perceived modularity and reliability issues in traditional "monolithic" operating systems. (i) Describe the typical architecture of a microkernel...
-
In your hometown what system is used to price the publicly supplied water? Why was that pricing system chosen? Would you recommend an alternative?
-
Refer back to Problem 3-72B. Requirements 1. Use the Spa View Services data in Problem 3-72B to prepare the company's classified balance sheet at January 31, 2016. Show captions for total assets,...
-
The germination time in days of a newly planted seed is exponentially distributed with parameter = 0.31. If the germination times of different seeds are independent of one another, estimate the...
-
Do stakeholders influence environmental accounting systems? Do environmental accounting systems influence stakeholders? Provide an example to illustrate your argument.
-
Rhodes, Inc., is a fast-growing start-up firm that manufactures bicycles. The following income statement is available for July: Sales revenue (200 units @ $500 per unit) . . . . . . . $100,000 Less...
-
Find /(524 + (5x4 + 4x6) dx Question Help: Video + C
-
Respond to the following: Tyler Corporation has provided you with the following budgeted income statement for one of their products: Tyler Corporation believes that 80% of the fixed costs would be...
-
What is the output of the following application? A. true B. false C. The code does not compile. D. The result is unknown until runtime. E. An exception is thrown. F. None of the above. import...
-
Suppose you have the following class in a module named animal.insect.impl. Which two most likely go in the module-info of the service locator? (Choose two.) A. requires animal.insect.api.bugs; B....
-
Which statement about the following method is correct? Assume the directory coffee exists and is able to be read. A. It does not compile. B. It compiles but does not print anything at runtime. C. It...
-
The following code snippet results in an exception at runtime. Which of the following is the most likely type of exception to be thrown? A. AtomicMoveNotSupportedException B....
-
Which of the following is most likely to be caused by a race condition? A. A thread perpetually denied access to a resource B. A program hanging indefinitely C. An int variable incorrectly reporting...
-
How would using malate for the energy source in the citric acid cycle effect hoe many ATP could be produced. please show work/ explain in detail.
-
Where are the olfactory sensory neurons, and why is that site poorly suited for their job?
-
A large multimegabyte L3 cache can take tens of cycles to access because of the long wires that have to be traversed. For example, it may take 20 cycles to access a 16 MB L3 cache. Instead of...
-
You are designing a PMD and optimizing it for low energy. The core, including an 8 KB L1 data cache, consumes 1 W whenever it is not in hibernation. If the core has a perfect L1 cache hit rate, it...
-
The TPU uses fixed-point arithmetic (sometimes also called quantized arithmetic, with overlapping and conflicting definitions), where integers are used to represent values on the real number line....
-
Review the sample scope statement in this chapter. Assume you are responsible for planning and then managing the deliverable called survey development. What additional information would you want to...
-
You are part of a team in charge of a project to help people in your company (500 people) lose weight. This project is part of a competition, and the top losers will be featured in a popular...
-
Your organization initiated a project to raise money for an important charity. Assume that there are 1,000 people in your organization. Also assume that you have six months to raise as much money as...
Study smarter with the SolutionInn App