mathematical methods of engineers

7/30/2019 Mathematical Methods of Engineers

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MME1 (MATH 1063) Molly QU, SAIBT, 20111

Mathematical Methods for Engineers 1 (MATH 1063)

Calculus 1

Assignment 2, Trimester 1, 2011

Due by 4pm on Wednesday, 11 May 2011.

Submit this assignment to Student Centre by 4pm on the due date. Your assignment must

have a cover sheet, yoursolutions and MATLAB code, graphs and output answers.

Also you need to publish your MATLAB code into a Word document and submit to

Assignment on Moodle. You will lose marks if just copy your code and graph into

Word without any comments for yourcode or output answers. The entire graph should

include the title, legend, and labels. Please ensure that all pages with page numbers are

securely attached and your full name is on the front page. Only A4 size papers should be

used. The solutions should be quite clear; you should not round off and give a decimal

approximation. Assignments handed up late without a previously negotiated extension

will be penalised at a rate of5% per day or part thereof. You may discuss your work

with others, but your written solutions should be your own work. Straight copying is

forbidden and is usually not helpful anyway, also your will get zero mark for your

assignment. Hence any joint work must be indicated. Dont forget to use the text,

Edwards and Penney, for reference. Make sure you learn any material from the text that

you use in this assignment. This assignment will be marked forcompleteness, accuracy,

clarity and correct conclusions. Bonus marks for good presentation and neat

handwriting.


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1. a) Use the scalar triple product to prove that the four points below are all coplanar:

b) Find the Volume of the parallelepiped with vertices:

2. a) Show that there is one and only one solution of the cubic equation

.

b) Demonstrate you understanding of the Mean Value Theorem by applying it to

the function , on the interval 0, 5[ ].

3. Find the point on the straight line closest to the point , using the

techniques of calculus.

4. Suppose

1)Expand manually and find all critical points where . All of these

critical points separate the x-axis into a few open intervals, find all the open

intervals and determine the increasing or decreasing behavior of on these

intervals.

2)Use MATLAB to check your solutions from 1).


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a.Use Matlab Polynomial commands to expand and convert it into an

equivalent Matlab array. Find the roots numerically and re-assemble the

polynomial coefficients from the roots and then change back into a symbolic

form. Plot where and .

b.Use Symbolic Toolbox commands to find the first derivative of and format

output to look like type-set mathematics and compare this with your

solution from 1). Factor then solve by using Symbolic

Toolbox commands to check all critical points you found manually.

c.Plot where and and compare your critical points

and increasing or decreasing intervals with the graph.

3)Manually find the second derivative and all points of inflection then find

intervals where the f(x) is concave up or down.

4)Use Symbolic Toolbox commands to find the second derivative of and

format output to look like type-set mathematics and compare this with

your solution from 3). Factor then solve by using

Symbolic Toolbox commands to check all points of inflection you found

manually. Plot where , .

5)Show all important features on graph . (All critical points and points of

inflection and local minima and maxima.)


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5. An object initially at the origin moves away with velocity .

1) Explain how we know the object is always moving to right, and what happens

to the velocity as ttends to infinity?

2) What is the average velocity during the first seconds? (Refer to Edwards

and Penney for the definition of the average value of a function.)

3) Find the distance traveled by the object as t tends to infinity. (Hint: find

and allow Tto tend to infinity.)

4) Use MATLAB to graph the physical location of the object for the first 10

seconds of its motion.

6. Evaluate the following limits by recognizing the sum as a Riemann sum

associated with a regular partition of and evaluating the associated integral.

1)

2)

7. Use the substitution to evaluate the given integral and check your solutions with

MATLAB integral command.


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1)

2)

3)

4)

Note: All graphs should have title, legend and labels for x-axis and y-axis.

mathematical methods of engineers

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