Chabot College
Fall 2010
Course Outline for Mathematics 25
COMPUTATIONAL METHODS FOR ENGINEERS AND SCIENTISTS
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Catalog Description:
MTH 25 - Computational Methods for Engineers and Scientists
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3.00 units
May be repeated 2 times
Methodology and techniques for solving engineering/science problems using numerical-analysis computerapplication programs MATLAB and EXCEL. Technical computing and visualization using MATLAB software.
Examples and applications from applied-mathematics, physical-mechanics, electrical circuits, biology,
thermal systems, fluid systems, and other branches of science and engineering.
Prerequisite: MTH 1
Strongly Recommended: CSCI 8 May not receive credit if Mathematics 25 or Physics 25 has been
completed.
Units
Contact Hours
Week
Term
3.00
Lecture
Laboratory
Clinical
Total
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3.00
2.00
3.00
0.00
5.00
35.00
52.50
0.00
87.50
Prerequisite Skills:
Before entry into this course, the student should be able to:
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use delta notation;
explain limits and continuity;
use Newton’s method;
apply the definition of the derivative of a function;
define velocity and acceleration in terms of mathematics;
differentiate algebraic and trigonometric functions;
apply the chain rule;
find all maxima, minima and points of inflection on an interval;
sketch the graph of a differentiable function;
apply implicit differentiation to solve related rate problems;
apply the Mean Value Theorem;
find the value of a definite integral as the limit of a Riemann sum;
differentiate appropriate functions using the Fundamental Theorem of Integral Calculus;
integrate using the substitution method;
find the volume of a solid of revolution using the shell, disc, washer methods;
find the volume of a solid by slicing;
find the work done by a force;
find the hydrostatic force on a vertical plate;
find the center of mass of a plane region;
approximate a definite integral using Simpson’s Rule and the Trapezoidal Rule.
Expected Outcomes for Students:
Upon completion of this course, the student should be able to:
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analyze engineering/science word problems to formulate a mathematical model of the problem;
express in MATLAB notation: scalars, vectors, matrices;
perform, using MATLAB or EXCEL, mathematical operations on vectors, scalars, and matrices
a. addition and subtraction;
b. multiplication and addition;
c. exponentiation;
compute, using MATLAB or EXCEL, the numerical-value of standard mathematical functions
a. trigonometric functions;
b. exponential functions;
c. square-roots and absolute values;
create, store, and run MATLAB script files;
import data to MATLAB for subsequent analysis from data-sources
a. data-acquisition-system data-files;
b. spreadsheet files;
construct graphical plots for mathematical-functions in two or three dimensions;
formulate a fit to given data in terms of a mathematical curve, or model, based on linear,
polynomial, power, or exponential functions
a. assess the goodness-of-fit for the mathematical model using regression analysis;
apply MATLAB to find the numerical solution to systems of linear equations
a. uniquely determined;
b. underdetermined;
c. overdetermined;
perform using MATLAB or EXCEL statistical analysis of experimental data to determine the mean,
median, standard deviation, and other measures that characterize the nature of the data ;
compute, for empirical or functional data, numerical definite-integrals and discrete-point derivatives;
solve numerically, using MATLAB, linear, second order, constant-coefficient, nonhomogenous
ordinary differential equations;
assess, symbolically, using MATLAB
a. the solution to transcendental equations;
b. derivatives, antiderivatives, and integrals;
c. solutions to ordinary differential equations;
apply, using EXCEL, linear regression analysis to xy data-sets to determine for the best-fit line the:
slope, intercept, and correlation-coefficient;
draw using MATLAB or EXCEL two-dimensional Cartesian (xy) line-plots with multiple data-sets
(multiple lines);
draw using EXCEL qualitative-comparison charts such as Bar-Charts and Column-Charts in two or
three dimensions;
perform, using MATLAB and EXCEL, mathematical-logic operations;
plan, conceptually, computer-solutions to engineering/science problems using psuedocode and/or
flow-chart methods;
compose MATLAB script files that employ FOR and WHILE loops to solve engineering/science
problems that require repetitive actions.
Course Content:
Course Content (Lecture):
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Engineering problem solving
A. construct Physical Model
B. construct Mathematical Model
C. solve using graphic-geometrical, analytical, or numerical methods
Using MATLAB
A. user interface and working-environment
B. writing MATLAB software-code using script or function files (“.m” files)
C. MATLAB Mathematical functions, including logical operations
D. graphical output
MATLAB linear algebra using arrays and matrices
A. array and Matrix mathematical operations: addition/subtraction, multiplication/division
exponentiation, transpose, inversion
MATLAB Files and Data Structures
A. importing data from ASCII and spreadsheet files
B. complex number formats, rectangular and polar
Programming with MATLAB
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A. psuedocoding (written-english description of the intended program-function)
B. basic flow charting
C. conditional Branching using if/then/else techniques
D. fixed-termination loops (FOR loops)
E. dynamically-terminated conditional loops (WHILE loops)
F. DeBugging MATLAB programs
MATLAB graphical-output and curve-fitting
A. two dimensional Cartesian (XY) plots using multiple data-sets with proper scaling and
labeling
a. linear-linear
b. log-linear (semilog)
c. log-log
B. data-set curve fitting with regression analysis
a. linear
b. polynomial/power function
c. exponential function
C. three dimensional plots: line, surface mesh, contour
solutions to systems of linear equations
A. gaussian elimination
B. matrix inversion decomposition
C. cramer’s method
D. underdetermined systems and the minimum-norm solution
E. overdetermined systems and the least-squares solution
MATLAB and EXCEL statistical analysis for empirical data
A. calculate standard statistical metrics: mean, median, mode, standard deviation, minimum,
maximum, range
B. generate random numbers
C. linear interpolation
MATLAB numerical integration and differentiation
A. trapezoidal integration
B. simpson’s rule integration
C. numerical differentiation
a. forward difference
b. backward difference
c. central difference
MATLAB solutions for ordinary differential equations
A. Runge-Kutta based ODE solvers
a. stiff and nonstiff systems
b. low, medium, and variable order solvers
MATLAB symbolic mathematics
A. mathematical-expressions and algebra
B. solve algebraic and transcendental equations
C. ordinary and partial differentiation
D. antiderivatives and definite integrals
E. solve linear and nonlinear ordinary differential equations
EXCEL user-interface and working-environment
EXCEL mathematical functions including logical operations
EXCEL graphical output
A. bar and column charts
B. line and xy plots
C. three dimensional surface plots
EXCEL statistical analysis, and curve fitting including linear regression
Course Content (Laboratory):
16. Use MATLAB’s interactive mode to compute numerical results for scalar, vector, matrix, can array
operations
17. compose user-defined, special-purpose, MATLAB programs
18. plan computer-code (“m” file) solutions to engineering/science problems using
A. psueodcode
B. diagrams and/or tables
C. flow-charts
19. design, in detail, computer-code solutions to engineering/science problems using – run, test, and
check the solutions
20. import into MATLAB and EXCEL data generated by data-acquisition systems to
A. analyze the data statistically
B. evaluate the data using visualization tools such as plots, charts, and tables
21. compose MATLAB script-files to solve engineering/science problems that employ
A. logical variables
B. relational operations
C. FOR and/or WHILE loops
D. user-defined MATLAB functions
22. Create from experimental or theoretical data visualization objects
A. XY Cartesian plots: linear, semilog, and log-log
B. XYZ three-dimensional plots
C. Comparison charts: bar, column, pie, area, line, radar
D. Histograms
23. Numerically Determine using MATLAB the maxima, minima, and zeros of single-variable functions
24. Use MATLAB to solve numerically and symbolically systems of linear equations
25. Use MATLAB to solve numerically and symbolically ordinary differential equations
26. Use MATLAB to perform numerically and symbolically mathematical differentiation and integration
27. Use MATLAB and EXCEL to analyze empirical data using linear regression analysis
28. Use MATLAB and EXCEL to analyze empirical data using higher-order regression analysis
29. Create MATLAB SIMULINK models for time-varying systems that employ feedback loops
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Methods of Presentation
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Lecture/Discussion
Laboratory
Demonstration/Exercise
Reading from the text
Engineering/Science Problem Solving tutorials using this progression a. Use diagrams, charts, and
tables to analyze the problem b. Formulation of a mathematical model for the problem c. Translate,
conceptually, the math model for computer implementation using psuedocode or flow-charts d.
Implement the solution using a MATLAB script-file e. Run and Check the MATLAB solution f.
Output the solution in the form of a properly constructed table, chart, or graph
Assignments and Methods of Evaluating Student Progress
1. Typical Assignments
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Read chapter-2 in the text on MATLAB’s Array and Matrix handling functions
Exercises from the text book, or those created by the instructor
A set of three equations defines the mesh currents in shown in Text Figure 5.5. Write at
MATLAB program to compute the mesh currents using the given resistor and voltagesource values
D. 2) The Useful life of a machine bearing depends on its operating temperature as indicated
by the following data: Obtain a functional description (curve fit) for this data. Plot the
function and the data on the SAME plot. Also estimate bearing life for a 150 °F operating
temperature.
E. Given that i = ?-1, let y = -3 + ix. For x = 0, 1, 2 use MATLAB to numerically evaluate the
following expressions. Hand check your answers. (a) |y| (b) ?y (c) (-5-7i)y (d) y/(6-3i)
F. Consider an Ordinary Differential Equation (ODE) and its analytical solution: Ordinary
Differential Equation Solution Verify the solution using two methods (a) By hand take the
analytical derivative of x(t) and substitute in to the ODE. (b) Solve the ODE symbolically
using MATLAB’s dsolve function
G. Given a set of deflection vs. load data for a cantilever beam use EXCEL to plot the data
and calculate the slope of the best straight line through the data.
H. 6) A DIODE is an electronic component that essentially acts as “check valve” for electrical
current; i.e., the diode allows current flow in one direction, but not the other. The electrical
circuit symbol, and the theoretical equation relating the diode current to the electrical
potential (or pressure) in applied across the diode: Circuit Symbol V-I Relation Where Va
the Applied Electrical Potential in Volts (V) I the Diode Current in Amps Isat the Diode
SATURATION Current, a PRACTICAL constant, in Amps q the ELECTRONIC CHARGE,
a UNIVERSAL constant = 1.6 x10-19 Coulombs k BOLTZMANN’S value, a UNIVERSAL
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constant = 1.3805 x10-23 J/K n the Diode IDEALITY FACTOR, a PRACTICAL constant
without units T the Thermodynamic Temperature in Kelvins (K)
An instrument called a Digital MultiMeter, or DMM (which we will use in ENGR43) can be
used to collect Va vs. I data. The CSV (Comma Separated Values) data file provided by
the instructor contains Va ? 0 data for a 1N4123 silicon diode. Use these data to create a
SOME TYPE of Plot to reveal the values of the practical constants: Isat n Based on your
analysis, comment on the quality/reliability of the values for Isat and n
7) The height, H, reached by a bubble rising through a liquid in time ?r may be calculated
using the double integral equation 1. Find the rise velocity, vr, from Using the instructorprovided values for the constants ro, CD, and g, determine for the specified rise time the
value of H.
Create a SimuLink model to plot the solution of the following equation for 0 < t < 10
2. Methods of Evaluating Student Progress
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Textbooks (Typical):
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Exams/Tests
Home Work
Smith, David M (2010). Engineering Computation with MATLAB (2nd/e). Addison-Wesley.
Palm, William (2005). Introduction To MATLAB 7 For Engineers McGraw-Hill.
Ogata, Katsuhiko (2008). MATLAB® Prentice Hall.
Hanselman, Duane C. and Bruce L. Littlefield (2005). Mastering MATLAB 7 Prentice Hall.
Gilat, Amos (2008). MATLAB: An Introduction with Applications (3rd/e). John-Wiley.
Musto, Joseph, William E. Howard, and Richard R. Williams (2009). Engineering Computation: An
Introduction Using MATLAB and Excel McGraw-Hill.
Special Student Materials
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Laboratory access to MATLAB and EXCEL Software
Software: MATLAB and Simulink student version with the Symbolic Math Toolbox
Software: Microsoft Office Student and Teacher Edition; includes EXCEL
USB Computer Storage Drive