Republic of the Philippines
NUEVA VIZCAYA STATE UNIVERSITY
Bambang, Nueva Vizcaya
INSTRUCTIONAL MODULE
IM No.: ME6-2S-2020-2021
College of Engineering
Bambang Campus
DEGREE
PROGRAM
SPECIALIZATION
YEAR LEVEL
Bachelor of Science in COURSE NO.
Mechanical Engineering
COURSE
TITLE
2nd Year
TIME FRAME
MECH02
Advanced Mathematics for ME
3
WK NO.
hrs/week
I.
UNIT TITLE/CHAPTER TITLE: Numerical Methods and Analysis
II.
LESSON TITLE: Numerical Analysis
III.
2-3
IM
NO.
1
II.1 Numerical Methods: Introduction, Discrete Algebra, Accuracy, Errors
II.2 Numerical analysis in Engineering: Taylor Series
LESSON OVERVIEW:
The course program differentiates accuracy from error approximation. It gives an overview to
students the significance of finding and computing for the error as one criteria to satisfy numerical
calculation. It also provide details on how the error is estimated in numerical analysis. The Taylor Series,
together with MacLaurin series are derived and discussed in estimating the truncation error and know its
applicability to other advanced topics such as Power series.
IV.
DESIRED LEARNING OUTCOMES:
1. Define accuracy and error estimation.
2. Discuss the different types of error encountered in numerical computation and modeling.
3. Solve problems involving Taylor and McLaurin series.
V.
LESSON CONTENT:
1. Introduction, Discrete Algebra, Accuracy and Error
The algebraic type of mathematics is different from the numerical type of mathematics. In solving
algebraic functions such as ODEs and calculus, there are few choices on how to solve the problem with
algebraic solution. The numerical method gives numerical solution that are tabulated and represented in
graphs, means that choice of numerical method or algorthim depends on its applicability. The results can
be selected based on its accuracy and when to start or end the computation. Thus, it is essential for
mechanical engineers to know and learn numerical methods, the estimation of errors and order of
convergence. Such numerical problems have complicated equation that finding its roots is a difficult task.
We begin the discussion with round-off errors. The round-off error is caused by chopping, or discarding
all the digits from some decimal or rounding of values. The rule of rounding the error to n significant
digit is same as replacing the the decimal with significant digits.
ROUNDOFF RULE: To round a number x to k no. of decimals, use
chop the signficant digits after the (k+1)st digit.
and add to x then
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Republic of the Philippines
NUEVA VIZCAYA STATE UNIVERSITY
Bambang, Nueva Vizcaya
INSTRUCTIONAL MODULE
IM No.: ME6-2S-2020-2021
Sample:
Round the number 1.23454621 to k=2 decimals, 3 decimals, 4 decimals and 5 decimals.
Using the roundoff rule,
a.)
.
, so that 1.23454621+0.005 becomes 1.23954621. Then chop off
the (k+1)st digit, which is starting 3rd digit, remove “954621” to get 1.23.
b.)
.
, so that 1.23454621+0.0005 becomes 1.23504621. Then chop
off the (k+1)st digit, which is starting 4th digit, remove “04621” to get 1.235.
c.)
.
, so that 1.23454621+0.00005 becomes 1.23459621. Then chop
th
off the (k+1)st digit, which is starting 5 digit, remove “9621” to get 1.2345.
d.)
.
, so that 1.23454621+0.000005 becomes 1.23455121. Then
chop off the (k+1)st digit, which is starting 6th digit, remove “121” to get 1.23455.
NOTE: The rounding of values may lead to round-off error which is very dangerous in the computation
of large data. It is therefore important to analyze some computational programs and expect some roundoff errors in-between the series of computation.
The experimental errors are simply caused by human errors or error specified in the tool for gathering
the needed data.
The truncation errors result from truncating or prematurely breaking off and usually computed when
using Taylor series and getting the difference with the supposed computation of sum of the terms.
The formula in getting the error depends on the true value and approximated value, getting the difference,
Sometimes, getting the absolute value of the difference is used in some experiments to denote absolute
error. For the relative error, get the ratio between computed error and true value, i.e.
Sometimes, the errors are limited with error bounds (β) depending on the practice used in the field of
study, that is,
| |
| |
In error propagation, when the computed errors with bounds are carried or propagated to the next
computation, usually affects the accuracy of overall computation. Thus, bounds for error is simply
getting the sum of error bounds while bounds for relative error are added under multiplication/division.
a.) In addition and subtraction, bound for the error of the results is given by the sum of the error bounds
for the terms.
b.) In multiplication and division, bound for relative error of the results is given by the sum of the bounds
for the relative errors.
Sample Problem 1:
Approximations of
3.14159265358979 … are 22/7 and 355/113. Determine the corresponding errors
and relative errors to 3 significant digits.
Solution: Since 22/7 is the irrational number that defines pi, it is the true value. The 355/113 is the
approximated value (AV). Thus, computing the errors gives
22 355
1
1.264222503 10
.
7 113 791
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works included in this material may be reproduced for educational purposes only and not for commercial distribution.
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Republic of the Philippines
NUEVA VIZCAYA STATE UNIVERSITY
Bambang, Nueva Vizcaya
INSTRUCTIONAL MODULE
1
355
113
22
7
1
2486
4.022526146 10
IM No.: ME6-2S-2020-2021
.
Note that we cannot apply the rounding-off rule since we only need to consider 3 significant digits in
exponential form and not the decimal form.
Sample Problem 2:
Determine the absolute error and relative error when approximating the derivative of
x=2 using the approximation formula,
5 at
′
with h=0.1
Solution: The approximation formula represents the approximated value while the true value is simply
the derivative ′
at x=2.
AV:
0.1
2
0.1
5 2
0.1
0.1
6.09 6
0.9
0.1
2
TV:
2
Thus,
5
|
2
2 2
.
. |
0.1
0.1
2
5
5 2
5
5 1
.
.
2. Taylor and Maclaurin Series
The Taylor series given a function f(x), a real value a for which the nth derivative
0, hence the Taylor series of f about a is defined as
Note:
≡
.
is defined for all
!
Moreover, we define the radius of convergence of the Taylor series as the largest
∈
,
, the series converges.
0 such that, for all
When the center a =0, the Taylor series is called Maclaurin series.
Sample Problem 1:
Determine the Taylor series for the function
Recalling the Taylor series form,
at a=0.
!
we express the terms when n=0, n=1, n=2, n=3, n=4 and so on.
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works included in this material may be reproduced for educational purposes only and not for commercial distribution.
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Republic of the Philippines
NUEVA VIZCAYA STATE UNIVERSITY
Bambang, Nueva Vizcaya
INSTRUCTIONAL MODULE
n=0:
0
n=1:
0
n=2:
0
n=3:
0
n=4:
0
0
!
1
IM No.: ME6-2S-2020-2021
(Note: 0! = 1)
(the derivative of ex is ex)
!
!
!
!
Now, adding the terms (n=1 to n=4) to form the Taylor series of ex is written in the form:
⋯
!
!
This is the fourth-degree Taylor series approximation since up to fourth derivative is solved.
Sample Problem 2:
Determine the Maclaurin series for the function
Recalling the Maclaurin series form,
.
0
0
!
we express the terms when n=0, n=1, n=2, n=3, n=4 and so on.
n=0:
0
n=1:
0
n=2:
0
n=3:
0
n=4:
0
!
!
!
!
!
0
sin 0
0 (Note: x0 = 1 and
≡
)
cos 0
sin 0
0
cos 0
sin 0
0
Therefore, the Maclaurin series of sinx is written in the form:
⋯
!
!
This is the fourth-degree Taylor series approximation since up to fourth derivative is solved.
Sample Problem 3:
Determine the Taylor series for the function
The Taylor series form,
at x=1.
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works included in this material may be reproduced for educational purposes only and not for commercial distribution.
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Republic of the Philippines
NUEVA VIZCAYA STATE UNIVERSITY
Bambang, Nueva Vizcaya
INSTRUCTIONAL MODULE
1
n=0:
1
n=1:
1
n=2:
1
n=3:
1
1
!
1
1
!
ln 1
1
0
IM No.: ME6-2S-2020-2021
1
!
1 (Note: first derivative of ln x = 1/x)
(Note: derivative 1/x is -1/x2)
!
(Note: derivative of -1/x2 is 2/x3)
!
Therefore, the Taylor series of lnx at x=1 is written in the form:
⋯
!
!
This is the third-degree Taylor series approximation since up to third derivative is solved.
VI.
LEARNING ACTIVITIES:
Seatwork #1: Solve the following problems and box your final answers. Pattern the step-by-step solution
in sample problems.
1.) Write the Maclaurin series form of cos 2x.
2.) Write the Taylor series form of
at x0=1.
3.) Use the formula in approximating the derivative (same as Sample Problem 2) of
at x=1 using h=0.5. Compute the absolute and relative errors.
4.) What is the greatest value of h that can be used to approximate the derivative of of
with an error of no more than 1%?
VII.
2
3
1
at x=4
5.) Determine the relative error when approximating e-1 using fifth-degree Taylor polynomial with respect
to ex compared to sixth-degree Taylor polynomial.
(Hint: express the degree series using x=-1 to get the exact values)
EVALUATION
To be given via MS Teams
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works included in this material may be reproduced for educational purposes only and not for commercial distribution.
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Republic of the Philippines
NUEVA VIZCAYA STATE UNIVERSITY
Bambang, Nueva Vizcaya
INSTRUCTIONAL MODULE
VIII.
IM No.: ME6-2S-2020-2021
ASSIGNMENT:
Assignment #1. Solve the following problems and box your final answers. Pattern the step-by-step solution
in sample problems.
1.) Determine the absolute and relative error when approximating 2
polynomial compared to seventh-degree Taylor polynomial.
using sixth-degree Taylor
2.) Determine the absolute and relative error that occurs when approximating the integral 3
using
three rectangles, each with width ∆
1 and for which the heights are f(0), f(1), f(2) where f(x)=3x.
(Hint: the expressed integral is TV while area (∆
is the AV)
3.) Determine the absolute and relative error that occurs when approximating the integral 2
using
three rectangles, each with width ∆
0.5 and for which the heights are f(0), f(1), f(2) where f(x)=2x2.
(Hint: the expressed integral is TV while area (∆
is the AV)
IX.
REFERENCES:
E. Kreyzsig, Advanced Engineering Mathematics, 10th Edition, John Wiley & Sons, 2011. United States
of America
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works included in this material may be reproduced for educational purposes only and not for commercial distribution.
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