This presentation is a part of Computer Oriented Numerical Method . Newton-Cotes formulas are an extremely useful and straightforward family of numerical integration techniques.
This presentation is a part of Computer Oriented Numerical Method . Newton-Cotes formulas are an extremely useful and straightforward family of numerical integration techniques.
Open newton cotes quadrature with midpoint derivative for integration of al...eSAT Journals
Abstract Many methods are available for approximating the integral to the desired precision in Numerical integration. A new set of numerical integration formula of Open Newton-Cotes Quadrature with Midpoint Derivative type is suggested, which is the modified form of Open Newton-Cotes Quadrature. This new midpoint derivative based formula increase the two order of precision than the classical Open Newton-Cotes formula and also gives more accuracy than the existing formula. Further the error terms are obtained and compared with the existing methods. Finally, the effectiveness of the proposed algorithm is illustrated by means of a numerical example.
Key Words: Numerical Integration, Open Newton-Cotes formula, Midpoint Derivative, Numerical Examples
A NEW STUDY OF TRAPEZOIDAL, SIMPSON’S1/3 AND SIMPSON’S 3/8 RULES OF NUMERICAL...mathsjournal
The main goal of this research is to give the complete conception about numerical integration including Newton-Cotes formulas and aimed at comparing the rate of performance or the rate of accuracy of Trapezoidal, Simpson’s 1/3, and Simpson’s 3/8. To verify the accuracy, we compare each rules demonstrating the smallest error values among them. The software package MATLAB R2013a is applied to determine the best method, as well as the results, are compared. It includes graphical comparisons mentioning these methods graphically. After all, it is then emphasized that the among methods considered, Simpson’s 1/3 is more effective and accurate when the condition of the subdivision is only even for solving a definite integral.
Open newton cotes quadrature with midpoint derivative for integration of al...eSAT Journals
Abstract Many methods are available for approximating the integral to the desired precision in Numerical integration. A new set of numerical integration formula of Open Newton-Cotes Quadrature with Midpoint Derivative type is suggested, which is the modified form of Open Newton-Cotes Quadrature. This new midpoint derivative based formula increase the two order of precision than the classical Open Newton-Cotes formula and also gives more accuracy than the existing formula. Further the error terms are obtained and compared with the existing methods. Finally, the effectiveness of the proposed algorithm is illustrated by means of a numerical example.
Key Words: Numerical Integration, Open Newton-Cotes formula, Midpoint Derivative, Numerical Examples
A NEW STUDY OF TRAPEZOIDAL, SIMPSON’S1/3 AND SIMPSON’S 3/8 RULES OF NUMERICAL...mathsjournal
The main goal of this research is to give the complete conception about numerical integration including Newton-Cotes formulas and aimed at comparing the rate of performance or the rate of accuracy of Trapezoidal, Simpson’s 1/3, and Simpson’s 3/8. To verify the accuracy, we compare each rules demonstrating the smallest error values among them. The software package MATLAB R2013a is applied to determine the best method, as well as the results, are compared. It includes graphical comparisons mentioning these methods graphically. After all, it is then emphasized that the among methods considered, Simpson’s 1/3 is more effective and accurate when the condition of the subdivision is only even for solving a definite integral.
Mean Value Theorem explained with examples.pptxvandijkvvd4
The Mean Value Theorem (MVT) is a crucial concept in calculus, connecting the average rate of change of a function to its instantaneous rate of change. It's a fundamental theorem that holds a significant place in calculus and has far-reaching implications across various mathematical fields. Exploring it through 3000 alphabets involves diving into its core principles, applications, and significance.
At its heart, the Mean Value Theorem asserts that if a function is continuous on a closed interval and differentiable on the open interval, there exists at least one point within that interval where the instantaneous rate of change (the derivative) equals the average rate of change of the function over that interval.
Geometrically, MVT can be visualized as a tangent line parallel to a secant line at a certain point within the function, signifying the equality between the average and instantaneous rates of change.
Understanding MVT involves grasping its conditions and implications. For a function
�
(
�
)
f(x), the prerequisites for applying MVT are continuity and differentiability within the specified interval
[
�
,
�
]
[a,b].
The theorem's application extends to various contexts in mathematics, science, and economics. It's utilized to prove the existence of solutions to equations, establish bounds for functions, and analyze behavior in optimization problems.
MVT plays a pivotal role in other fundamental theorems of calculus like the Fundamental Theorem of Calculus, aiding in the development of integral calculus and its applications in areas such as physics, engineering, and economics.
Beyond its practical applications, the Mean Value Theorem's elegance lies in its capacity to capture the essence of rates of change, providing a bridge between local and global behavior of functions.
Mathematicians and scientists rely on MVT to understand and model real-world phenomena, utilizing its principles to analyze trends, make predictions, and solve problems across diverse disciplines.
In essence, the Mean Value Theorem stands as a cornerstone of calculus, fostering a deeper comprehension of the relationship between a function and its derivatives while serving as a powerful tool in mathematical analysis and problem-solving.
The Mean Value Theorem (MVT) in calculus asserts that for a continuous and differentiable function on a closed interval, there exists at least one point within that interval where the derivative (instantaneous rate of change) of the function equals the average rate of change of the function over that interval. It's a fundamental concept connecting the behavior of functions locally and globally, pivotal in calculus, and extensively applied in various fields like physics, engineering, and economics. MVT's essence lies in relating the function's behavior at specific points to its overall behavior, aiding in problem-solving, equation-solving, and understanding rates of change in real-world scenarios.
MVT relates function's average to in
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
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Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
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1. 2.4 Rates of Change and Tangent Lines
Devil’s Tower, Wyoming
2. 2 1
2 1
Average rate of change
change in y y y y
change in x x x x
If f(t) represents the position of an object as a function of time,
then the rate of change is the velocity of the object.
Average rate of change (from bc)
Average rate of change of f(x) over the interval [a,b]
a
b
a
f
b
f
)
(
)
(
3. Find the average rate of change of f (t) = 2 + cost on [0, π]
F(b)= f(π) = 2 + cos (π) = 2 – 1 = 1
F(a)= f(0) = 2 + cos (0) = 2 + 1 = 3
1. Calculate the function value (position) at each
endpoint of the interval
The average velocity on [0, π] is 0.63366
2. Use the slope formula
63366
.
0
2
0
3
1
)
(
)
(
a
b
a
f
b
f
4. Consider a graph of displacement (distance traveled) vs. time.
time (hours)
distance
(miles)
Average velocity can be
found by taking:
change in position
change in time
s
t
t
s
A
B
ave
f t t f t
s
V
t t
The speedometer in your car does not measure average
velocity, but instantaneous velocity.
(The velocity at one
moment in time.)
The velocity problem
5. The slope of a line is given by:
y
m
x
x
y
The slope of a curve at (1,1) can be
approximated by the slope of the secant line
through (1,1) and (4,16).
5
We could get a better approximation if we
move the point closer to (1,1). ie: (3,9)
y
x
9 1
3 1
8
2
4
Even better would be the point (2,4).
y
x
4 1
2 1
3
1
3
y
x
16 1
4 1
15
3
6. The slope of a line is given by:
y
m
x
x
y
If we got really close to (1,1), say (1.1,1.21),
the approximation would get better still
y
x
1.21 1
1.1 1
.21
.1
2.1
How far can we go?
2
f x x
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1 2 3 4
7. If we try to apply the same formula to find The instantaneous
velocity
and evaluate the velocity at an instant (a,f(a)) not an interval
, we will find it
0
0
)
(
)
(
t
a
a
a
f
a
f
s
Which is undefined, so the best is to make Δx as small as
experimentally possible Δx 0
a
x
a
f
x
f
a
x
)
(
)
(
lim
The instantaneous velocity at the point (a,f(a)) =
0
lim
t
f t t f t
ds
V t
dt t
8. The slope of the secant line= the average rate of change
= The average velocity
The slope of the curve at a point
= the slope of the tangent line of the curve at this point
= instantaneous velocity
=
2 1
2 1
Average rate of change
change in y y y y
change in x x x x
h
x
f
h
x
f
h
)
(
)
(
lim
0
a
x
a
f
x
f
a
x
)
(
)
(
lim
9. Other form for Slope of secant
line of tangent line
sec
( ) ( )
y f a h f a
m
x h
tan 0
( ) ( )
limh
f a h f a
m
h
Let h = x - a Then x = a + h
10. Rates of Change:
Average rate of change =
f x h f x
h
Instantaneous rate of change =
0
lim
h
f x h f x
f x
h
These definitions are true for any function.
( x does not have to represent time. )
11. Analytic Techniques
Rewrite algebraically if direct substitution
produces an indeterminate form such as
0/0
• Factor and reduce
• Rationalize a numerator or denominator
• Simplify a complex fraction
When you rewrite you are often producing another function that
agrees with the original in all but one point. When this happens
the limits at that point are equal.
12. Find the indicated limit
2
3
6
lim
3
x
x x
x
3
lim ( 2)
x
x
3
( 3)( 2)
lim
3
x
x x
x
= - 5
direct substitution fails
Rewrite and cancel
now use direct sub.
0
0
13.
1
f
1 1 h
1
f h
h
slope
y
x
1 1
f h f
h
slope at
1,1
2
0
1 1
lim
h
h
h
2
0
1 2 1
lim
h
h h
h
0
2
lim
h
h h
h
2
The slope of the curve at the point is:
y f x
,
P a f a
0
lim
h
f a h f a
m
h
14. The slope of the curve at the point is:
y f x
,
P a f a
0
lim
h
f a h f a
m
h
f a h f a
h
is called the difference quotient of f at a.
If you are asked to find the slope using the definition or using
the difference quotient, this is the technique you will use.
15. In the previous example, the tangent line could be found
using .
1 1
y y m x x
The slope of a curve at a point is the same as the slope of
the tangent line at that point.
If you want the normal line, use the negative reciprocal of
the slope. (in this case, )
1
2
(The normal line is perpendicular.)
16. Example 4:
a Find the slope at .
x a
0
lim
h
f a h f a
m
h
0
1 1
lim
h
a h a
h
0
1
lim
h
h
a a h
a a h
0
lim
h
a a h
h a a h
2
1
a
Let
1
f x
x
On the TI-89:
limit ((1/(a + h) – 1/ a) / h, h, 0)
F3 Calc
Note:
If it says “Find the limit”
on a test, you must
show your work!
a a h
a a h
a a h
0
17. Example 4:
b Where is the slope ?
1
4
Let
1
f x
x
2
1 1
4 a
2
4
a
2
a
On the TI-89:
Y= y = 1 / x
WINDOW
6 6
3 3
scl 1
scl 1
x
y
x
y
GRAPH
18. Example 4:
b Where is the slope ?
1
4
Let
1
f x
x
On the TI-89:
Y= y = 1 / x
WINDOW
6 6
3 3
scl 1
scl 1
x
y
x
y
GRAPH
We can let the calculator
plot the tangent:
F5 Math
A: Tangent ENTER
2 ENTER
Repeat for x = -2
tangent equation
19. Review:
average slope:
y
m
x
slope at a point:
0
lim
h
f a h f a
m
h
average velocity: ave
total distance
total time
V
instantaneous velocity:
0
lim
h
f t h f t
V
h
If is the position function:
f t
These are
often
mixed up
by
Calculus
students!
So are these!
velocity = slope
Editor's Notes
For analytic techniques you will need a good algebraic background.