Lecture #: 06: "Physical Meanings/Applications of Derivatives" with in a course on Applied Calculus offered at Faculty of Engineering, University of Central Punjab
By: Prof. Muhammad Rafiq.
Applied Calculus: An Introduction to Derivativesbaetulilm
Lecture #: 05: "An introduction to Derivatives" with in a course on Applied Calculus offered at Faculty of Engineering, University of Central Punjab
By: Prof. Muhammad Rafiq.
Applied Calculus: An Introduction to Derivativesbaetulilm
Lecture #: 05: "An introduction to Derivatives" with in a course on Applied Calculus offered at Faculty of Engineering, University of Central Punjab
By: Prof. Muhammad Rafiq.
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
How to handle Initial Value Problems using numerical techniques?
#WikiCourses
https://wikicourses.wikispaces.com/Topic+Initial+Value+Problems
https://eau-esa.wikispaces.com/Topic+Initial+Value+Problems
2 Dimensional Wave Equation Analytical and Numerical SolutionAmr Mousa
2 Dimensional Wave Equation Analytical and Numerical Solution
This project aims to solve the wave equation on a 2d square plate and simulate the output in an user-friendly MATLAB-GUI
you can find the gui in mathworks file-exchange here
https://www.mathworks.com/matlabcentral/fileexchange/55117-2d-wave-equation-simulation-numerical-solution-gui
ENGINEERING SYSTEM DYNAMICS-TAKE HOME ASSIGNMENT 2018musadoto
1. Read Chapter 4 – System Dynamics for Mechanical Engineers by Matthew Davies and Tony L. Schmitz and implement Examples 4.1 to 4.12 in Matlab.
2. Read Chapter 7 – System Dynamics for Mechanical Engineers by Matthew Davies and Tony L. Schmitz and implement Examples 7.1 to 7.11 in Matlab.
3. Read Chapter 9 – System Dynamics for Mechanical Engineers by Matthew Davies and Tony L. Schmitz and implement Examples 9.1 to 9.6 in Matlab.
4. Read Chapter 11 – System Dynamics for Mechanical Engineers by Matthew Davies and Tony L. Schmitz and implement Examples 11.1 to 11.7 in Matlab.
5. Read Chapter 2 - System Dynamics for Engineering Students: Concepts and Applications by Nicolae Lobontiu and attempt problem 2.18 (page 63).
6. Read Chapter 3 - System Dynamics for Engineering Students: Concepts and Applications by Nicolae Lobontiu and attempt problem 3.13 (pp 98 - 100).
7. Read Chapter 4 - System Dynamics for Engineering Students: Concepts and Applications by Nicolae Lobontiu and attempt problem 4.20 (page 146).
8. Read Chapter 5 - System Dynamics for Engineering Students: Concepts and Applications by Nicolae Lobontiu and attempt problems 5.15 (page 198), 5.21 (pp 199 - 200) and 5.27 (pp 201 – 202).
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
How to handle Initial Value Problems using numerical techniques?
#WikiCourses
https://wikicourses.wikispaces.com/Topic+Initial+Value+Problems
https://eau-esa.wikispaces.com/Topic+Initial+Value+Problems
2 Dimensional Wave Equation Analytical and Numerical SolutionAmr Mousa
2 Dimensional Wave Equation Analytical and Numerical Solution
This project aims to solve the wave equation on a 2d square plate and simulate the output in an user-friendly MATLAB-GUI
you can find the gui in mathworks file-exchange here
https://www.mathworks.com/matlabcentral/fileexchange/55117-2d-wave-equation-simulation-numerical-solution-gui
ENGINEERING SYSTEM DYNAMICS-TAKE HOME ASSIGNMENT 2018musadoto
1. Read Chapter 4 – System Dynamics for Mechanical Engineers by Matthew Davies and Tony L. Schmitz and implement Examples 4.1 to 4.12 in Matlab.
2. Read Chapter 7 – System Dynamics for Mechanical Engineers by Matthew Davies and Tony L. Schmitz and implement Examples 7.1 to 7.11 in Matlab.
3. Read Chapter 9 – System Dynamics for Mechanical Engineers by Matthew Davies and Tony L. Schmitz and implement Examples 9.1 to 9.6 in Matlab.
4. Read Chapter 11 – System Dynamics for Mechanical Engineers by Matthew Davies and Tony L. Schmitz and implement Examples 11.1 to 11.7 in Matlab.
5. Read Chapter 2 - System Dynamics for Engineering Students: Concepts and Applications by Nicolae Lobontiu and attempt problem 2.18 (page 63).
6. Read Chapter 3 - System Dynamics for Engineering Students: Concepts and Applications by Nicolae Lobontiu and attempt problem 3.13 (pp 98 - 100).
7. Read Chapter 4 - System Dynamics for Engineering Students: Concepts and Applications by Nicolae Lobontiu and attempt problem 4.20 (page 146).
8. Read Chapter 5 - System Dynamics for Engineering Students: Concepts and Applications by Nicolae Lobontiu and attempt problems 5.15 (page 198), 5.21 (pp 199 - 200) and 5.27 (pp 201 – 202).
Many problems in science are about rates of change. They boil down to the mathematical question of finding the slope of a line tangent to a curve. We state this quantity as a limit and give it a name: the derivative
Response of dynamic systems to harmonic excitation is discussed. Single degree of freedom systems are considered. For general damped multi degree of freedom systems, see my book Structural Dynamic Analysis with Generalized Damping Models: Analysis (e.g., in Amazon http://buff.ly/NqwHEE)
1.1/3 points | Previous AnswersSerPSE8 2.P.010.My Notes |
Question Part
Points
Submissions Used
A car travels along a straight line at a constant speed of 41.5 mi/h for a
distance d and then another distance d in the same direction at another constant
speed. The average velocity for the entire trip is 25.0 mi/h.
(a) What is the constant speed with which the car moved during the second
distance d?
Your response is within 10% of the correct value. This may be due to roundoff
error, or you could have a mistake in your calculation. Carry out all intermediate
results to at least four-digit accuracy to minimize roundoff error. mi/h
(b) Suppose the second distance d were traveled in the opposite direction; you
forgot something and had to return home at the same constant speed as found
in part (a). What is the average velocity for this trip?
Your response differs significantly from the correct answer. Rework your solution
from the beginning and check each step carefully. mi/h
(c) What is the average speed for this new trip?
mi/h
2.–/3 pointsSerPSE8 2.P.013.My Notes |
Question Part
Points
Submissions Used
A velocity—time graph for an object moving along the x axis is shown in the
figure. Every division along the vertical axis corresponds to 2.00 m/s and each
division along the horizontal axis corresponds to 2.50 s.
(a) Plot a graph of the acceleration versus time.
This answer has not been graded yet.
(b) Determine the average acceleration of the object in the following time
interval t = 12.5 s to t = 37.5 s. m/s2
(c) Determine the average acceleration of the object in the following time
interval t = 0 to t = 50.0 s.
m/s2
3.–/3 pointsSerPSE8 2.P.016.WI.My Notes |
A particle starts from rest and accelerates as shown in the figure below.
(a) Determine the particle's speed at t = 10.0 s.
m/s
Determine the particle's speed at t = 20.0 s?
m/s
(b) Determine the distance traveled in the first 20.0 s. (Enter your answer to one
decimal places.)
m
4.–/3 pointsSerPSE8 2.P.017.MI.My Notes |
A particle moves along the x axis according to the equation
x = 1.99 + 2.99t − 1.00t2,
where x is in meters and t is in seconds.
(a) Find the position of the particle at t = 2.50 s.
m
(b) Find its velocity at t = 2.50 s.
m/s
(c) Find its acceleration at t = 2.50 s.
m/s2
5.–/2 pointsSerPSE8 2.P.020.My Notes |
Draw motion diagrams for the following items. (Do this on paper. Your instructor
may ask you to turn in your work.)
(a) an object moving to the right at constant speed
(b) an object moving to the right and speeding up at a constant rate
(c) an object moving to the right and slowing down at a constant rate
(d) an object moving to the left and speeding up at a constant rate
(e) an object moving to the left and slowing down at a constant rate
This answer has not ...
SolutionsPlease see answer in bold letters.Note pi = 3.14.docxrafbolet0
Solution
s:
Please see answer in bold letters.
Note pi = 3.1415….
1. The voltage across a 15Ω is as indicated. Find the sinusoidal expression for the current. In addition, sketch the v and i waveform on the same axis.
Note: For the graph of a and b please see attached jpg photo with filename 1ab.jpg and for c and d please see attached photo with filename 1cd.jpg.
a. 15sin20t
v= 15sin20t
By ohms law,
i = v/r
i = 15sin20t / 15
i = sin20t A
Computation of period for graphing:
v= 15sin20t
i = sin20t
w = 20 = 2pi*f
f = 3.183 Hz
Period =1/f = 0.314 seconds
b. 300sin (377t+20)
v = 300sin (377t+20)
i = 300sin (377t+20) /15
i = 20 sin (377t+20) A
Computation of period for graphing:
v = 300sin (377t+20)
i = 20 sin (377t+20)
w = 377 = 2pi*f
f = 60 Hz
Period = 1/60 = 0.017 seconds
shift to the left by:
2pi/0.017 = (20/180*pi)/x
x = 9.44x10-4 seconds
c. 60cos (wt+10)
v = 60cos (wt+10)
i = 60cos (wt+10)/15
i = 4cos (wt+10) A
Computation of period for graphing:
let’s denote the period as w sifted to the left by:
10/180*pi = pi/18
d. -45sin (wt+45)
v = -45sin (wt+45)
i = -45sin (wt+45) / 15
i = -3 sin (wt+45) A
Computation of period for graphing:
let’s denote the period as w sifted to the left by:
45/180 * pi = 1/4*pi
2. Determine the inductive reactance (in ohms) of a 5mH coil for
a. dc
Note at dc, frequency (f) = 0
Formula: XL = 2*pi*fL
XL = 2*pi* (0) (5m)
XL = 0 Ω
b. 60 Hz
Formula: XL = 2*pi*fL
XL = 2 (60) (5m)
XL = 1.885 Ω
c. 4kHz
Formula: XL = 2*pi*fL
XL = = 2*pi* (4k)(5m)
XL = 125.664 Ω
d. 1.2 MHz
Formula: XL = 2*pi*fL
XL = 2*pi* (1.2 M) (5m)
XL = 37.7 kΩ
3. Determine the frequency at which a 10 mH inductance has the following inductive reactance.
a. XL = 10 Ω
Formula: XL = 2*pi*fL
Express in terms in f:
f = XL/2 pi*L
f = 10 / (2pi*10m)
f = 159.155 Hz
b. XL = 4 kΩ
f = XL/2pi*L
f = 4k / (2pi*10m)
f = 63.662 kHz
c. XL = 12 kΩ
f = XL/2piL
f = 12k / (2pi*10m)
f = 190.99 kHz
d. XL = 0.5 kΩ
f = XL/2piL
f = 0.5k / (2pi*10m)
f = 7.958 kHz
4. Determine the frequency at which a 1.3uF capacitor has the following capacitive reactance.
a. 10 Ω
Formula: XC = 1/ (2pifC)
Expressing in terms of f:
f = 1/ (2pi*XC*C)
f = 1/ (2pi*10*1.3u)
f = 12.243 kΩ
b. 1.2 kΩ
f = 1/ (2pi*XC*C)
f = 1/ (2pi*1.2k*1.3u)
f = 102.022 Ω
c. 0.1 Ω
f = 1/ (2pi*XC*C)
f = 1/ (2pi*0.1*1.3u)
f = 1.224 MΩ
d. 2000 Ω
f = 1/ (2pi*XC*C)
f = 1/ (2pi*2000*1.3u)
f = 61.213 Ω
5. For the following pairs of voltage and current, indicate whether the element is a capacitor, an inductor and a capacitor, an inductor, or a resistor and find the value of C, L, or R if insufficient data are given.
a. v = 55 sin (377t + 50)
i = 11 sin (377t -40)
Element is inductor
In this case voltage leads current (ELI) by exactly 90 degrees so that means the circuit is inductive and the element is inductor.
XL = 55/11 = 5 Ω
we know the w=2pif so
w= 377=2pif
f= 60 Hz
To compute for th.
Similar to Applied Calculus: Physical Meanings/Applications of Derivatives (20)
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The purpose of on-line aptitude test system is to take online test in an efficient manner and no time wasting for checking the paper. The main objective of on-line aptitude test system is to efficiently evaluate the candidate thoroughly through a fully automated system that not only saves lot of time but also gives fast results. For students they give papers according to their convenience and time and there is no need of using extra thing like paper, pen etc. This can be used in educational institutions as well as in corporate world. Can be used anywhere any time as it is a web based application (user Location doesn’t matter). No restriction that examiner has to be present when the candidate takes the test.
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Power plants release a large amount of water vapor into the
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2. PHYSICAL MEANINGS OF DERIVATIVE
Definition: The average rate of change of a function f(x) with
respect to ‘x’ over the interval from ° to ° is given by:
Average rate of change ° °
Definition: The instantaneous rate of change of ‘f’ with respect
to ‘x’ is given by:
Instantaneous rate of change at °
° °
Conclusion: Physically, derivative measures the instantaneous
rate of change of a function with respect to some independent
variable.
3. Note: We often use the word rate of change instead of
instantaneous rate of change.
EXAMPLE NO 1
The free fall motion of a heavy ball released from rest at
t = 0 is given by:
S = 4.9
(a) How many meters do the ball fall in first 2 seconds?
(b) What are its velocity, speed and acceleration then?
Solution:
S = 4.9
(a) The ball falls in first 2 seconds :
4. S (2) = 4.9 ( =19.8 m
(b) The velocity and acceleration is given by:
V (t) = = 9.8 t
At t = 2
V (2) = 9.8 (2) = 19.6
a= = 9.8
Speed = = 19.6
EXAMPLE NO 2
A dynamite blast a heavy rock straight up with a launch velocity
of 160 , it reaches a height of S = 160 t 16 feet
after‘t’ seconds.
5. (a) How high does the rock go?
(b) What is the velocity and speed of the rock when it is 256ft
above the ground on the way up on the way down?
(c) What is the acceleration of the rock at any time‘t’ during
its flight after the blast?
(d) When does the rock hit the ground?
Solution:
(a)
For maximum height
32 t = 160
7. Either
V (2) = 160
V (8) = 160
Speed at t = 2 and t = 8 is given by
Speed = =
= 96
(c) a = =
a =
(d) put S = 0
8. ,
,
At t = 0 the blast occur and at t = 10 the
rock hits the ground again.
Marginal Analysis:
(Application of derivative in business and economics)
The marginal cost at some level of production ‘x’ is the cost of
producing item.
The marginal cost can be approximated by taking derivative of
cost function.
9. EXAMPLE NO 3
The cost C(x) of producing ‘x’ units of some items is given by
(a)Find the actual cost of producing unit.
(b)Find the marginal cost when x = 1000
Solution:
(a)The actual cost of producing unit is
=
=
= 1.7999
10. (a) The marginal cost is given by
(1000)=2
=1.80
Conclusion:
(a) Part a is the average rate of change of cost function over
the interval 1000 to 1001
(b) Part b is the instantaneous rate of change of c(x) at x 1000
, thus the marginal cost at x 1000 is approximately equal
to the actual cost of producing units
11. EXAMPLE NO 4
Given the position, , of a body
moving on a coordinate line for , with ‘s’ in meters
and ‘t’ in seconds
a) Find the body’s displacement and average velocity for
the given time interval.
b) Find the body’s speed and acceleration at the endpoints
of the interval.
c) When in the interval does the body change direction (if
ever)?
Solution:
(a) S= ,
13. ( c)
v=0 2t-3=0
t=3/2
‘v’ is negative in the interval 0<t<3/2 and ‘v’ is positive when
3/2 <t<2
the body changes direction at t=3/2
EXERCISE 2.3
Q(1-16)