Convolution and Fourier Transforms
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SAMPLE QUESTIONExercise 1 Consider the functionf (x,C).docx
SAMPLE QUESTION:
Exercise 1: Consider the function
f (x,C)=
sin(C x)
Cx
(a) Create a vector x with 100 elements from -3*pi to 3*pi. Write f as an inline or anonymous function
and generate the vectors y1 = f(x,C1), y2 = f(x,C2) and y3 = f(x,C3), where C1 = 1, C2 = 2 and
C3 = 3. Make sure you suppress the output of x and y's vectors. Plot the function f (for the three
C's above), name the axis, give a title to the plot and include a legend to identify the plots. Add a
grid to the plot.
(b) Without using inline or anonymous functions write a function+function structure m-file that does
the same job as in part (a)
SAMPLE LAB WRITEUP:
MAT 275 MATLAB LAB 1 NAME: __________________________
LAB DAY and TIME:______________
Instructor: _______________________
Exercise 1
(a)
x = linspace(-3*pi,3*pi); % generating x vector - default value for number
% of pts linspace is 100
f= @(x,C) sin(C*x)./(C*x) % C will be just a constant, no need for ".*"
C1 = 1, C2 = 2, C3 = 3 % Using commans to separate commands
y1 = f(x,C1); y2 = f(x,C2); y3 = f(x,C3); % supressing the y's
plot(x,y1,'b.-', x,y2,'ro-', x,y3,'ks-') % using different markers for
% black and white plots
xlabel('x'), ylabel('y') % labeling the axis
title('f(x,C) = sin(Cx)/(Cx)') % adding a title
legend('C = 1','C = 2','C = 3') % adding a legend
grid on
Command window output:
f =
@(x,C)sin(C*x)./(C*x)
C1 =
1
C2 =
2
C3 =
3
(b)
M-file of structure function+function
function ex1
x = linspace(-3*pi,3*pi); % generating x vector - default value for number
% of pts linspace is 100
C1 = 1, C2 = 2, C3 = 3 % Using commans to separate commands
y1 = f(x,C1); y2 = f(x,C2); y3 = f(x,C3); % function f is defined below
plot(x,y1,'b.-', x,y2,'ro-', x,y3,'ks-') % using different markers for
% black and white plots
xlabel('x'), ylabel('y') % labeling the axis
title('f(x,C) = sin(Cx)/(Cx)') % adding a title
legend('C = 1','C = 2','C = 3') % adding a legend
grid on
end
function y = f(x,C)
y = sin(C*x)./(C*x);
end
Command window output:
C1 =
1
C2 =
2
C3 =
3
More instructions for the lab write-up:
1) You are not obligated to use the 'diary' function. It was presented only for you convenience. You
should be copying and pasting your code, plots, and results into some sort of "Word" type editor that
will allow you to import graphs and such. Make sure you always include the commands to generate
what is been asked and include the outputs (from command window and plots), unless the pr.
Signal Processing Assignment Help
I am Geoffrey J. I am a Signal Processing Assignment Expert at matlabassignmentexperts.com. I hold a Ph.D. in Matlab, Arizona State University. I have been helping students with their homework for the past 8 years. I solve assignments related to Signal Processing.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com. You can also call on +1 678 648 4277 for any assistance with Signal Processing Assignments.
math 111 assignment 4 fall 2014 2 T4. This is.docx
math 111 assignment 4 fall 2014 2
T
4. This is a multipart graphical problem. You should print the following five pages (parts a – f)
and use them to present your solution.
In the diagram below, the 2×2 shaded square centred at the origin is bounded by the lines x = ±1 and y =
±1. Let T denote the transformation that maps the square into the 1×2 rectangle centred at (–1, 1.5).
(a) Use the structure theorem for affine transformations to find a matrix A and a vector b for which
T(x) = Ax + b.
[Check your work carefully with a suitable point, e.g. x = [1, 1], as you will be using this formula later.]
math 111 assignment 4 fall 2014 3
T
D
R
F
L
y
x
(b) Use the graphs at the right to show how T can be
decomposed as a composition of four basic transformations in
the following order: first a dilation D (using positive numbers),
followed by a rotation R through 90° (counter-clockwise),
followed by a reflection F, followed by a translation L. At the
very right, provide an expression for a general point
y
x
as it
passes through each of these steps so that at the end you get the
formula for T(x) you found in (a).
math 111 assignment 4 fall 2014 4
T
(c) This question (and part (d) below) are about TT , the composite of T with itself. Find a formula for
TT (x) = T(T(x)) of the form
T(T(x)) = Cx + d
f
e
y
x
dc
ba
y
x
TT
Do this working not with the diagram, but with the formula T(x) = Ax + b that you derived in (a).
[Hint: to keep your notation simple, work first with the vector form T(x) = Ax + b finding expressions for
C and d in terms of A and b, then calculate C and d.]
math 111 assignment 4 fall 2014 5
T
(d) Now solve the problem of (c) graphically. To do this, start with the T-image of the central square
(that’s the 2×1 rectangle that came from the first application of T) and track it through D, R, F and L to
get the result of TT . [You can check that you get the same result as (c)]. It turns out that you can
nicely do the tracking for this on one diagram, drawing the resulting rectangle after each of the four steps.
Be sure you include the black shading.
PROVIDE THE IMAGE OF THE RECTANGLE
AFTER EACH OF THE FOUR STEPS
DRAW ONLY THE FINAL IMAGE:
THE IMAGE OF THE ORIGINAL CENTRAL
RECTANGLE UNDER TT
math 111 assignment 4 fall 2014 6
invL
invF
invR
invD
y
x
T
(e) This problem asks you to find the inverse of T. Your job
now is to draw a shaded rectangle with the property that if you
apply T to it you will get the original 2×2 square. Use the
following geometric strategy: start with the original 2×2 square
and apply the inverses of component transformations D, R, F
and L in rev ...
Signals and System Assignment Help
I am John G. I am a Signals and Systems Assignment Expert at matlabassignmentexperts.com. I hold a Master's in Matlab, from Glasgow University. I have been helping students with their assignments for the past 9 years. I solve assignments related to Signals and Systems.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com.
You can also call on +1 678 648 4277 for any assistance with Signals and Systems assignments.
Lab 5 template Lab 5 - Your Name - MAT 275 Lab The M.docx
Lab 5 template
%% Lab 5 - Your Name - MAT 275 Lab
% The Mass-Spring System
%% EX 1 10 pts
%A) 1 pts | short comment
%
%B) 2 pts | short comment
%
%C) 1 pts | short comment
%
%D) 1 pts
%E) 2 pts | List the first 3-4 t values either in decimal format or as
%fractions involving pi
%F) 3 pts | comments. | (1 pts for including two distinct graphs, each with y(t) and v(t) plotted)
%% EX 2 10 pts
%A) 5 pts
% add commands to LAB05ex1 to compute and plot E(t). Then use ylim([~,~]) to change the yaxis limits.
% You don't need to include this code but at least one plot of E(t) and a comment must be
% included!
%B) 2 pts | write out main steps here
% first differentiate E(t) with respect to t using the chain rule. Then
% make substitutions using the expression for omega0 and using the
% differential equation
%C) 3 pts | show plot and comment
%% EX 3 10 pts
%A) 3 pts | modify the system of equations in LAB05ex1a
% write the t value and either a) show correponding graph or b) explain given matlab
% commands
%B) 2 pts | write t value and max |V| value; include figure
%note: velocity magnitude is like absolute value!
%C) 3 pts | include 3 figures here + comments.
% use title('text') to attach a title to the figure
%D) 2 pts | What needs to happen (in terms of the characteristic equation)
%in order for there to be no oscillations? Impose a condition on the
%characteristic equation to find the critical c value. Write out main steps
%% EX4 10 pts
% A) 5 pts | include 1 figure and comment
%B) 2 pts
% again find dE/dt using the chain rule and make substitutions based on the
% differential equation. You should reach an expression for dE/dt which is
% in terms of y'
%C) 3 pts | include one figure and comment
Exercise (1):
function LAB05ex1
m = 1; % mass [kg]
k = 9; % spring constant [N/m]
omega0=sqrt(k/m);
y0=0.4; v0=0; % initial conditions
[t,Y]=ode45(@f,[0,10],[y0,v0],[],omega0); % solve for 0<t<10
y=Y(:,1); v=Y(:,2); % retrieve y, v from Y
figure(1); plot(t,y,'b+-',t,v,'ro-'); % time series for y and v
grid on;
%------------------------------------------------------
function dYdt= f(t,Y,omega0)
y = Y(1); v= Y(2);
dYdt = [v; -omega0^2*y];
Exercise (1a):
function LAB05ex1a
m = 1; % mass [kg]
k = 9; % spring constant [N/m]
c = 1; % friction coefficient [Ns/m]
omega0 = sqrt(k/m); p = c/(2*m);
y0 = 0.4; v0 = 0; % initial conditions
[t,Y]=ode45(@f,[0,10],[y0,v0],[],omega0,p); % solve for 0<t<10
y=Y(:,1); v=Y(:,2); % retrieve y, v from Y
figure(1); plot(t,y,'b+-',t,v,'ro-'); % time series for y and v
grid on
%------------------------------------------------------
function dYdt= f(t,Y,omega0,p)
y = Y(1); v= Y(2);
dYdt = [v; ?? ]; % fill-in dv/dt
More instructions for the l ...
Digital Signal Processing Assignment Help
This document contains 10 problems (P.1 through P.10) related to signal modulation and filtering systems. Each problem provides a figure and asks the reader to analyze and/or design some aspect of the system. For example, problem P.1 asks the reader to sketch the Fourier transforms of signals in a given modulation system and indicate whether one signal is equal to another. The document also contains solutions (S.1 through S.10) that analyze each problem in detail through mathematical analysis and additional figures.
Matlab Assignment Help
I am Green J. I am a Signal Processing Assignment Expert at matlabassignmentexperts.com. I hold a Ph.D. in Matlab, University of Oxford, UK. I have been helping students with their assignments for the past 10 years. I solve assignments related to Signal Processing.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com. You can also call on +1 678 648 4277 for any assistance with Signal Processing Assignments.
Final
This document contains a multi-part exam problem analyzing control systems. Part 1 involves finding the transfer function and difference equation for a system with an impulse response of 1 for the first N points. Part 2 analyzes a continuous-time system with two real poles, one in the right half-plane, and finds its zero-order hold equivalent discrete-time system. The remaining parts involve analyzing the stability of this system using root loci, Nyquist, and designing a lead compensator to stabilize it with over 45 degrees of phase margin.
Recommended
SAMPLE QUESTIONExercise 1 Consider the functionf (x,C).docx
SAMPLE QUESTION:
Exercise 1: Consider the function
f (x,C)=
sin(C x)
Cx
(a) Create a vector x with 100 elements from -3*pi to 3*pi. Write f as an inline or anonymous function
and generate the vectors y1 = f(x,C1), y2 = f(x,C2) and y3 = f(x,C3), where C1 = 1, C2 = 2 and
C3 = 3. Make sure you suppress the output of x and y's vectors. Plot the function f (for the three
C's above), name the axis, give a title to the plot and include a legend to identify the plots. Add a
grid to the plot.
(b) Without using inline or anonymous functions write a function+function structure m-file that does
the same job as in part (a)
SAMPLE LAB WRITEUP:
MAT 275 MATLAB LAB 1 NAME: __________________________
LAB DAY and TIME:______________
Instructor: _______________________
Exercise 1
(a)
x = linspace(-3*pi,3*pi); % generating x vector - default value for number
% of pts linspace is 100
f= @(x,C) sin(C*x)./(C*x) % C will be just a constant, no need for ".*"
C1 = 1, C2 = 2, C3 = 3 % Using commans to separate commands
y1 = f(x,C1); y2 = f(x,C2); y3 = f(x,C3); % supressing the y's
plot(x,y1,'b.-', x,y2,'ro-', x,y3,'ks-') % using different markers for
% black and white plots
xlabel('x'), ylabel('y') % labeling the axis
title('f(x,C) = sin(Cx)/(Cx)') % adding a title
legend('C = 1','C = 2','C = 3') % adding a legend
grid on
Command window output:
f =
@(x,C)sin(C*x)./(C*x)
C1 =
1
C2 =
2
C3 =
3
(b)
M-file of structure function+function
function ex1
x = linspace(-3*pi,3*pi); % generating x vector - default value for number
% of pts linspace is 100
C1 = 1, C2 = 2, C3 = 3 % Using commans to separate commands
y1 = f(x,C1); y2 = f(x,C2); y3 = f(x,C3); % function f is defined below
plot(x,y1,'b.-', x,y2,'ro-', x,y3,'ks-') % using different markers for
% black and white plots
xlabel('x'), ylabel('y') % labeling the axis
title('f(x,C) = sin(Cx)/(Cx)') % adding a title
legend('C = 1','C = 2','C = 3') % adding a legend
grid on
end
function y = f(x,C)
y = sin(C*x)./(C*x);
end
Command window output:
C1 =
1
C2 =
2
C3 =
3
More instructions for the lab write-up:
1) You are not obligated to use the 'diary' function. It was presented only for you convenience. You
should be copying and pasting your code, plots, and results into some sort of "Word" type editor that
will allow you to import graphs and such. Make sure you always include the commands to generate
what is been asked and include the outputs (from command window and plots), unless the pr.
Signal Processing Assignment Help
I am Geoffrey J. I am a Signal Processing Assignment Expert at matlabassignmentexperts.com. I hold a Ph.D. in Matlab, Arizona State University. I have been helping students with their homework for the past 8 years. I solve assignments related to Signal Processing.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com. You can also call on +1 678 648 4277 for any assistance with Signal Processing Assignments.
math 111 assignment 4 fall 2014 2 T4. This is.docx
math 111 assignment 4 fall 2014 2
T
4. This is a multipart graphical problem. You should print the following five pages (parts a – f)
and use them to present your solution.
In the diagram below, the 2×2 shaded square centred at the origin is bounded by the lines x = ±1 and y =
±1. Let T denote the transformation that maps the square into the 1×2 rectangle centred at (–1, 1.5).
(a) Use the structure theorem for affine transformations to find a matrix A and a vector b for which
T(x) = Ax + b.
[Check your work carefully with a suitable point, e.g. x = [1, 1], as you will be using this formula later.]
math 111 assignment 4 fall 2014 3
T
D
R
F
L
y
x
(b) Use the graphs at the right to show how T can be
decomposed as a composition of four basic transformations in
the following order: first a dilation D (using positive numbers),
followed by a rotation R through 90° (counter-clockwise),
followed by a reflection F, followed by a translation L. At the
very right, provide an expression for a general point
y
x
as it
passes through each of these steps so that at the end you get the
formula for T(x) you found in (a).
math 111 assignment 4 fall 2014 4
T
(c) This question (and part (d) below) are about TT , the composite of T with itself. Find a formula for
TT (x) = T(T(x)) of the form
T(T(x)) = Cx + d
f
e
y
x
dc
ba
y
x
TT
Do this working not with the diagram, but with the formula T(x) = Ax + b that you derived in (a).
[Hint: to keep your notation simple, work first with the vector form T(x) = Ax + b finding expressions for
C and d in terms of A and b, then calculate C and d.]
math 111 assignment 4 fall 2014 5
T
(d) Now solve the problem of (c) graphically. To do this, start with the T-image of the central square
(that’s the 2×1 rectangle that came from the first application of T) and track it through D, R, F and L to
get the result of TT . [You can check that you get the same result as (c)]. It turns out that you can
nicely do the tracking for this on one diagram, drawing the resulting rectangle after each of the four steps.
Be sure you include the black shading.
PROVIDE THE IMAGE OF THE RECTANGLE
AFTER EACH OF THE FOUR STEPS
DRAW ONLY THE FINAL IMAGE:
THE IMAGE OF THE ORIGINAL CENTRAL
RECTANGLE UNDER TT
math 111 assignment 4 fall 2014 6
invL
invF
invR
invD
y
x
T
(e) This problem asks you to find the inverse of T. Your job
now is to draw a shaded rectangle with the property that if you
apply T to it you will get the original 2×2 square. Use the
following geometric strategy: start with the original 2×2 square
and apply the inverses of component transformations D, R, F
and L in rev ...
Signals and System Assignment Help
I am John G. I am a Signals and Systems Assignment Expert at matlabassignmentexperts.com. I hold a Master's in Matlab, from Glasgow University. I have been helping students with their assignments for the past 9 years. I solve assignments related to Signals and Systems.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com.
You can also call on +1 678 648 4277 for any assistance with Signals and Systems assignments.
Lab 5 template Lab 5 - Your Name - MAT 275 Lab The M.docx
Lab 5 template
%% Lab 5 - Your Name - MAT 275 Lab
% The Mass-Spring System
%% EX 1 10 pts
%A) 1 pts | short comment
%
%B) 2 pts | short comment
%
%C) 1 pts | short comment
%
%D) 1 pts
%E) 2 pts | List the first 3-4 t values either in decimal format or as
%fractions involving pi
%F) 3 pts | comments. | (1 pts for including two distinct graphs, each with y(t) and v(t) plotted)
%% EX 2 10 pts
%A) 5 pts
% add commands to LAB05ex1 to compute and plot E(t). Then use ylim([~,~]) to change the yaxis limits.
% You don't need to include this code but at least one plot of E(t) and a comment must be
% included!
%B) 2 pts | write out main steps here
% first differentiate E(t) with respect to t using the chain rule. Then
% make substitutions using the expression for omega0 and using the
% differential equation
%C) 3 pts | show plot and comment
%% EX 3 10 pts
%A) 3 pts | modify the system of equations in LAB05ex1a
% write the t value and either a) show correponding graph or b) explain given matlab
% commands
%B) 2 pts | write t value and max |V| value; include figure
%note: velocity magnitude is like absolute value!
%C) 3 pts | include 3 figures here + comments.
% use title('text') to attach a title to the figure
%D) 2 pts | What needs to happen (in terms of the characteristic equation)
%in order for there to be no oscillations? Impose a condition on the
%characteristic equation to find the critical c value. Write out main steps
%% EX4 10 pts
% A) 5 pts | include 1 figure and comment
%B) 2 pts
% again find dE/dt using the chain rule and make substitutions based on the
% differential equation. You should reach an expression for dE/dt which is
% in terms of y'
%C) 3 pts | include one figure and comment
Exercise (1):
function LAB05ex1
m = 1; % mass [kg]
k = 9; % spring constant [N/m]
omega0=sqrt(k/m);
y0=0.4; v0=0; % initial conditions
[t,Y]=ode45(@f,[0,10],[y0,v0],[],omega0); % solve for 0<t<10
y=Y(:,1); v=Y(:,2); % retrieve y, v from Y
figure(1); plot(t,y,'b+-',t,v,'ro-'); % time series for y and v
grid on;
%------------------------------------------------------
function dYdt= f(t,Y,omega0)
y = Y(1); v= Y(2);
dYdt = [v; -omega0^2*y];
Exercise (1a):
function LAB05ex1a
m = 1; % mass [kg]
k = 9; % spring constant [N/m]
c = 1; % friction coefficient [Ns/m]
omega0 = sqrt(k/m); p = c/(2*m);
y0 = 0.4; v0 = 0; % initial conditions
[t,Y]=ode45(@f,[0,10],[y0,v0],[],omega0,p); % solve for 0<t<10
y=Y(:,1); v=Y(:,2); % retrieve y, v from Y
figure(1); plot(t,y,'b+-',t,v,'ro-'); % time series for y and v
grid on
%------------------------------------------------------
function dYdt= f(t,Y,omega0,p)
y = Y(1); v= Y(2);
dYdt = [v; ?? ]; % fill-in dv/dt
More instructions for the l ...
Digital Signal Processing Assignment Help
This document contains 10 problems (P.1 through P.10) related to signal modulation and filtering systems. Each problem provides a figure and asks the reader to analyze and/or design some aspect of the system. For example, problem P.1 asks the reader to sketch the Fourier transforms of signals in a given modulation system and indicate whether one signal is equal to another. The document also contains solutions (S.1 through S.10) that analyze each problem in detail through mathematical analysis and additional figures.
Matlab Assignment Help
I am Green J. I am a Signal Processing Assignment Expert at matlabassignmentexperts.com. I hold a Ph.D. in Matlab, University of Oxford, UK. I have been helping students with their assignments for the past 10 years. I solve assignments related to Signal Processing.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com. You can also call on +1 678 648 4277 for any assistance with Signal Processing Assignments.
Final
This document contains a multi-part exam problem analyzing control systems. Part 1 involves finding the transfer function and difference equation for a system with an impulse response of 1 for the first N points. Part 2 analyzes a continuous-time system with two real poles, one in the right half-plane, and finds its zero-order hold equivalent discrete-time system. The remaining parts involve analyzing the stability of this system using root loci, Nyquist, and designing a lead compensator to stabilize it with over 45 degrees of phase margin.
Differentiation and applications
The document discusses several key points about derivatives:
1) The definition of a derivative directly from limits is not usually the most efficient way to find derivatives except for simple functions. General rules like the power rule and sum rule make derivatives easier to find.
2) Derivatives can be represented with different notations like f'(x) or dy/dx.
3) Functions being derived are assumed to be differentiable. Examples of finding derivatives using the power rule are shown for simple polynomial functions.
Etienne_ME_a241ms
This document contains a student's coursework for a class on Matlab and Simulink. It includes several assignments:
1. Analyzing respiratory data from a case study, including loading specific data, plotting cycles, and normalizing the data.
2. Creating a Matlab function to model the respiratory motion with a cosine wave formula and fitting the function to the data.
3. Building two Simulink models - a first order model and one based on rearranging the differential equation - to simulate the response of a mass to an input force based on given parameters.
4. Performing image processing on a CT scan image, including determining pixel values of different structures, applying a mesh filter
25285 mws gen_int_ppt_trapcontinuous
The document discusses the trapezoidal rule of integration. It begins by defining integration as measuring the area under a function plotted on a graph. It then explains that the trapezoidal rule approximates this area as a trapezoid by assuming the integrand can be approximated by a linear polynomial. The rule takes the average of the function values at the limits of integration. The document provides an example problem and solution to demonstrate applying the single-segment and multiple-segment trapezoidal rule. It shows that dividing the interval into more segments reduces the true error of the approximation.
Online Signals and Systems Assignment Help
I am Grey Nolan. Currently associated with matlabassignmentexperts.com as an assignment helper. After completing my master's from the University of British Columbia, I was in search for an opportunity that expands my area of knowledge hence I decided to help students with their Signals and Systems assignments. I have written several assignments till date to help students overcome numerous difficulties they face in Signals and Systems Assignments.
From the Front LinesOur robotic equipment and its maintenanc.docx
From the Front Lines
Our robotic equipment and its maintenance represent a fixed cost of $23,320 per month. The cost-effectiveness of robotic-assisted surgery is related to patient volume: With only 10 cases, the fixed cost per case is $2,332, and with 40 cases, the fixed cost per case is $583.
Source: Alemozaffar, Chang, Kacker, Sun, DeWolf, & Wagner (2013).
MATLAB sessions: Laboratory 5
MAT 275 Laboratory 5
The Mass-Spring System
In this laboratory we will examine harmonic oscillation. We will model the motion of a mass-spring
system with differential equations.
Our objectives are as follows:
1. Determine the effect of parameters on the solutions of differential equations.
2. Determine the behavior of the mass-spring system from the graph of the solution.
3. Determine the effect of the parameters on the behavior of the mass-spring.
The primary MATLAB command used is the ode45 function.
Mass-Spring System without Damping
The motion of a mass suspended to a vertical spring can be described as follows. When the spring is
not loaded it has length ℓ0 (situation (a)). When a mass m is attached to its lower end it has length ℓ
(situation (b)). From the first principle of mechanics we then obtain
mg︸︷︷︸
downward weight force
+ −k(ℓ − ℓ0)︸ ︷︷ ︸
upward tension force
= 0. (L5.1)
The term g measures the gravitational acceleration (g ≃ 9.8m/s2 ≃ 32ft/s2). The quantity k is a spring
constant measuring its stiffness. We now pull downwards on the mass by an amount y and let the mass
go (situation (c)). We expect the mass to oscillate around the position y = 0. The second principle of
mechanics yields
mg︸︷︷︸
weight
+ −k(ℓ + y − ℓ0)︸ ︷︷ ︸
upward tension force
= m
d2(ℓ + y)
dt2︸ ︷︷ ︸
acceleration of mass
, i.e., m
d2y
dt2
+ ky = 0 (L5.2)
using (L5.1). This ODE is second-order.
(a) (b) (c) (d)
y
ℓ
ℓ0
m
k
γ
Equation (L5.2) is rewritten
d2y
dt2
+ ω20y = 0 (L5.3)
c⃝2011 Stefania Tracogna, SoMSS, ASU
MATLAB sessions: Laboratory 5
where ω20 = k/m. Equation (L5.3) models simple harmonic motion. A numerical solution with ini-
tial conditions y(0) = 0.1 meter and y′(0) = 0 (i.e., the mass is initially stretched downward 10cms
and released, see setting (c) in figure) is obtained by first reducing the ODE to first-order ODEs (see
Laboratory 4).
Let v = y′. Then v′ = y′′ = −ω20y = −4y. Also v(0) = y′(0) = 0. The following MATLAB program
implements the problem (with ω0 = 2).
function LAB05ex1
m = 1; % mass [kg]
k = 4; % spring constant [N/m]
omega0 = sqrt(k/m);
y0 = 0.1; v0 = 0; % initial conditions
[t,Y] = ode45(@f,[0,10],[y0,v0],[],omega0); % solve for 0<t<10
y = Y(:,1); v = Y(:,2); % retrieve y, v from Y
figure(1); plot(t,y,’b+-’,t,v,’ro-’); % time series for y and v
grid on;
%-----------------------------------------
function dYdt = f(t,Y,omega0)
y = Y(1); v = Y(2);
dYdt = [ v ; -omega0^2*y ];
Note that the parameter ω0 was passed as an argument to ode45 rather than set to its value ω0 = 2
directly in the funct ...
Ch 7 tutoring notes quadratics
The document discusses transformations of quadratic functions in vertex form f(x) = a(x-h)2 + k. It explains how changing the coefficients a, h, and k affects the graph of the quadratic function. Specifically, it states that changing a widens or narrows the graph, changing h shifts the graph left or right, and changing k shifts the graph up or down. It also provides examples of writing equations for quadratic functions based on given graphs and finding the vertex of a quadratic function in standard form.
Machnical Engineering Assignment Help
I am Terry K . I am a Mechanical Engineering Assignment Expert at matlabassignmentexperts.com. I hold a Ph.D. Matlab, University of North Carolina, USA. I have been helping students with their homework for the past 5 years. I solve assignments related to Mechanical Engineering.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com.
You can also call on +1 678 648 4277 for any assistance with Mechanical Engineering Assignments.
Chemistry Exam Help
I am Luther H. I am a Chemistry Exam Helper at liveexamhelper.com. I hold a Masters' Degree in Chemistry, from the University of Illinois, USA. I have been helping students with their exams for the past 10 years. You can hire me to take your exam in Chemistry.
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EE380-4 Course project Experimental determination of a Ser.docx
EE380-4 Course project
Experimental determination of a Servo-Motor State Space Model
Dr. A. Masoud, Course Project, Posted Thursday, October 8, 2015, Due a week before the end of the semester 151.
Objective: To determine experimentally the state space model and the transfer function of the
control laboratory servo-trainer using physical measurements instead of mathematical
derivations. In addition, to be familiarized with the mathematical tools needed for doing this task
Background: The servo-process you will be examining in the EE380 laboratory is the DC
motor (figure-1). By now, you are familiar with how to model theoretically this process. To do
this, you need to know beforehand all the parameters of the system. These parameters are
usually not readily available. Computing them may be difficult or not possible. Therefore, the
only other alternative is to determine the model of the servo-process experimentally.
Figure-1: Equivalent circuit of a DC motor
The motor used in the laboratory servo-trainer is a permanent magnet DC motor. The field is
generated by the permanent magnet is a constant. This makes the armature voltage (Va) the
only means of control available, i.e the motor is in an armature control mode.
The state of the overall system (X) consists of the state of the electrical part which is the current
in the inductor (the armature current Ia), the angular position ( )θ and the velocity of the
mechanical part (θ ). As for the output, it can be selected as anyone of these variables. This
makes writing the output equation of the state space model of the motor a simple and a
straightforward task. The state vector of the overall system is: X=[ Ia θ θ ]T. However, we are
going to use a simplified model that neglects the electrical component and focus only on the
mechanical one. The state vector of the simplified model is: X=[θ θ ]T. The state equation of
the simplified model in (1)
Va,
dc
ba
⎥
⎦
⎤
⎢
⎣
⎡
+⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎦
⎤
⎢
⎣
⎡
=⎥
⎦
⎤
⎢
⎣
⎡
f
e
θ
θ
θ
θ (1)
Determining the model means that the input (Va(t)), the states (θ (t), θ (t)) and their derivatives
can be directly measured or reliably computed. The parameters (a,b,c,d,e,f) are the unknowns
that need to be determined from the measurements.
To determine the parameters, we first need to change the form of the state space equations into
the form in (2)
,
f
e
d
c
b
a
Va0(t)(t)00
0Va00(t)(t)
(t)
(t)
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
⎥
⎦
⎤
⎢
⎣
⎡
=⎥
⎦
⎤
⎢
⎣
⎡
θθ
θθ
θ
θ
(2)
Now take “n” measurements of the states, their derivatives and the input and form the equations
in (3). Determining the time instants (t1, t2,..,tn) at which measurements are to be taken are left
up to the student. However, one should make sure that the measurements are distinct, i.e. do
not take the same measurement more than once. ...
Week 6
The document discusses differentiation rules for various functions. It begins by discussing the derivatives of polynomials and exponential functions. The power rule is introduced, which states the derivative of x^n is nx^{n-1}. It then covers the derivatives of exponential functions f(x)=ax, proving the formula f'(x)=af(x). The product rule and quotient rule are also introduced. Finally, it discusses the derivatives of trigonometric functions, proving that the derivative of sin(x) is cos(x) and the derivative of cos(x) is -sin(x).
lecture8-derivativerules-140925171214-phpapp01.ppt
The document discusses rules for taking derivatives of various functions including:
1) The power rule which states that the derivative of x^n is nx^{n-1}.
2) The constant multiple rule which states that the derivative of c*f(x) is c*f'(x).
3) The sum and difference rules which allow you to take the derivative of a sum or difference of functions.
It also discusses higher order derivatives, the product rule, and quotient rule for more complex functions.
Lesson 2: A Catalog of Essential Functions (slides)
The document provides an overview of Calculus I taught by Professor Matthew Leingang at New York University. It outlines key topics that will be covered in the course, including different classes of functions, transformations of functions, and compositions of functions. The first assignments are due on January 31 and February 2, with first recitations on February 3. The document uses examples to illustrate concepts like linear functions, other polynomial functions, and trigonometric functions. It also explains how vertical and horizontal shifts can transform the graph of a function.
Lesson 2: A Catalog of Essential Functions (slides)
This document provides an overview of different types of functions including: linear, polynomial, rational, power, trigonometric, and exponential functions. It discusses representing functions verbally, numerically, visually, and symbolically. Key topics covered include transformations of functions through shifting graphs vertically and horizontally, as well as composing multiple functions.
Heaviside's function
The document discusses Heaviside's unit step function, which is used to model abrupt changes in functions at specific times. It has a value of 0 for negative time and 1 for positive time. Shifted and delayed step functions are also discussed. The Laplace transform of various step functions is provided. Applications include modeling the switching on/off of voltages in electrical circuits and representing signals that switch at a given time.
17, r) -,r I -l19.t... 121.2t-314 23. ^t -rr - .docx
17, r) -,r I : -l
19.t:...: 1
21.2t-31:4
/ 23. ^t: -rr - 1)t I,r r.= ll-vl
11 1
Evaluating a Function In Exercises 29-14, evaluate the
function at each specified value of the independent
variable and simplify.
29.fO-3t+t
(a) f(2) (b) /(-4) (c) f(r + 2)
30. s(y) :1 - 3y
(a) s(o) tul s(l) (c) s(s + 2)
tlzt.t(t):t2-2t
(a) h(.2) ft) /,(1.s) (c) h(x + 2)
32. v(r) - !rr3
(a) v(3) (b) Y(;) k) v(2r)
33./(.-r)::-./y
@) f(a) (u) l(0.2s) (c') [email protected])
3a.f(x)- aE+8+2
(a) f (-+) (b) /(8)
1
' x'-9
(a) q(-3) (b) q(z)
)t2+\
36. ./(r)- t'
(il qQ) 0) q(o)
lrl
37. i(r) : "'x
ar f(e) (b) /(-e)
38. -.,. : -r *4
: , -i (b) /(_5)
-, - l. x<0
'lq -, - l. .r > 0'\-
'\-
6r t(0)
', - -:. -r < 0t.
> |
1..
1OG Chapter I Functjons and Their Graphs
Testing for Functions Represented Algebraically In
Exercises 77-28. determine whether the equation
represents,r' as a function of r.
18.x:]2+1
20. y :-lf+ 5
22.r:-.y+5
24.r'l!2:3
20. lyl :1-x
28.r,:8
(c) /(x - 8)
(c) q(y + 3)
(.c) q(.- x)
(c) /(t)
(c) f(t)
(c) JQ)
,cl .f(l)
Evalr.rating a Fr.lnction In Exercises 45-48, assume that
the domain of/is the setA = {-2, - 1, 0, 1, 2}. Determine
the set of ordered pairs representing the function/.
[.rr-4. x<o42.fG)-1r_r,., r>0L1 - i.\
(il f?2) (b) /(0)
[.r- ]. t<0
I
a3.f(x)-1a. 0<r<2
L*, + t. r > 2
(a) .f(.-2) (b) /(1)
(s - ), r < otJ
44. ffrr --]s. us r < I
l.+*- r, r2 l
(a) J(. 2) (b) /(])
(c) /(1)
@ f(a)
(c) f(t)
a6. .f(.x) : x2 - 3
as. /(x) : lx + 1l
45. f(x) - x2
a7. f(x): lxl + z
Evaluating a Function In Exercises 49 and 50, complete
the tahle.
4s. h(t): llr + :l
l" - ?l50..f(r) -:
Finding the lnputs That Have Outputs of Zers In
Exercises 51-54, find all values of x such that/(r) = g'
st. 16) : 15 - 3x 52. f(x): 5r * I
3r-4
sa. f(x') - 2r-3s3. /(x) :
Finding the Dornain of a Function In Exercises 55-6J.
find the domain of the function.
,l ss. fG): 5x2 + 2x - | s6. s(ir) : 7 - 2x2
4-3v
57. hhl - ' 58. ,s( r') -I y-)
se. /(x) - 1C - 1 60. /(x) : X/" + 3x
. t 3 l0
{ el. gtrt - ' - 62. h(r) - .., 1..I f t- i LA
r'*2 -,8+6
64./(:r) :--' o f .t
t -5 -4 -3 -1
It(r)
,t 0 l2
I
2
4
/(')
63. s(.v) : 5- 10
the Domain and Range of a Function In 1)
mriffs 65-68, use a graphing utility to graph the
hhu Find the domain and range of the function.
. ,.-
-E' - \
+ i 66. f(x): 1F I 1
68. g(x) : I, - sl.j1- -r,-; : i1r + 3l
I. Geometry Write the areaA of a circle as a function of
rs --ircumference C.
il" Cmmetry Write the arca A of an equilaterai tiangle
"ts i tunction of the length s of its sides.
1!- E4loration An open box of maximum volume is to
s made from a square piece of mateial, 24 centimeters
cm a side, by cutting equal squares from the corners and
uuia-e up the sides (see figure).
,"1 , The table shows the volume 7 (in cubic centimeters)
of the box for various heights x (in centimeters).
L-se the table to estimate the maximum volume.
i Plot the points (x, I/) from the table in part (a). Does
rtre relation defined.
1. (5 pts) Which of these graphs represent a one-to-one function .docx
1. (5 pts) Which of these graphs represent a one-to-one function?
Answer(s): ____________
(
no explanation required
.)
(There may be more than one graph that qualifies.)
(A)
(B)
(C)
(D)
2. (5 pts) Convert to a logarithmic equation: 7
x
= 2401.
(no explanation required)
2. ______
A.
B.
C.
D.
3. (10 pts) Based on data about the growth of a variety of ornamental cherry trees, the following logarithmic model about these trees was determined:
h
(
t
) = 6.47 ln(
t
) + 2.83,
where
t
= age of tree in years and
h
(
t
) = height of tree, in feet.
(Note that "ln" refers to the natural log function) (explanation optional)
Using the model,
(a) At age 3 years, how tall is this type of ornamental cherry tree, to the nearest tenth of a foot?
(b) At age 12 years, how tall is this type of ornamental cherry tree, to the nearest tenth of a foot?
4.
(5 pts)
Solve the equation.
Check
all proposed solutions.
Show work in solving and in checking, and state your final conclusion.
5. (10 pts)
(a)
_______ (fill in the blank)
(b) Let
State the exponential form of the equation.
(c) Determine the numerical value of
, in simplest form. Work optional.
6. (10 pts) Let
f
(
x
) = 2
x
2
–
x
– 10 and
g
(
x
) = 3
x
+ 1
(a) Find the composite function
and simplify the results. Show work.
(b) Find
. Show work.
7. (15 pts) Let
(a) Find
f
-
1
, the inverse function of
f
.
Show work
.
(b) What is the domain of
f
? What is the domain of the inverse function?
(c) What is
f
(2) ?
f
(2) = ______
work/explanation optional
(d) What is
f
-
1
( ____ ), where the number in the blank is your answer from part (c)?
work/explanation optional
8. (15 pts) Let
f
(
x
) =
e
x
– 1
+ 4.
Answers can be stated without additional work/explanation.
(a) Which describes how the graph of
f
can be obtained from the graph of
y
=
e
x
? Choice: ________
A. Shift the graph of
y
=
e
x
to the left by 1 unit and up by 4 units.
B. Shift the graph of
y
=
e
x
to the right by 1 unit and up by 4 units.
C. Reflect the graph of
y
=
e
x
across the
x
-axis and shift up by 4 units.
D. Reflect the graph of
y
=
e
x
across the
y
-axis and shift up by 4 units.
(b) What is the domain of
f
?
(c) What is the range of
f
?
(d) What is the horizontal asymptote?
(e) What is the
y
-intercept? State the approximation to 2 decimal places (i.e., the nearest hundredth).
(f) Which is the graph of
f
?
GRAPH A GRAPH B GRAPH C GRAPH D
NONLINEAR MODELS - For the latte.
Unit step function
The document discusses unit step functions and their use in describing abrupt changes in function values that occur at specific times, such as a voltage being switched on or off in an electrical circuit. It defines the unit step function u(t) as having a value of 0 for negative t and 1 for positive t. Shifted and rectangular pulse functions are also described. Examples are provided of writing functions in terms of unit step functions and sketching the corresponding waveforms.
Seismic data processing introductory lecture
This document provides a syllabus for a course on seismic data processing. The syllabus outlines topics that will be covered, including the mathematical foundations of Fourier transforms, sampling considerations for seismic time series, basic processing sequences, velocity analysis, filtering and migration techniques, acquisition of seismic data both on land and at sea, 3D seismic data processing, and other advanced topics such as Radon transforms and AVO analysis. References for the course include books on seismic data processing and digital signal processing. The document explains that seismic data processing is important to remove unwanted signals and noise and enhance signal-to-noise ratios, as reflection seismic signals may be obscured by other seismic arrivals like ground roll and direct waves.
Signals and Systems Homework Help.pptx
I am Jayson L. I am a Signals and Systems Homework Expert at matlabassignmentexperts.com. I hold a Master's in Matlab, from the University of Sheffield. I have been helping students with their homework for the past 7 years. I solve homework related to Signals and Systems.
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From the Front LinesOur robotic equipment and its maintenanc.docx
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Our robotic equipment and its maintenance represent a fixed cost of $23,320 per month. The cost-effectiveness of robotic-assisted surgery is related to patient volume: With only 10 cases, the fixed cost per case is $2,332, and with 40 cases, the fixed cost per case is $583.
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MATLAB sessions: Laboratory 5
MAT 275 Laboratory 5
The Mass-Spring System
In this laboratory we will examine harmonic oscillation. We will model the motion of a mass-spring
system with differential equations.
Our objectives are as follows:
1. Determine the effect of parameters on the solutions of differential equations.
2. Determine the behavior of the mass-spring system from the graph of the solution.
3. Determine the effect of the parameters on the behavior of the mass-spring.
The primary MATLAB command used is the ode45 function.
Mass-Spring System without Damping
The motion of a mass suspended to a vertical spring can be described as follows. When the spring is
not loaded it has length ℓ0 (situation (a)). When a mass m is attached to its lower end it has length ℓ
(situation (b)). From the first principle of mechanics we then obtain
mg︸︷︷︸
downward weight force
+ −k(ℓ − ℓ0)︸ ︷︷ ︸
upward tension force
= 0. (L5.1)
The term g measures the gravitational acceleration (g ≃ 9.8m/s2 ≃ 32ft/s2). The quantity k is a spring
constant measuring its stiffness. We now pull downwards on the mass by an amount y and let the mass
go (situation (c)). We expect the mass to oscillate around the position y = 0. The second principle of
mechanics yields
mg︸︷︷︸
weight
+ −k(ℓ + y − ℓ0)︸ ︷︷ ︸
upward tension force
= m
d2(ℓ + y)
dt2︸ ︷︷ ︸
acceleration of mass
, i.e., m
d2y
dt2
+ ky = 0 (L5.2)
using (L5.1). This ODE is second-order.
(a) (b) (c) (d)
y
ℓ
ℓ0
m
k
γ
Equation (L5.2) is rewritten
d2y
dt2
+ ω20y = 0 (L5.3)
c⃝2011 Stefania Tracogna, SoMSS, ASU
MATLAB sessions: Laboratory 5
where ω20 = k/m. Equation (L5.3) models simple harmonic motion. A numerical solution with ini-
tial conditions y(0) = 0.1 meter and y′(0) = 0 (i.e., the mass is initially stretched downward 10cms
and released, see setting (c) in figure) is obtained by first reducing the ODE to first-order ODEs (see
Laboratory 4).
Let v = y′. Then v′ = y′′ = −ω20y = −4y. Also v(0) = y′(0) = 0. The following MATLAB program
implements the problem (with ω0 = 2).
function LAB05ex1
m = 1; % mass [kg]
k = 4; % spring constant [N/m]
omega0 = sqrt(k/m);
y0 = 0.1; v0 = 0; % initial conditions
[t,Y] = ode45(@f,[0,10],[y0,v0],[],omega0); % solve for 0<t<10
y = Y(:,1); v = Y(:,2); % retrieve y, v from Y
figure(1); plot(t,y,’b+-’,t,v,’ro-’); % time series for y and v
grid on;
%-----------------------------------------
function dYdt = f(t,Y,omega0)
y = Y(1); v = Y(2);
dYdt = [ v ; -omega0^2*y ];
Note that the parameter ω0 was passed as an argument to ode45 rather than set to its value ω0 = 2
directly in the funct ...
Ch 7 tutoring notes quadratics
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EE380-4 Course project Experimental determination of a Ser.docx
EE380-4 Course project
Experimental determination of a Servo-Motor State Space Model
Dr. A. Masoud, Course Project, Posted Thursday, October 8, 2015, Due a week before the end of the semester 151.
Objective: To determine experimentally the state space model and the transfer function of the
control laboratory servo-trainer using physical measurements instead of mathematical
derivations. In addition, to be familiarized with the mathematical tools needed for doing this task
Background: The servo-process you will be examining in the EE380 laboratory is the DC
motor (figure-1). By now, you are familiar with how to model theoretically this process. To do
this, you need to know beforehand all the parameters of the system. These parameters are
usually not readily available. Computing them may be difficult or not possible. Therefore, the
only other alternative is to determine the model of the servo-process experimentally.
Figure-1: Equivalent circuit of a DC motor
The motor used in the laboratory servo-trainer is a permanent magnet DC motor. The field is
generated by the permanent magnet is a constant. This makes the armature voltage (Va) the
only means of control available, i.e the motor is in an armature control mode.
The state of the overall system (X) consists of the state of the electrical part which is the current
in the inductor (the armature current Ia), the angular position ( )θ and the velocity of the
mechanical part (θ ). As for the output, it can be selected as anyone of these variables. This
makes writing the output equation of the state space model of the motor a simple and a
straightforward task. The state vector of the overall system is: X=[ Ia θ θ ]T. However, we are
going to use a simplified model that neglects the electrical component and focus only on the
mechanical one. The state vector of the simplified model is: X=[θ θ ]T. The state equation of
the simplified model in (1)
Va,
dc
ba
⎥
⎦
⎤
⎢
⎣
⎡
+⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎦
⎤
⎢
⎣
⎡
=⎥
⎦
⎤
⎢
⎣
⎡
f
e
θ
θ
θ
θ (1)
Determining the model means that the input (Va(t)), the states (θ (t), θ (t)) and their derivatives
can be directly measured or reliably computed. The parameters (a,b,c,d,e,f) are the unknowns
that need to be determined from the measurements.
To determine the parameters, we first need to change the form of the state space equations into
the form in (2)
,
f
e
d
c
b
a
Va0(t)(t)00
0Va00(t)(t)
(t)
(t)
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
⎥
⎦
⎤
⎢
⎣
⎡
=⎥
⎦
⎤
⎢
⎣
⎡
θθ
θθ
θ
θ
(2)
Now take “n” measurements of the states, their derivatives and the input and form the equations
in (3). Determining the time instants (t1, t2,..,tn) at which measurements are to be taken are left
up to the student. However, one should make sure that the measurements are distinct, i.e. do
not take the same measurement more than once. ...
Week 6
The document discusses differentiation rules for various functions. It begins by discussing the derivatives of polynomials and exponential functions. The power rule is introduced, which states the derivative of x^n is nx^{n-1}. It then covers the derivatives of exponential functions f(x)=ax, proving the formula f'(x)=af(x). The product rule and quotient rule are also introduced. Finally, it discusses the derivatives of trigonometric functions, proving that the derivative of sin(x) is cos(x) and the derivative of cos(x) is -sin(x).
lecture8-derivativerules-140925171214-phpapp01.ppt
The document discusses rules for taking derivatives of various functions including:
1) The power rule which states that the derivative of x^n is nx^{n-1}.
2) The constant multiple rule which states that the derivative of c*f(x) is c*f'(x).
3) The sum and difference rules which allow you to take the derivative of a sum or difference of functions.
It also discusses higher order derivatives, the product rule, and quotient rule for more complex functions.
Lesson 2: A Catalog of Essential Functions (slides)
The document provides an overview of Calculus I taught by Professor Matthew Leingang at New York University. It outlines key topics that will be covered in the course, including different classes of functions, transformations of functions, and compositions of functions. The first assignments are due on January 31 and February 2, with first recitations on February 3. The document uses examples to illustrate concepts like linear functions, other polynomial functions, and trigonometric functions. It also explains how vertical and horizontal shifts can transform the graph of a function.
Lesson 2: A Catalog of Essential Functions (slides)
This document provides an overview of different types of functions including: linear, polynomial, rational, power, trigonometric, and exponential functions. It discusses representing functions verbally, numerically, visually, and symbolically. Key topics covered include transformations of functions through shifting graphs vertically and horizontally, as well as composing multiple functions.
Heaviside's function
The document discusses Heaviside's unit step function, which is used to model abrupt changes in functions at specific times. It has a value of 0 for negative time and 1 for positive time. Shifted and delayed step functions are also discussed. The Laplace transform of various step functions is provided. Applications include modeling the switching on/off of voltages in electrical circuits and representing signals that switch at a given time.
17, r) -,r I -l19.t... 121.2t-314 23. ^t -rr - .docx
17, r) -,r I : -l
19.t:...: 1
21.2t-31:4
/ 23. ^t: -rr - 1)t I,r r.= ll-vl
11 1
Evaluating a Function In Exercises 29-14, evaluate the
function at each specified value of the independent
variable and simplify.
29.fO-3t+t
(a) f(2) (b) /(-4) (c) f(r + 2)
30. s(y) :1 - 3y
(a) s(o) tul s(l) (c) s(s + 2)
tlzt.t(t):t2-2t
(a) h(.2) ft) /,(1.s) (c) h(x + 2)
32. v(r) - !rr3
(a) v(3) (b) Y(;) k) v(2r)
33./(.-r)::-./y
@) f(a) (u) l(0.2s) (c') [email protected])
3a.f(x)- aE+8+2
(a) f (-+) (b) /(8)
1
' x'-9
(a) q(-3) (b) q(z)
)t2+\
36. ./(r)- t'
(il qQ) 0) q(o)
lrl
37. i(r) : "'x
ar f(e) (b) /(-e)
38. -.,. : -r *4
: , -i (b) /(_5)
-, - l. x<0
'lq -, - l. .r > 0'\-
'\-
6r t(0)
', - -:. -r < 0t.
> |
1..
1OG Chapter I Functjons and Their Graphs
Testing for Functions Represented Algebraically In
Exercises 77-28. determine whether the equation
represents,r' as a function of r.
18.x:]2+1
20. y :-lf+ 5
22.r:-.y+5
24.r'l!2:3
20. lyl :1-x
28.r,:8
(c) /(x - 8)
(c) q(y + 3)
(.c) q(.- x)
(c) /(t)
(c) f(t)
(c) JQ)
,cl .f(l)
Evalr.rating a Fr.lnction In Exercises 45-48, assume that
the domain of/is the setA = {-2, - 1, 0, 1, 2}. Determine
the set of ordered pairs representing the function/.
[.rr-4. x<o42.fG)-1r_r,., r>0L1 - i.\
(il f?2) (b) /(0)
[.r- ]. t<0
I
a3.f(x)-1a. 0<r<2
L*, + t. r > 2
(a) .f(.-2) (b) /(1)
(s - ), r < otJ
44. ffrr --]s. us r < I
l.+*- r, r2 l
(a) J(. 2) (b) /(])
(c) /(1)
@ f(a)
(c) f(t)
a6. .f(.x) : x2 - 3
as. /(x) : lx + 1l
45. f(x) - x2
a7. f(x): lxl + z
Evaluating a Function In Exercises 49 and 50, complete
the tahle.
4s. h(t): llr + :l
l" - ?l50..f(r) -:
Finding the lnputs That Have Outputs of Zers In
Exercises 51-54, find all values of x such that/(r) = g'
st. 16) : 15 - 3x 52. f(x): 5r * I
3r-4
sa. f(x') - 2r-3s3. /(x) :
Finding the Dornain of a Function In Exercises 55-6J.
find the domain of the function.
,l ss. fG): 5x2 + 2x - | s6. s(ir) : 7 - 2x2
4-3v
57. hhl - ' 58. ,s( r') -I y-)
se. /(x) - 1C - 1 60. /(x) : X/" + 3x
. t 3 l0
{ el. gtrt - ' - 62. h(r) - .., 1..I f t- i LA
r'*2 -,8+6
64./(:r) :--' o f .t
t -5 -4 -3 -1
It(r)
,t 0 l2
I
2
4
/(')
63. s(.v) : 5- 10
the Domain and Range of a Function In 1)
mriffs 65-68, use a graphing utility to graph the
hhu Find the domain and range of the function.
. ,.-
-E' - \
+ i 66. f(x): 1F I 1
68. g(x) : I, - sl.j1- -r,-; : i1r + 3l
I. Geometry Write the areaA of a circle as a function of
rs --ircumference C.
il" Cmmetry Write the arca A of an equilaterai tiangle
"ts i tunction of the length s of its sides.
1!- E4loration An open box of maximum volume is to
s made from a square piece of mateial, 24 centimeters
cm a side, by cutting equal squares from the corners and
uuia-e up the sides (see figure).
,"1 , The table shows the volume 7 (in cubic centimeters)
of the box for various heights x (in centimeters).
L-se the table to estimate the maximum volume.
i Plot the points (x, I/) from the table in part (a). Does
rtre relation defined.
1. (5 pts) Which of these graphs represent a one-to-one function .docx
1. (5 pts) Which of these graphs represent a one-to-one function?
Answer(s): ____________
(
no explanation required
.)
(There may be more than one graph that qualifies.)
(A)
(B)
(C)
(D)
2. (5 pts) Convert to a logarithmic equation: 7
x
= 2401.
(no explanation required)
2. ______
A.
B.
C.
D.
3. (10 pts) Based on data about the growth of a variety of ornamental cherry trees, the following logarithmic model about these trees was determined:
h
(
t
) = 6.47 ln(
t
) + 2.83,
where
t
= age of tree in years and
h
(
t
) = height of tree, in feet.
(Note that "ln" refers to the natural log function) (explanation optional)
Using the model,
(a) At age 3 years, how tall is this type of ornamental cherry tree, to the nearest tenth of a foot?
(b) At age 12 years, how tall is this type of ornamental cherry tree, to the nearest tenth of a foot?
4.
(5 pts)
Solve the equation.
Check
all proposed solutions.
Show work in solving and in checking, and state your final conclusion.
5. (10 pts)
(a)
_______ (fill in the blank)
(b) Let
State the exponential form of the equation.
(c) Determine the numerical value of
, in simplest form. Work optional.
6. (10 pts) Let
f
(
x
) = 2
x
2
–
x
– 10 and
g
(
x
) = 3
x
+ 1
(a) Find the composite function
and simplify the results. Show work.
(b) Find
. Show work.
7. (15 pts) Let
(a) Find
f
-
1
, the inverse function of
f
.
Show work
.
(b) What is the domain of
f
? What is the domain of the inverse function?
(c) What is
f
(2) ?
f
(2) = ______
work/explanation optional
(d) What is
f
-
1
( ____ ), where the number in the blank is your answer from part (c)?
work/explanation optional
8. (15 pts) Let
f
(
x
) =
e
x
– 1
+ 4.
Answers can be stated without additional work/explanation.
(a) Which describes how the graph of
f
can be obtained from the graph of
y
=
e
x
? Choice: ________
A. Shift the graph of
y
=
e
x
to the left by 1 unit and up by 4 units.
B. Shift the graph of
y
=
e
x
to the right by 1 unit and up by 4 units.
C. Reflect the graph of
y
=
e
x
across the
x
-axis and shift up by 4 units.
D. Reflect the graph of
y
=
e
x
across the
y
-axis and shift up by 4 units.
(b) What is the domain of
f
?
(c) What is the range of
f
?
(d) What is the horizontal asymptote?
(e) What is the
y
-intercept? State the approximation to 2 decimal places (i.e., the nearest hundredth).
(f) Which is the graph of
f
?
GRAPH A GRAPH B GRAPH C GRAPH D
NONLINEAR MODELS - For the latte.
Unit step function
The document discusses unit step functions and their use in describing abrupt changes in function values that occur at specific times, such as a voltage being switched on or off in an electrical circuit. It defines the unit step function u(t) as having a value of 0 for negative t and 1 for positive t. Shifted and rectangular pulse functions are also described. Examples are provided of writing functions in terms of unit step functions and sketching the corresponding waveforms.
Seismic data processing introductory lecture
This document provides a syllabus for a course on seismic data processing. The syllabus outlines topics that will be covered, including the mathematical foundations of Fourier transforms, sampling considerations for seismic time series, basic processing sequences, velocity analysis, filtering and migration techniques, acquisition of seismic data both on land and at sea, 3D seismic data processing, and other advanced topics such as Radon transforms and AVO analysis. References for the course include books on seismic data processing and digital signal processing. The document explains that seismic data processing is important to remove unwanted signals and noise and enhance signal-to-noise ratios, as reflection seismic signals may be obscured by other seismic arrivals like ground roll and direct waves.
Signals and Systems Homework Help.pptx
I am Jayson L. I am a Signals and Systems Homework Expert at matlabassignmentexperts.com. I hold a Master's in Matlab, from the University of Sheffield. I have been helping students with their homework for the past 7 years. I solve homework related to Signals and Systems.
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Convolution and Fourier Transforms
- 1. MECHANICAL ENGINEERING Convolution and Fourier Transforms matlabassignmentexperts.com info@matlabassignmentexperts.com +1 (315) 557-6473
- 2. Problem 1: (a) Plot the result of convolving a one-dimensional function consisting of three impulses with a triangular function as shown below: Answer: Use the convolution definition y(t) = f x h = ʃf (Ʈ )h(t - Ʈ)d Ʈ and using the sifting property of the impulse function, y (t) = h(t + 1.5) +h (t- 0.75) matlabassignmentexperts.com
- 3. Answer: h(t) is the as same used in part (a) (b) Plot the result of convolving a one-dimensional function consisting of a rectangular function with the same triangular function used in part(a), as shown below: matlabassignmentexperts.com
- 4. The result of the convolution, y(t), is plotted in the following figure Four basic cases can be observed while varying t (sliding the triangle waveform)., matlabassignmentexperts.com
- 5. (c) Plot the result of convolving a pair of identical even “top-hat” (pulse) functions: f(t) = 1 for |t| < T/2 and f(t) = 0 otherwise. Use your result show that the Fourier transform 2 of a triangular pulse (such as used in parts (a) and (b)) is of the form (sin(x)/x)2 . matlabassignmentexperts.com
- 7. Problem 2: Show that when two gaussian functions f1(x) = e−ax2 f2(x) = e−bx2 are convolved, the result is another gaussian function. Hint: Complete the square in the exponent, change the variable of integration, and recognize that matlabassignmentexperts.com