This is the first installment of Maple Assignment #2 It is due on Friday December 14.
You may work in teams of not more than three students. Each team to submit one report. All arithmetic and Algebra is to be done by Maple. Evidence of pencil or calculator calculations will result in deductions. What you submit must be printed Maple Worksheets..e.g. photo copies of Maple Worksheets are not acceptable. Write to me if you get stuck and I will help you.
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1. Page 249 #26 of our Calculus textbook. (Not our Maple Lab Manual)
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2. A mass on a spring vibrates vertically in such a way that its distance from its equilibrium position as a function of time is given by d(t)=exp(-0.5t)*sin(2t). Distance is measured in inches and time in seconds. Make a plot of the distance from equilibrium vs time .In fact plot altogether the three functions{exp(-0.5t) , - exp(-0.5t), exp(-0.5t)*sin(2t).}
What does this graph tell us about the behavior of the spring? Is this
reasonable? Use Maple to find the last time that the spring is 0.1 inches from it's
equilibrium. How do you know that the time you submit is the last time?
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3. Use Maple to find all points on the parabola y = x^2 - 2*x + 1 whose tangent lines
pass through the point ( 2 , -3 ). In other words , find all values of a such that the
line tangent to f(x)= x^2 - 2*x + 1 at point ( a , f(a) ) passes through ( 2 , -3 ).
Illustrate with a plot which includes the parabola and the relevant tangent lines. The
tangent lines should of course intersect at ( 2 , -3 ). Be sure that your plot shows this.
Whatever Algebra you use should be done by Maple.
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4. Adapted from Page 276 # 11 of our Calculus textbook.
Use Maple to find the point on the parabola y = x^2 that is closest to
the point ( 1, 0 ). When you set this up and take the appropriate derivative you will
find that you have to solve a polynomial of degree three. Use Maple to do that.
Then plot , on the same axes , (i) the parabola y = x^2 , (ii) the line that connects
the point ( 1, 0 ) to the point on the parabola closest to it ( closest to ( 1, 0 ) that is )
and (iii) the line that is tangent to the parabola at the point closest to ( 1, 0 ).
Notice anything?
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5. Use Maple to do problem 7 on page 158 of our text. Here the integrand is sin (1/x).
Why were we not asked to integrate with a lower limit x=0 ? What happens to
this integrand as x gets close to zero?
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6. Use Maple to investigate the family of functions f(x) = x^3+ a*x. This is a parametrized family,
parametrized by a. This means that we don't have an honest member of the family until we specify
a numerical value for a. Plot the function that results for these choices of a:
{ -9 , -4 , -1 , 0 , 1 , 4 , 9 }
For what values of a do you get a function with a local max AND a local min.?
For what values of a wll you get a function that is always increasing?
For what values of a will you get a function that is always decreasing?
Of course you could use any number for a..not just the ones in the set { -9 , -4 , -1 , 0 , 1 , 4 , 9 }
Try a = 0.1 and a = - 0.1 Does the resulting graph show what you expect to see? If not , adjust
the plot range.
What will happen if you use very small positive values for a ?
What will happen if you use very small negative values for a ?
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Here are the five exercises in Maple Assignment #1. Most have several parts.
The complete assignment is due on Tuesday October 30 at lecture.
No teamwork on this one. Each student to submit one report and what you submit must be printed Maple Worksheets..e.g. photo copies of Maple worksheets are not acceptable.
Teamwork will be OK on the second.
All arithmetic and algebra is to be done by Maple. Evidence of pencil or calculator calculations will result in deductions. What you submit must be printed Maple worksheets..e.g. photo copies of Maple worksheets are not acceptable.
Here is the complete set of exercises.
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1.
A. Plot all of the linear functions listed below on the SAME set of axes. Explain what is being
illustrated , if anything.
-x-5, 4x-5, 10x-5 , -3x-5, -4x-5, -10x-5, -5, -0.5x-5 , 0.5x-5
B. Plot all of the linear functions on the new list below on the SAME set of axes. Explain what is being illustrated , if anything.
0.5x-4, 0.5x+4, 0.5x-2, 0.5x+10, 0.5x, 0.5x-17, 0.5x+13, 0.5x-2.5
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2. Do Problem 8 page 100 of our Math 130 Laboratory Manual. Follow the instructions there
very carefully. Especially , plot both relevant exponential functions on the same set of axes
in a manner which shows the point of intersection.
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3. Exponential Growth Functions vs Power Functions: The Exponentials Win
Every exponential growth function eventually overtakes and beats every power
function , no matter how small the exponential growth rate and no matter how
large the power . Illustrate this by setting
y1:= (1.005) ^x y2: 2 000 * x y3 :=x^4 y4 := x^9 and then using Maple to find the point where
y1 overtakes y2 once and for all , finally and forever; the point where y1 overtakes y3 once and for all ,
finally and forever; and the point where y1 overtakes y4 once and for all , finally and forever;
You may need to do some subs experimenting. and some fsolve playing. Illustrate each point of
overtake by an appropriately restricted plot. So you will need to include three plots: The first showing y1
overtaking y2 , the second showing y1 overtaking y3 , and the third showing y1 overtaking y4.
Make it clear what you are doing and thinking by liberal use of comments.
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4. When we wish to find the equation of the line connecting two given points using Maple , we need
only tell Maple to solve for m ( the slope ) and b ( the intercept) with the two given points
substituted. For example , to have Maple find the equation of the line connecting ( 5 , 23 ) and
( 11 , 71 ) we need only ask
> solve({m*5+b=23,m*11+b=71},{m,b});
To which Maple replies
{m = 8, b = -17}
( Be sure you understand everything in the solve command before proceeding.)
A. (i) Exponential functions are , as we know, also completely determined by two points.
Use Maple to find the unique exponential function y= f(x)=Q0 * a ^x which connects the points
( 3.4 , 34.25876446 ) and (7.8 , 38.19061956). Of course you will have to ask Maple to solve for
Q0 and a now instead of m and b . If the solve command appears to misbehave you might try
using fsolve instead .
(ii) After you have found Q0 and a , plot the function and verify that the two points (3.4 , 34.25876446)
and ( 7.8 , 38.19061956) are on the graph , as they should be.
B. (i) Pure power functions are , as we know, also completely determined by two points.
Use Maple to find the unique exponential function y= f(x)=A * (x ^p )which connects the points
(2.4, 48.69890911) and(4.8 , 147.6274866) Of course you will have to ask Maple to solve for
A and p now instead of Q0 and a . If the solve command appears to misbehave you might try
using fsolve instead .
(ii) After you have found A and p , plot the function and verify that the two points (2.4, 48.69890911)
and (4.8 , 147.6274866) are on the graph , as they should be.
C. Quadratic functions (parabolas) have the form y = f(x)= a*x^2+b*x+c . They are determined
by three points . Use an approach similar to that in problem A and B above to find
the equation of the parabola which connects the three points ( 3 , 6) , ( 5 , 24) and (7 , 58)
(ii) After you have found a , b and c plot the parabola that results and verify that the points
( 3 , 6) , ( 5 , 24) and (7 , 58) are on the graph , as they should be.
(ii) The equation of the "parabola" which connects the three points ( 2 , 9 ) ( 5 , 24 )
and (7 , 34). Plot the "parabola" that results. Explain what is happening here and
why.
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5. Problem 2 (a) , page 97 of the Maple/Calculus Lab Manual:
You are about to buy your first used car. You have a choice of buying a 1986 Chevy
for $3650 or a 1986 Toyota $4400. Your mechanic has assured you that both cars
are in good shape and should give you many repair free miles. The Chevy gets 18
miles per gallon of gasoline and the Toyota gets 35 miles per gallon . Assume that
gasoline costs $1.20 per gallon. How many miles would you have to drive before the
cost of buying and fueling the Chevy becomes greater than the cost of buying
and fueling the Toyota ?