It is due on Friday December 13 by 5:00 PM
You may work with a (single) partner. Each team of two to submit one report All arithmetic and algebra is to be done by Maple. What you submit must be printed Maple worksheets..e.g. photo copies of Maple worksheets are not acceptable.
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1. Antiderivatives
Have Maple compute the antiderivatives (= indefinite integrals) for each of the followiong functions
( you can simply call on
> int (f(x) , x);
to do this , where you type in the appropriate formula for f(x)
(a) f(x):= 3*x^2 -2*x;
(b) f(x) := (sec(x))^2 ;
(c) f(x) := sec(x) ;
(d) f(x) := 3*x^2 -2*x;
(e) f(x): = 2^x;
(f) f(x): = x^x ;
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2. Numerical Approximation in the Absence of an Antiderivaive.
Note Maple's failure to integrate f(x) = x^x in #1 (f) above. Nonetheless we can evaluate definite
integrals whose integrand is x^x by numerical approximation , for example by actual left and right sums.
(a) Use the left sum and the right sum commands with 2, 10 , 50, and finally 250 subdivisions to estimate
the integral of x^x from x=1 to x=2.
(b) Take the average of the left and right sums with 250 subdivisions to get a better estimate.
(c) Check your estimate by having Maple give its estimate via
> evalf(int(x^x,x=1..2));
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3. An Optimization Exercise
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 ). Include the instruction
" scaling = CONSTRAINED" as part of youtr plot command. This will show the geometry of
the situation more clearly.
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4. An Area Problem (Adapted from # 25 , page 372 of our textbook)
Find the area of the region enclosed by the parabola y1:=(x^2/4)-1; and the
line y2:=(x/4)+4; Sketch. Have Maple do the algebra that is needed to determine the limits of
integration. Check your answer in the back of the textbook.
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5. Fundamental Theorem Workout
Work problem #3 of page 293 of our Maple Calculus Lab Manual
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The complete assignment is due on November 19 at lecture.
You may work with a (single) partner. Each team of two 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.
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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. The Beachlover Swim Club sells memberships at $100 per season. Members must pay $3.00
for each visit. Non members are permitted but they must pay $7.00 per vsit.
(a) Write down the expression which gives the total cot of making x visits for members.
In Maple , define the variable m to be equal to this expression.
(b) Write down the expression which gives the total cot of making x visits for non members.
In Maple , define the variable n t o be equal to this expression.
(c) Use Maple to plot m and n for x varying from 0 to 30 on the same set of axes.
(d) Use Maple's solve command to determine the value of x for which m and n are equa.
(e) How many visits would a beachlover have to make in order to make it worthwhile to purchase a
membership?
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3. Problem 2 (a) , page 256 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 ?
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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.)
(i) . 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 above to find the equation of the parabola which
connects the three points ( 3 , 6) , ( 5 , 24) and (7 , 58)
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) Now find , by the same method again, 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. Use Maple to do problem #60 of page 153 of Thomas. Have Maple do the derivatives and all
calculations and relvant algebra. Include a plot showing exctly what is shown at the bottom of page 153.
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6. Problem 4 page 264 of the Maple/Calculus Lab Manual: The plot required in part (b)
is absolutely essential. include several plots over smaller and smaller intervals containing x=2.
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7. 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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8. Problem 3 page 276 of the Maple/Calculus Lab Manual:
END OF ASSIGNMENT #1
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