restart;Math 151 Lab 10/01/09 Michael Pelsmajer and Jiariu Yang
Instructions: Work through the following problems in a Maple notebook.
Save often! When you are done, revise your work so that it is easy for Jiariu (the TA) to find and understand your solutions. Then drop offthe assignment in Blackboard. Your work is due by 5pm on Wednesday.1. Maple can be made to work with implicitly defined functions, finding derivatives and so forth.
Example Find the slopes of the tangent lines to the graph of
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
at all points for which LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbW5HRiQ2JFEiMUYnRj5GPi1GIzYkLUZaNiRRIzEwRidGPkY+LyUubGluZXRoaWNrbmVzc0dGZm4vJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGYG8vJSliZXZlbGxlZEdGQkYrRj5GK0Y+ .
Then plot the curve together with these tangent lines.First, let's plot the equation, and find all the (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUSJ5RidGNEY3Rj5GK0Y+) points on the graph with 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 .
We start by naming the equation (not a function):eqn := x^3 + y^3 = 2*x*y;and drawing the graph. with(plots):with(plottools):
implicitplot(eqn, x=-1.2..1.2, y=-1.2..1.2, grid=[90,90], color="Green");In the plot, grid[90,90] works like numpoints=700 from the first assignment.Try changing it to grid[10,10] to see what happens.(Don't worry about "with(plots)" and "with(plottools)".)We'll substitute 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 into eqn,
then ask Maple for approximate solutions which we'll name approx_sols.eqnOneTenth:=subs(x=1/10,eqn);
approx_sols:=fsolve(eqnOneTenth,y);Apparently, there are three LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-values for 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 . (Look at the plot; this makes sense, right?)
They are called "approx_sols[1]", "approx_sols[2]", "approx_sols[3]".Let's focus on the third one.We can plot the point, using "display" to combine it with the graph of the curve.TheCurve:= implicitplot(eqn, x=-1.2..1.2, y=-1.2..1.2, grid=[52,52], color="Green"):
Point3:= point([1/10, approx_sols[3]], color = "Indigo", symbolsize = 15):
display(TheCurve,Point3);Next, the derivative of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= with respect to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. I'll give it the name dydx.dydx := implicitdiff(eqn,y,x);Now we're ready. I'll find the tangent line to one of the points and graph it. (Later, you'll find the others.)
I'll find the tangent line to the point ( 1/10 , approx_sols[3] ) .The slope:slope3 := subs( x=1/10, y=approx_sols[3], dydx);The tangent line (right?):tangent3 := x -> slope3*(x-1/10) + approx_sols[3];Plot it together with the previous stuff:TangentLine3:= plot(tangent3,-1.2..1.2,y=-1.2..1.2,color="MediumSpringGreen"):
display(TheCurve,Point3,TangentLine3);1.a. The curve has three points with x=1/10, and each has a tangent line.
Find the other tangent lines, and produce one plot that shows all three tangent lines, together with the graph of the equation, and the three points.
(Tip: To avoid making those little annoying errors, "Copy and Paste" as much as you can!)(Tip: If you end a "plot" with a colon ":", the plot is not actually shown. Later, use "display" to see the plot and other plots, together. See above for examples!)(Tip: All the "plot"s must have -1.2 < x < 1.2 , -1.2 < y < 1.2 , or it won't look right.)2. Consider the equation 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. The graph of this equation is a rather strange looking curve.The main goal of this problem is to find the tangent line to the curve at the point ( 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 , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21vR0YkNi1RKiZ1bWludXMwO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZJLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbW5HRiQ2JFEiNUYnRjVGNS1GIzYkLUZSNiRRIjNGJ0Y1RjUvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmluLyUpYmV2ZWxsZWRHRjpGK0Y1RitGNQ== ), then plot the curve, point, and tangent line in a single plot.(The following might be useful to you.)another_eqn := x*cos(Pi*y) = (x+y)^2;
pointX := ( 23+sqrt(129) )/12;
pointY := -5/3;2.a. But first, verify that the point ( 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 , 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 ) is on the graph of the curve, by plugging it into the equation and showing that both sides are equal. Use Maple to help you do this!2.b. Plot a graph of the equation for -8 < x < 8 , -5 < y < 6 , together with the point.
(This will again verify that the point is on the curve, visually.)2.c. Use Maple to find LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEjZHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjtRJ25vcm1hbEYnLUYjNiQtRiw2JVEjZHhGJ0Y3RjpGPS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGSS8lKWJldmVsbGVkR1EmZmFsc2VGJ0YrRj0= , and then find the slope of the tangent line to the curve at the given point.2.d. Plot a graph of the curve with the point and the tangent, all together.(All the plots must use -8 < x < 8 , -5 < y < 6 , or it won't look right.)3. In this problem, we'll look at an example where we get an answer that seems to be different than the well-known correct answer. Then we'll see a couple ways to check whether the two answers are actually equal.Let's start by reviewing how Maple finds a derivative. (It was covered in "newintro".) It's different depending on whether we're given a function or an expression.expr1 := 3*cos(x);
funct1 := x-> 3*tan(x);To take the derivative of an expression, you have to specify the variable.diff(expr1,x);(Looks right.)To take the derivative of a function:D(funct1);(Looks... uh....)Is that what you were expecting to see? I think not.3.a. What is the formula you memorized for 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 ? According to that, what should 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 be?
(Yes, this is different than what Maple gives.)3.b. Explain why it is that your answer and Maple's answer are equal. (Tip: To figure this out, you don't need Maple.You will need to remember your trignometry, however.)(Since your answers are equal, there is no problem here: both you and Maple are correct.)Maple can sometimes help with this sort of thing.
Maple has a few tools for rewriting an expression or function differently. Fasshauer's Introduction mentioned simplify and factor, but not "expand".Let's see what Maple has to say about it. (Don't read every detail. Just toget an idea, read the first paragraph and look at the first few examples.)?expandThis has a nice side effect.If you have two expressions that ought to to be equal, try expanding bothof them. Maple will probably rewrite them in the same way, which verifies that the two expressions are in fact the same. For example:(x-1)^2 - 1; x*(x-2);expand( (x-1)^2 - 1);expand( x*(x-2) );An example with numbers:(1+sqrt(2))^2; 2*(1+sqrt(2)) +1;expand( (1+sqrt(2))^2 );expand( 2*(1+sqrt(2)) +1 ); Another example:expand(cos(x)^2);
expand(1-sin(x)^2);Okay, so that didn't work. Here's another way to see it:simplify(cos(x)^2 - (1-sin(x)^2));3.c. Redo problem 3.b with these methods. In other words: First try to use expand on each quantity, and see if they're equal. Then try using simplify on the difference, and see if it's zero.