Math 152-02 Lab #2, February 11, 2009 Michael Pelsmajer and Allen Flavell
Try to finish this by the end of class, or Friday. The due date isTuesday at 5pm, so that in case you run into trouble, you have
Monday and Tuesday to get help. If you haven't already, save this file as "[your-name]-Feb11lab".
Save often! When you are done, revise your work so that it is easy for Allen to find and understand your solutions. (Also, for yourself, since you may need to come back to your earlier work.)restart;The Benefits and Pitfalls of Doing Calculus with Technology1. Using technology, some things become much easier. For example,antiderivatives and integrals of 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foo := arctan(x)/(1+x^2);
integrate(foo,x);
integrate(foo,x=-1..sqrt(3));Of course, you can't just copy down the output. You have to correctly understand the question, first of all.1.a. State two antiderivatives of 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 (This should be very easy, using the above Maple output, if you understand the question.)Also, you have to realize that technology may give you an answer that isn't useful or meaningful to you, since it wasn't necessarily designed for exactly what you're studying in your class right now.For example:integrate(arcsin(x),x=Pi..2*Pi);If I asked you for the antiderivative of arcsin(x), and you copied the above expression, I'd give you zero points.1.b. What's the answer you should give? And, more importantly, why? (The answer to "why?" is not because of what Maple showed you.) (Big hint: Plot arcsin.)That all being said, technology is very nice. Do the following problem using Maple:1.c. Evaluate 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 . 2. As far as I can tell, Maple is unable to solve the following antiderivative:int( (1+ln(x))*sqrt(1+ (x*ln(x))^2 ) ,x);(That's why it rewrites the problem instead of evaluating it.)However, if you help Maple get started, Maple can help you finish the problem.2.a. Come up with a clever u-substitution that makes the problem look much easier. 2.b. Get Maple to evaluate the indefinite integral from part 2.a. (The solution will probably contain an inverse hyperbolic function, which you may never have seen before.)2.c. Substitute, to express the answer in terms of the original variable, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= . (Now you have an answer that is presumably correct, even thought you might not know what parts of it mean.)2.d. Check you answer by taking its derivative, then simplify it using "factor", as suggested below.diff( YOUR-FUNCTION-GOES-HERE, x);factor(%);3. Finally, let's consider a case where Maple gives you a correct answer, but you can get a nicer answer by hand.3.a. Find an antiderivative of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1JJW1zdXBHRiQ2JS1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiNRIUYnLUYjNiQtRlA2JUYrLUkjbW5HRiQ2JFEiMkYnRjkvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjktRjY2LVEiK0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZjby1GaW42JFEiNUYnRjlGOUY5LUZpbjYkUSI4RidGOUZcby1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS1GLDYlUSZ1c2luZ0YnL0YwRj1GOUZbcC1GLDYlUSZNYXBsZUYnRmFwRjktRjY2LVEiLkYnRjlGO0Y+RkBGQkZERkZGSEZKRk1GOQ==3.b. Find it by hand. (You should get a much nicer looking answer!). Enter in the answer here:version2 := PUT-YOUR-ANSWER-HERE ;3.c. Find the difference of the two answers, and use Maple to simplify. Explain.