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Here are some tips as you prepare for the Final Exam
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One prof has been very forthcoming about the kinds of things to expect on the final exam, here are some tidbits. Do realize that I have no idea myself, and this is just what he has said, or what I have read between the lines. - There will be a couple of improper
integrals to solve (Lesson 12, #1 in my book). He said that, in order to solve the limit you set up in the improper integral, in one case, you will need to use L'Hôpital's Rule. I am predicting that it will be a question like the December 1997 exam in my book, #2(e) on page 211. Also look at April 2000, #7(a) on page 220 and April 2001, #7(b) on page 223. Of course, I
have solutions to all of those questions in the solutions section of my book.
- There will also be one or two improper integrals where you will use the Comparison Theorem, so look at all my examples in Lesson 12, #2. Make sure that you always use a test point from the region of integration to check that your statement is correct about who is bigger between f(x), the original function, and g(x), the function you used as a
comparison.
- Of course, you will be tested on all the techniques of integration. I hope you have made the index cards as I suggested on page 6, and have been shuffling, and checking yourself throughout. Be ready for at least one of each technique (u substitution, trig integrals, integration by parts, trigonometric substitution, and partial fractions). Study all of my examples in Lessons 4, 7, 8, 9, and 10.
- Again, he
has assured everyone that there will be no graphing of Parametric or Polar equations. You will be doing arc length and/or surface area with a parametric equation, and arc length with a polar curve. Sounds very likely that you will be given a cardioid and asked to find the circumference of it (Lesson 15). Almost certainly, you will be told to setup the integral but do not solve. Remember that a cardioid is fully drawn from 0 to
2π. If it is a cosine cardioid you can integrate from 0 to π and double the integral. If it is a sine cardioid, you can integrate from π/2 to 3π/2 and double it.
- Be ready for the perfect square type of arc length like I show in Lesson 13, #1.
- Be ready to use the Disc or Washer method and the Cylindrical Shell method for volume. You will certainly be doing one of each. It is also likely that they
will have you sketch the region (not the 3-D volume, just the graphs that are being rotated). As always, that is just a matter of finding the points of intersection for your endpoints, then plotting those points and one more point in between for each curve to draw the graphs.
- Most of the applications (volume, arc length, and surface area) will be setup but do not solve integrals. So, be careful to make no mistakes in the setup. Get
the correct endpoints. Don't forget your dx or dy or dt or dθ on the end. Make sure you are using x values for the endpoints if it is a dx integral; y values if it is a dy integral; t values if it is a dt value; θ values if it is a dθ integral. Always use a test point between the two endpoints to confirm that each part of your integral is positive, adjusting if necessary.
- Best wishes and STUDY HARD!
The department posted SOLUTIONS for Assignment 1 (these are not my solutions). Click here.The department posted SOLUTIONS for Assignment 2 (these are not my solutions). Click here.The department posted SOLUTIONS for Assignment 3 (these are not my solutions). Click here.The department posted SOLUTIONS for Assignment 4 (these are not my solutions). Click
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