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Try a Free Sample of Grant's Book and Audio Lectures
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Don't have my book or audio? You can download a free sample of my book and audio lectures containing Lessons 1 and 11:
Did you read my tips on how to study and learn this course? If not, here is a link to those important suggestions: Here is a link to the actual assignment, for those of you who don't have it: Study Lesson 1 (Inverse Trigonometric Functions), Lesson 2 (The Fundamental Theorem of Calculus), Lesson 11 (L'Hopital's Rule), Lesson 14 (Parametric Equations: ONLY THE GRAPHING
AND DERIVATIVES PART, my Lesson 14, #1, #2, #3, #5(a,b,c) and #6), and Lesson 15 (Polar Curves: ONLY THE GRAPHING AND DERIVATIVES PART, my Lesson 15, #1 and #4) from my Calculus 2 book to prepare for this assignment. You will also need Lesson 3 (Riemann Sums) but this lesson is very unlikely to ever be tested, so I suggest you just read the lesson, but don't worry about memorizing all of the complicated formulas in it.
Of course, always seek out assistance from my book, your course notes, etc. if you ever hit a question you don't understand, but try not to be learning things as you do an assignment. Learn first, then put your learning to the test.
These are all classic L'Hopital's Rule questions. Study my Lesson 11 thoroughly to prepare.
Study Lesson 1 in my book first. Especially my examples in #1. I don't do one exactly like this. Fact is, the concept comes into play much later when I teach you how to solve integrals by trigonometric substitution. I will use
arctan to represent inverse tangent, and arcsin to represent inverse sine, and arccos to represent inverse cosine, since showing the -1 superscript for inverse functions is undependable online.
- Let arctanx = θ then you know tanθ = x. And you want to find cosθ since you want cos(arctanx).
- Any trig function represents the ratio of two sides of a right triangle. Here, if tanθ = x = x/1, that is like saying the opposite side is x and the adjacent side is 1,
since tan is opposite/adjacent.
- If you know two sides of a right triangle, you should also be able to figure out the third side by Pythagorean Theorem.
- Now that you know all three sides of the triangle, you can determine what any trig function is by computing the appropriate ratio. Find cosθ.
Study Lesson 14 but only study pages 157-161 (#1 to #3), pages 164-166 (#5, parts (a), (b) and (c), and pages 168-175 (#6) at this time. You are not ready to do the integral applications of Parametric Equations yet. You can also ignore my
discussion about increasing/decreasing and concave up/concave down charts in #6. Thankfully, they have phased that rather pointless exercise out of the course.
Study Lesson 15 but only study pages 176-185 (#1) and page 188 (#4) at this time. You are not ready to do the integral applications of Polar Curves yet.
This is quite similar to my #4. I think there is a typo here
that you may want to ask the prof about. It doesn't make sense to figure out where the tangent line is "vertical, horizontal or neither at θ = 3π/2." It can't be all of those things at that one value. I think they just meant find the tangent line at θ = 3π/2. Or maybe they meant to ask two questions. First, find the points where the tangent line is horizontal or vertical. Second, find the tangent line at θ = 3π/2. Ask the prof to check the
wording of this question.
You can look for horizontal and vertical tangent lines for polar just like you do for parametric. The variable θ in polar is just like the variable t in parametric. But, the difference is that you are not given equations for x and y in polar, so you must create your own equations as I illustrate in my #4.
Again, make sure you have studied Lesson 15, #1 before attempting this question.
This is using the method I illustrate in my Lesson 3, #3. I recommend you do this question open book, just following my examples. Note, because you have been given an x-cubed in your function, that will cause i-cubed to show up among other things, so you
will need the summation formula for i-cubed. That link shows you the formula, they just use k-cubed instead of i-cubed, but that's irrelevant. The answer in terms of n they give you is exactly what you need. |
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