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I have given some extra hints for questions 3 and 4 below.
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Don't have my book or audio lectures? You can download a sample containing some of these lessons here: Did you read my tips on how to study and learn Math
1700? If not, here is a link to those important suggestions: These are tips for the first assignment in the Distance/Online Math 1700 course, but I strongly recommend that you do this assignment as homework even if you are taking the classroom lecture section of the course. These assignments are very good (and challenging) practice.
It is possible that you are doing the topics in a different order in the classroom lecture sections, so you may need to wait until later before tackling this assignment. Here is a link to the actual assignment, in case you don't have it: Study Lesson 14 (Parametric Equations) and Lesson 15 (Polar Curves) from my Calculus 2 book to prepare for this assignment.
Part (a) is asking you to isolate t in one of the parametric equations and substitute into the other equation to remove t from the problem. Depending on how you isolate t, you either end up with y as a function of x, or x
as a function of y. I think you should find it easier to isolate t in the second equation and substitute into the first, to create x as a function of y.
Part (b) expects you to draw the graph strictly by making a table of values, just like my first example I do at the start of Lesson 14. I suggest you find the values of t that give horizontal or vertical tangent lines
(there aren't many), then pick at least one t value that is lower than the t values you find, and at least one t value that is higher, and plot your points. Remember, always connect the dots in order of increasing t, and don't forget to put arrows on the curve to show the direction of motion. The Cartesian equation you found in part (a) may help you draw the graph, but you still need to use the t table of values to clearly identify the
direction of motion.
Similar to my Lesson 14, questions 1, 5(b) and 6(i).
Use the arc length formula for parametric equations that I illustrate in Lesson 14, question 5(d), part (i). You will find the identity sin(2z) = 2 sin(z)cos(z) of use.
Use the surface area formula for parametric equations that I illustrate in Lesson 14, question 5(d), part (iii). Keep your eyes open for a perfect square.
Similar to my Lesson 15, question 1 (b) and (c).
Similar to my Lesson 15, question 4.
Use the area formula for polar curves I use in my Lesson 15, question 2. Note that you have to find the area inside one curve and outside the other, so first find where the two curves intersect to get your integral's endpoints. With trig, this is
often most easily done by inspection. Try theta = 0, pi/6, pi/4, etc. to find the angles where the two graphs intersect. That is your endpoints, of course. Then it is the area of the outer curve minus the area of the inner curve.
I strongly recommend you sketch the two curves, using tables of values as I teach you in this lesson, so that you can properly visualize the area they describe. The outer curve is the
one further from the origin; the inner curve is nearer to the origin. Draw the angle lines representing the angles of intersection then visualize rotating counterclockwise from the smaller angle to the larger angle, like a radar screen in an air traffic control centre. Visualize the angle line sweeping around the curves.
Use the arc length formula for polar curves I use in my Lesson 15, question 3. Keep an eye open for a common factor that can be pulled out of the square root completely to be able to solve the
integral.
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