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MAT187-WRITTEN-HOMEWORK IV, April 6, 11:59 PM

Instructions:

(1)This assignment should be submitted individually to Gradescope.

(2) You should write your answer in the provided space in the submission template and submit it. If you need more space for any of the questions, you should use the blank space at the end. You should submit the exact template with all the pages.

(3) You are responsible for the legibility of your work.

(4) No late assignment is accepted.

(5) A subset of the questions in the assignment are chosen to be graded. Some of the questions may be graded for completeness and quality of writing only and some for correctness. You will not know which questions will be graded before the due date.

(6) Unless you are explicitly told otherwise for a particular problem, you are always expected to show your work and to give justification for your answers.

(7) You are explicitly told whether you are allowed to use technology (Desmos, MATLAB, or a calculator) in your solution. If you are not asked to use technology, you may not use the results you get from any type of technology to justify of your answer.

(8) Write down your solutions in full sentences, as if you were writing them for another student in the class to read and understand. You can use examples in your textbook as a model for your writing. Only writing mathematical symbols or final answers is not considered a full sentence.

(9) Don’t be sloppy, since your solutions will be judged on precision and completeness and not merely on "basically getting it right". If you need help with writing your homework you can come to any of our course office hours, or go to Math Learning Centre for help.

(10) When working on this assignment, you are encouraged to work with your peers, and ask your questions during question hours or on piazza or tutorials. Your tutorial TA may allocate some time to get you started on the homework with your tutorial group. However, you should write your final solution entirely in your own words. You should be able to explain your solution if asked.

Problem 1. In this question, we will see a technique commonly used in mathematics: comparison. We compute the same quantity in two different ways and compare the result. In general, this method reveals some cool mathematics. In our case, we will indirectly show that converges and find the value it converges to.

You can use the following two identities without proof: For any nonzero real number α, we have

(1)

And

(2)

(1) Use the identity (1) to show that

(2) Use the result in part (a) to find the Maclaurin series for arcsin(x) and determine its radius of convergence.

(3) Use the Maclaurin series for arcsin(x) to express as an infinite power series.

(Hint: your series will have infinitely many terms of the form where cn is not a function of x.)

(4) In this step, we will evaluate using the power series we developed. With the help of the identity (2) show that

(5) Now evaluate using the integration techniques you learned in module A.

Compare your result with the one in part (d).

Is the series convergent? Yes No

If yes, What does it converge to?

(6) Using part (e), compute

converges to

Problem 2. Two polar curves are sketched below, one is r = 1 + 2 sin(θ), shown in blue, and the other is r = √ 2 − 1, shown in orange.

(a) Find the points of intersection between the two polar curves. The points must be expressed in xy-coordinates.

points of intersections are

(b) Find an integral which describes the area of the shaded region. You do not need to compute this integral yet. In your solution, include written justification for how you arrived at this integral.

Problem 3. Consider the parametric curve c(t) shown below. The parameter t varies in [0, 1], with c(0) = (0, 0) and c(1) = (1, 1).

Our goal in this problem is to find a parametric quadratic polynomial,

where ai , bi , ci ∈ R, for i = 1, 2, 3, that approximates c(t). We want to match the endpoint data provided in the plot. Find values for the parameters ai , bi and ci which allows to approximate this curve for t ∈ [0, 1].

a1 =   b1 =   c1 =

a2 =   b2 =   c2 =