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