Calculus 1501 - Homework #3
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Calculus 1501 - Homework #3
Each question should be submitted on Gradescope (www.gradescope.ca) with your handwritten work clearly shown, and it is expected that homework is done individuallg. your work should refer to the material learned in this course and the material presented in your previous math courses.
Remember to justify your calculations and conclusions. A poorly justiied solution will not receive many marks. A solution with just a inal answer and no work shown will recieve O. Be sure to clearly state your reasoning "since the function satisfes the conditions of. . . ", " ...which we evaluate using the identitg. . . ". calculators are not allowed in this course, so there may be a loss of marks if a simpliication is done without showing your rough work.
This assignment is due by 11:oo pm Eastern Time on Thursday August 1Oth , and each question is to be submitted via Gradescope.
Question # 1. (6 marks)
(a) Recall the defnition of (n(T)) from the lessons in 4c. There is a f从ed real number k so that ( 
= 
for all n. Find k, making sure to justifg gour answer.
(b) suppose p and q are f从ed positive integers. Rewrite ^q 1 - 从p as an e从pression of the form, (1 十 f (从)) , bg fnding (and justfging) suitable function f (从) and suitable real number T in terms of p and q.
(c) Find a power series for arcsin(从) centered around O (gour fnal answer should have no binomial coe历cients in it). Hint: use an approach similar to how we handled arctan(从) in our lessons.
(d) specifcallg bg using the ratio test, determine the radius of convergence for the power series gou created for arcsin(从) .
Question # 2. (5 marks)
(a) Find each of the slopes of the two tangent lines that e从ist at the unique point where the polar curve, T = sin (
) crosses over itself.
(b) Find all the points on the parametric curve, (从, g) =(cos(t) 十 tsin(t), sin(t) - t cos(t)), t 持 O, where the tangent lines are vertical.
Question # 3. (6 marks)
(a) Find the area of the region enclosed bg the polar curve T = cos(3θ) + 2 .
(b) suppose that n is some f①ed positive integer. In terms of nfnd the area enclosed bg one loop of T = 4 sin(nθ) .
(c) Fi① a real number q with 1 < q < 2 . Give an e①pression (involving a defnite integral) representing the area of the overlap between the two loops of the curve’ T = cos(qθ) that are present over O < θ < 
. You do not
need to evaluate this integral.
T = cos(qθ)
Question # 4. (7 marks)
(a) calculate the arc-length of the curve g = e22(从)e
1 over [ln(3), ln(5)] .
(b) calculate the arc-length of the curve (①, g) =(1 + 3t2 , 4 + 2t3 ) for O < t < 1 .
(c) The polar curve’ T = 1 ’when graphed looks like a spiral. As we follow the spiral counterclockwise (θ 喻 …)
the radial distance of the points on the curve decreases (T 喻 O) meaning that the pole is on the spiral in the limit. show that the arc-length along the spiral from the polar point (1, 1) to the pole is fnite. Hint: set up an improper integral to represent the arc-length. Remember’we onlg need to show the arc-length is fnite’ we don,t need the specifc value to show this. so can gou recall ang results in 2I that could help?
Question # 5. (4 marks) In this question we,ll be working with the centroid of a loop in a curve. If we have a parametric curve, (①, g) =(f (t), g(t)) which has a loop from t = a to t = b, then the centroid of that loop is the point, (① , g) given bg the following two formulas:
① = lab f (t)g(t)f、(t)dt
labg(t)f、(t)dt
g = lab b(g)(t)f (t)g、(t)dt
la f (t)g、(t)dt
(a) For the curve given bg (①, g) = (t3 - t, t2 ) fnd the unique point of self-intersection and specifcallg mention
the two parameters (t values) which produce this point on the curve.
(b) Find the centroid of the loop of the curve (①, g) = (t3 - t, t2 ) .
Question # 6. (6 marks)
…
(a) suppose we have a function represented bg a power series, f (①) = 
an ①n and we know that f (0) = -5,
f、(0) = 3, and f、、(0) = 8 . Find the value of a2 .
…
(b) suppose we have a function represented bg a power series, g(①) = 
bn ①n . write out the frst fve terms of
the power series centered at 0 for cos(①)g(①) .
(c) consider the diferential equation,
g、、+ e① g、+ cos(①)g = ①
suppose g(①) = 
cn ①n is a solution to this DE, represented bg a Maclaurin series with a nonzero radius of convergence. suppose that we know: g(0) = 5 and g、(0) = 7 . Determine c2 , c3 , and c4 .
2023-08-08