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The instructions and problems for a university-level mathematics final exam, including short-answer and full-solution problems covering topics such as integration, differentiation, power series, and limits.
Typology: Exams
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Name: Student Number:
There are 11 pages in this test including this cover page. No calculators, books, notes, or electronic devices of any kind are permitted. Unless otherwise indicated, show all your work.
Rules governing formal examinations:
Problem # Value Grade
Total 100
I have read and understood the instructions and agree to abide by them.
Signed:
(a) Evaluate
(2y + 1)^5 dy.
(b) Evaluate
e
1 x(ln(x))^2 dx.
(c) Let f (x) =
ex^ cos
(^3) (t)dt. Find f ′(x).
(d) Calculate the volume of the solid obtained by rotating the region above the x-axis, below the curve y = sin(x)/x, and between the lines x = π/2 and x = π about
the y-axis.
(i) Evaluate limn→∞
∑n j=
1 n cos(^
jπ 2 n ).
(j) Find the midpoint rule approximation to
1
1 x dx^ with^ n^ = 3.
(k) Evaluate
− 1
1 x^2 /^3 dx^ or show that it diverges.
(l) Evaluate limx→ (^0) x^13
∫ (^2) x x sin(t
(^2) )dt.
Full-Solution Problems. In questions 2-6, justify your answers and show all your work.
(a) ∫ x √ 1 − x^4
dx.
(b) ∫ (^1)
0
2 x + 3 (x + 1)^2
dx.
(b) ([5 marks]) Let R be the region under the curve y = e−x^ and above the x-axis, for 0 ≤ x ≤ 1. Find the x-coordinate of the centroid of R.
j=1 aj^ x j (^) converges.
(a) Prove that
j=1 aj^ (x/2)
j (^) converges absolutely.
(b) Must
j=1 a
2 j x j (^) converge? (Either prove that this is the case, or else provide a counterexample.)