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An old university examination paper from the university of wales, aberystwyth, institute of mathematics and physics, for the algebra and calculus course, specifically for the calculus topic. The exam covers various concepts such as limits, derivatives, taylor series expansion, integration by parts, inverse functions, and trigonometric identities. The questions require the application of mathematical concepts and techniques to solve problems. This document could be useful for students preparing for calculus exams or for self-study.
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MA10020 - Algebra and Calculus: Calculus Paper
Time allowed - 2 hours
Section A
f (x) = x
x^3 − 4 x. Compute limx→− 10 f (x). [5 marks]
[5 marks]
(i) (^) dxd ln tan^2 2 x, (ii) (^) dxd √ 1 −^1 cos x [5 marks]
f (x) = esin−^1 x. [5 marks]
ex^2 /^3 cos 2x =
n=
cnxn, determine the coefficients c 0 , c 1 and c 2. [5 marks]
0 x cos 5xdx = − 252. [5 marks]
Show that ∫^0 ∞ e−^7 x^2 xdx = 141 [5 marks]
Show that (^) ∫ (^) e
1 x ln xdx =^14 e^2 +^14. [5 marks]
f (x) = ex^ ln (1 + x) about x = 0 vanishes. [10 marks] (ii) Use Taylor series expansion to write
x^4 + 5x^3 + 10x^2 + 6 =
n=
cn(x − 1)n,
and determine the coefficients cn for n = 0, 1 , 2 , 3 , 4. [10 marks]
I =
∫ (^2) π 0 ex^ sin 5xdx, J =
∫ (^2) π 0 ex^ cos 5xdx.
[10 marks]
f (x) = x^3 + ax^2 + 3x + 10 have a single stationary point. For these values of a, classify the nature of the stationary point. [10 marks] (ii) Determine the stationary points of the function f (x) = x
(^2) + 3x x^2 + 4 and determine their nature. Use this to plot the graph of the function. [10 marks]
