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LANCASTER UNIVERSITY
2012 EXAMINATIONS
PART II
MATHEMATICS & STATISTICS
Math 271 : Minor Course in Mathematics Calculus and Integration 2 hours
Answer TWO questions in Section A and TWO questions in Section B. Use separate answer booklets for Section A and Section B. SECTION A: Calculus
A1. (i) The logistic function f is defined as f (x) = (^) 1 +^1 e−x (−∞ < x < ∞), and f has inverse function g. (a) Sketch graphs of f (x) and g(x), with asymptotes, on the same axes. [6] (b) Derive a formula for g, and state its domain of definition. [5] (c) Calculate the derivatives f ′(x) and g′(x). [5] (ii) Prove by induction that 12 + 2^2 +... + n^2 =^16 n(n + 1)(2n + 1) for all positive integers n. [6] (iii) Using partial fractions or otherwise, obtain a simple formula for Sn = (^1 1) · 4 + (^2 1) · 5 + (^3 1) · 6 +... + (^) n · (n^1 + 3) , and hence find the sum to infinity 1 1 · 4 +^
2 · 5 +^
3 · 6 +^....
Here a · b denotes the product of a and b.
[8]
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SECTION A continued
A2. (i) State the formula for the Maclaurin Series of a function f (x), and calculate the series for f (x) = (1 − x)−^1 /^2 , up to and including the term in x^4. [10] (ii) Let In =
∫ (^) π 0 cos
n (^) t dt. (a) Establish the recurrence relation In = n^ − n 1 In− 2 for all integers n ≥ 1. [8] (b) Evaluate I 6 and I 7. [4] (c) Hence find (^) ∫ (^1) − 1 √^ x^6 dx 1 − x^2 and
− 1 x
6 √ 1 − x (^2) dx. [8]
A3. (i) (a) By considering eiθ^ = cos θ + i sin θ, prove that sin^6 θ = − 32 1 cos 6θ + 16 3 cos 4θ − 1532 cos 2θ + 165. (^) [6]
(b) Hence find (^) ∫ sin^6 θ dθ. [3] (ii) By integrating by parts, evaluate I =
∫ (^) π 0 sinh 2x^ cos 3x dx.^ [10] (iii) Let sinh−^1 be the inverse of the hyperbolic function sinh x. (a) Calculate (^) dxd sinh−^1 x. [5] (b) By integrating by parts, find (^) ∫ sinh−^1 x dx. [6]
please turn over
SECTION B continued
B2. (i) Let C be the curve given by 3 y^2 = x^2 (1 − x) (0 ≤ x ≤ 1). (a) Verify that x = sin^2 t and y = √^13 sin^2 t cos t gives a point on C for 0 ≤ t ≤ π and hence sketch C. [6] (b) Find the gradients of the tangents to C at x = 34 and plot the tangents on your diagram. [8] (c) Show that the arclength of C is L = √^13
∫ (^) π 0 (3 cos
(^2) t + 1) sin t dt
and evaluate this integral. [8] (ii) Let T be the triangular region that is bounded by the lines x = 0, y = 1 and y = x. (a) Write down both repeated integrals for ∫ ∫ T
xn y dxdy.^ [2] (b) By considering these repeated integrals, or otherwise, show that ∫ (^1) 0 x
n (^) log x dx = −^1 (n + 1)^2.^ [6]
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SECTION B continued
B3. (i) Let four points (X, Y ) be listed in the table below X 1 3 4 7 Y 10 8 5 2. By considering Q(U, V ) = ∑(Y − V X − U )^2 , briefly derive the normal equations and hence find the straight line y = V x + U that passes closest to the four points. You should express the solution as V = a b and U = c d with a, b, c and d integers. [12] (ii) (a) The ellipse E with equation E : x 2 a^2 +^
y^2 b^2 = 1 encloses a region that has area A = 4a
∫ (^) b 0
1 − y 2 b^2 dy. Evaluate this integral. [8] (b) An elliptical cone has equation 3 x^2 + 2y^2 = z for 0 ≤ z ≤ 1. By using the result of 3(ii)(a) and slicing principle, find the volume of the cone. [10]
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