Mathematics Exam Questions: Calculus and Complex Analysis, Exams of Mathematics

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Candidates should answer the WHOLE of Section A and THREE questions from Section B. Section A carries 55% of the available marks.

SECTION A

1. A function is defined by f (x) = −|x + 2|.

Sketch the function f (x). [3 marks]

2. State the domain of the function y = f (x) = (x^ + 1) (x − 3)

Given that f is a one-to-one function, find f −^1 (x).

[3 marks]

3. Differentiate with respect to x (i) x^4 sinh x , (ii) (1 + x^3 )^10 , (iii)

1 + esin^ x^. [9 marks]

4. Given that y^4 − 2 x^2 y^2 + x

1 + x = 4 ,

find dy dx in terms of x and y. [4 marks]

5. Determine the following indefinite integrals (i)

(e^4 x^ + x^3 ) dx , (ii)

∫ (^) x 2 x + 5 dx.

[6 marks]

6. Evaluate (i)

0

x (x^2 + 1) dx , (ii)

∫ (^) π 0

x cos x dx.

[7 marks]

SECTION B

12. (a) Find the constants A and B for which 1 (x − 2)(x + 2) =^

A

x − 2 +^

B

x + 2 ,

is true. [3 marks] (b) Evaluate (^) ∫ 1 0

(x^2 − 4) dx ,

using the result in part (a) of this question. [3 marks] (c) Use a suitable substitution to determine the indefinite integral ∫ (^1) e^2 x^ − 3 ex^ dx. [6 marks] (d) Determine whether the integrand below is even, odd, or neither, and use that observation to evaluate the integral ∫ (^5) − 5

(x^3 + x^6 sin x + cos^3 x tan x) dx.

[3 marks]

13. (a) Evaluate ∫ (^1) 0

0

(x^2 y + xy^2 ) dy

dx. [5 marks] (b) Using polar coordinates, or otherwise, integrate f (x, y) = (x^2 + y^2 ) 32

over the area enclosed by the curve x^2 + y^2 = 4 and with the condition y > 0.

[5 marks] (c) Evaluate (^) ∫ ∫

A

(6y^2 cos x) dxdy ,

where A is the region of the xy-plane bounded by the lines y = sin x, x = π 2 and the x axis. [5 marks]

14. (a) Find in polar form all the roots of the equation z^3 = − 8 i ,

and draw a diagram showing their position in the complex plane.

[4 marks] (b) Use sin θ = e iθ−e−iθ 2 i to show that sin^5 θ = 16 1 (sin 5θ − 5 sin 3θ + 10 sin θ).

[4 marks] (c) Using the result in part (b) determine ∫ sin^5 x dx.

[3 marks] (d) Write (^) √ 1 + i ,

in the form a + ib. [4 marks]

15. (a) Sketch the graph y = x x^ + 4− 3. [5 marks] (b) Use the power series expansion of cos √x up to order x^3 to obtain an approximate estimate for the integral ∫ (^1) 0

cos

x dx ,

correct to three decimal places. [5 marks] (c) Given that the slope for sec−^1 x is positive for x > 0, show that d sec−^1 x dx =^

x

x^2 − 1

[5 marks]