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PAPER CODE NO. MATH
EXAMINER: Prof A.C. Irving DEPARTMENT: Math Sciences TEL.NO. 43782
JANUARY 2008 EXAMINATIONS
Bachelor of Arts: Year 1 Bachelor of Science : Year 1 Master of Mathematics : Year 1 Master of Physics : Year 1
FOUNDATION MODULE I
TIME ALLOWED : Two Hours and a Half
INSTRUCTIONS TO CANDIDATES
Answer all of Section A and THREE questions from Section B. The marks shown against questions, or parts of questions, indicate their relative weight. Section A carries 55% of the available marks.
S E C T I O N A
- Sketch the graph of the function f defined by:
f (x) =
−x for x < −^12 , 1 − |x| for −^12 < x < 1 , x^2 for x ≥ 1. Hence determine, giving reasons, whether or not f is continuous at the points x = −^12 , 0 and 1.
[6 marks]
- Find the general solution of
2 sin
( θ +
π 4
) = −
[3 marks]
- State the natural domain of the function
f (x) =
3 x − 2 1 + x
Find the inverse function f −^1 (x) of this function and state its natural domain and range.
[6 marks]
- Which of the following functions are odd or even?
(a) g(x) = x sin(x) + cos(x^3 ) , (b) s(t) =
1 + 2t^2 t^3 − 3 t
Give reasons.
[4 marks]
- Evaluate the following limits, where they exist:
(a) lim x→ (^23)
6 x^2 − 7 x + 2 6 x^2 + 8x − 8
, (b) lim x→ π 4
1 − tan x cos(2x)
, (c) lim x→ 0
3 x − 1 1 − ex^
[5 marks]
S E C T I O N B
- Given that
F (x) =
2 x^2 − 3 x x − 2
find constants A, B and C such that
F (x) = A + Bx +
C
x − 2
Find intervals of x on which the function is (i) increasing (ii) decreasing (iii) concave up and (iv) concave down. Locate any zeros, asymptotes, extrema and inflection points of F (x). Classify the extrema and then provide a sketch of the graph.
[15 marks]
- (i) Given that x = c satisfies the equation f (x) = x, verify that x = c also satisfies the equation f (f (x)) = x. (ii) The function f is defined by
f (x) = 2x(2x − 1).
Find the solutions of the equation
f (x) = x.
(iii) Show that f (f (x)) = 64x^4 − 64 x^3 + 8x^2 + 4x and use the result of (i) to find all the solutions of the equation
f (f (x)) = x.
(iv) Apply the Newton-Raphson method with initial value x 0 = 1/ 2 directly to f (f (x)) − x = 0 to find a solution of this equation correct to 4 decimal places. Compare your answer to that obtained in part (iii).
[15 marks]
- A new electrical supply cable is to be laid underground between the supply substation S shown in the accompanying diagram and the re- distribution point R which is on the opposite side of a river. The width of the river is 120m and the point R is 800m downstream as shown.
Since the cost of burying the cable under the river is 5 times the cost of burying it on land, the plan is to lay most of the cable under the river bank and then lay the rest in a straight line under the river between points P and R as shown. If the cost of burying on land is £80 per metre, find the minimum total cost of the operation and the distance laid under water. Give your answer to the nearest £100. How much money (to the nearest £100) is being saved by using this optimal route as opposed to simply minimising the length of cable laid under water?
[15 marks]
- Obtain approximate values for the integral
∫ (^3)
1
x 1 + x^2
dx
using (i) the trapezoidal rule and (ii) Simpsons’ rule with the interval [1, 3] divided into 10 equal sub-intervals in each case. Give your answers correct to 5 decimal places. Compare your answer with the exact result and comment very briefly on your findings.
[15 marks]
