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PAPER CODE NO. MATH
EXAMINER: Prof A.C. Irving DEPARTMENT: Math Sciences TEL.NO. 43782
JANUARY 2010 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
- Write down the natural domain of the function
f (x) =
1 − x 3 x − 2 and make a rough sketch of the function. Obtain the inverse function f −^1 (x) and identify its natural domain and range.
[7 marks]
- Solve the inequalities
(i) | 1 − x| ≤ 2 , (ii) − 1 <
x − 1
[6 marks]
- Determine, giving reasons, whether any of the following functions are odd, or even functions of x
(i) | sin(3x)| , (ii) x cos(x) − x , (iii)
x^2 − 2 |x − 1 |
If any are periodic functions of x, give the the corresponding period.
[5 marks]
- Find the following limits, where they exist:
(i) lim x→ 1
x^2 − 4 x + 3 x − 1
, (ii) lim x→ 2
x^2 − x − 2 ln(x − 1)
, (iii) lim θ→ π 2
cos θ 1 + cos 2θ
[6 marks]
- Differentiate 1 x − 1 with respect to x from first principles, i.e. by using an appropriate limiting process. [4 marks]
S E C T I O N B
- (a) Draw the graph of the function f defined by
f (x) =
0 for x ≤ − 1 , |x| for − 1 < x < 0 , 2 x^2 for 0 ≤ x < 1 , − 3 x^2 + 8x − 3 for 1 < x ≤ 2 , 1 /(x − 2) for x > 2.
Use this to help you determine whether or not f is continuous at the points x = − 1 , 0 , 1 , 2 and 4, giving your arguments. [8 marks] (b) The function R(x) is defined by
R(x) =
1 − 4 x for x ≤ 0 , 1 /(bx + c) for x > 0.
Find, giving reasons, the values of the constants b and c so that R is differentiable at x = 0. Does R(x), so defined, have any points of discontinuity and, if so, where?
[7 marks]
- Construct a sketch of the function
F (x) =
xex 1 + ex
making use of the Newton Raphson method, if necessary, to locate the position of any local extrema to 4 decimal places. Identify any asymptotes or zeros of F (x).
[15 marks]
- Evaluate the following integrals, using the hints provided or another method which you should identify:
(i) ∫ (^1)
1 +
2 x
dx (substitution) ;
(ii) ∫ (^) x + 4
x^2 − 5 x + 6
dx (partial fractions) ;
(iii) ∫ (^8) x (^2) − 3 x + 8
4 x^3 − 8 x^2 + x − 2
dx (partial fractions and substitution).
[15 marks]
- (a) Find the largest possible area of an isosceles triangle if the length of each of its two equal sides is 10m. [6 marks] (b) A one-metre length of stiff wire is cut into two pieces which are then bent into closed shapes. One piece is bent into a circle and the other is bent into a square. Find the length of the piece used for the square if the sum of the areas of the circle and square is (i) a maximum and (ii) a minimum.
[9 marks]
