Q1: Consider the piecewise defined function. Analyze the continuity of this function (4C)
Q2: Differentiate
a) b) (6K/U)
Q3: (4A)
Q4: An isosceles trapezoidal drainage gutter is to be made so that the angles at A and B in the
cross-section ABCD are each 120 degree. If the 5 m long sheet of metal that has to be bent to form the
open-topped gutter and the width of the sheet of metal is 60 cm, then determine the dimensions so that
the cross-sectional area will be a maximum.
(6T/I)
Q5: Determine f
11
(x).
a) f(x) = 4 sin
2
(x-2) b) f(x) = 2(cos x) (sec
2 x) (6K/U)
Q6: If y = e
2x – 1, prove that dy = 1-y
2
.
———– —– (5A)
e
2x +1 dx
(3T/I)
12. Find the points on the graph of y = 1 x
3 – 5x – 4 at which the tangent is horizontal. (3K/U)
—- —
3 x
13. Determine the slope of the tangent to h(x) = 2x(x + 1)
3
(x
2 + 2x + 1)
2 at x = -2.Explain how to find the
equation of the normal at x = -2. (4C)
14. Show that there are no tangents to graph to graph of f(x) = 5x + 2 that have a negative slope. (3T/I)
——–
X + 2
15. A swimming pool is treated periodically to control the growth of bacteria. Suppose that t days after a
treatment, the number of bacteria per cubic centimeter is N (t) = 30 t
2
– 240t + 500. Determine the lowest
number of bacteria during the first week after the treatment. (4A)
16. Find values of a, b, c, and d such that g(x) = ax
3 +bx
2 +cx+d has a local maximum at (2, 4) and a local
minimum at (0,0). (4T/I)
17. Find the value of the constant b such that the function f(x) =(X + 1)
1/2 + b/x has a point of inflection at
x = 3.
(5K/U)
1. (5T/I)
2.
3.
4. (K/U 8Marks)
Q5:
(4C)