3.
Evaluate the following integrals: (a)
2x 3
3
x 8 dx 7
(b)
(x x
(c)
(3
(4 marks)
)dx
(4 marks)
t 2e t )dt
(4 marks)
2
(d)
(sin3 2 cos3) d
(7 marks)
0
(e) 4.
(a)
1 1
(3e 2 y
3 )dy e2y
(6 marks)
A curve is described by the equation:
y x 4 2x 2 1
(b)
(i)
Find the gradient of the curve at the point (2,9).
(5 marks)
(ii)
Hence find the equation of the tangent to the curve at the same point. (4 marks)
(iii)
State the gradient of the normal at the same point.
(2 marks)
A rectangular piece of land of area 25m2 is to be marked out and a perimeter fence built around it (i)
Sketch a diagram of the piece of land showing its dimensions. (1 mark)
(ii)
Show that the perimeter (P) of the rectangle can be expressed in terms of one of its dimensions (x) as:
P 2x (iii)
50 x
(3 marks)
Given that x can vary, use differentiation to find the dimensions of the piece of ground that requires the minimum length of perimeter fence. You need to justify that your answer leads to the minimum perimeter length. (10 marks) Continued…
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