1.Let f(x) = cos(x2) and g(x) = ex, for –1.5 ≤ x ≤ 0.5.
Find the area of the region enclosed by the graphs of f and g. (Total 6 marks)
2.Let f(x) = Aekx + 3. Part of the graph of f is shown below.
The y-intercept is at (0, 13).
(a)Show that A =10.
(b)Given that f(15) = 3.49 (correct to 3 significant figures), find the value of k. (3)
(c)(i)Using your value of k, find f′(x).
(ii)Hence, explain why f is a decreasing function.
(iii)Write down the equation of the horizontal asymptote of the graph f. (5)
Let g(x) = –x2 + 12x – 24.
(d)Find the area enclosed by the graphs of f and g.
(Total 16 marks)
3.The following diagram shows the graphs of f (x) = ln (3x – 2) + 1 and g (x) = – 4 cos (0.5x) + 2, for 1 £ x £ 10.
(a)Let A be the area of the region enclosed by the curves of f and g. (i)Find an expression for A.
(ii)Calculate the value of A.
(b)(i)Find f ′ (x).
(ii)Find g′ (x).
(c)There are two values of x for which the gradient of f is equal to the gradient of g. Find both these values of x. (4)
(Total 14 marks)
4.The graph of f(x) = , for –2 ≤ x ≤ 2, is shown below.
The region enclosed by the curve of f and the x-axis is rotated 360° about the x-axis. Find the volume of the solid formed.
(Total 6 marks)
5.The graph of y = between x = 0 and x = a is rotated 360° about the x-axis. The volume of the solid formed is 32π. Find the value of a. (Total 7 marks)