Calculus 2
1.	Let g(x) = , for x > 0.
(a)	Use the quotient rule to show that .
(4)
 
(b)	The graph of g has a maximum point at A. Find the x-coordinate of A.
(3)
(Total 7 marks)

2.	Let f(x) = kx4. The point P(1, k) lies on the curve of f. At P, the normal to the curve is parallel to y = . Find the value of k.
(Total 6 marks)

3.	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.
(2)

(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.
(6)
(Total 16 marks)

4.	Consider f(x) = x ln(4 – x2), for –2 < x < 2. The graph of f is given below.

(a)	Let P and Q be points on the curve of f where the tangent to the graph of f is parallel to the x-axis.
(i)	Find the x-coordinate of P and of Q.
(ii)	Consider f(x) = k. Write down all values of k for which there are exactly two solutions.
(5)
 
	Let g(x) = x3 ln(4 – x2), for –2 < x < 2.
(b)	Show that g′(x) = .
(4)
 
(c)	Sketch the graph of g′.
(2)
 
(d)	Consider g′(x) = w. Write down all values of w for which there are exactly two solutions.
(3)
(Total 14 marks)

5.	Let f(x) = x ln(4 – x2), for –2 < x < 2. The graph of f is shown below.
		
	The graph of f crosses the x-axis at x = a, x = 0 and x = b.
 
(a)	Find the value of a and of b.
(3)
 
	The graph of f has a maximum value when x = c.
(b)	Find the value of c.
(2)
 
(c)	The region under the graph of f from x = 0 to x = c is rotated 360° about the x-axis. Find the volume of the solid formed.
(3)
 
(d)	Let R be the region enclosed by the curve, the x-axis and the line x = c, between x = a and x = c.
	Find the area of R.
(4)
(Total 12 marks)

6.	Let f (x) = ex (1 – x2).
(a)	Show that f ′ (x) = ex (1 – 2x – x2).
(3)
 
	Part of the graph of y = f (x), for – 6 £ x £ 2, is shown below. The x-coordinates of the local minimum and maximum points are r and s respectively.

(b)	Write down the equation of the horizontal asymptote.
(1)
 
(c)	Write down the value of r and of s.
(4)
 
(d)	Let L be the normal to the curve of f at P(0, 1). Show that L has equation x + y = 1.
(4)
 
(e)	Let R be the region enclosed by the curve y = f (x) and the line L.
(i)	Find an expression for the area of R.
(ii)	Calculate the area of R.
(5)
(Total 17 marks)