1. A curve C has the equation a) (i) Show that: (ii) Hence find the coordinates of the stationary point on the curve C (iii) Show that this stationary point is a point of inflection. b) (i) Show that: where a and b are constants to be determined. (ii) Deduce that the curve has another point of inflection. c) Sketch the curve C, indicating the two points of inflection. (Marks available: 11) Answer Answer outline and marking scheme for question: 1 Give yourself marks for mentioning any of the points below: a) (i) Using the product rule gives: dy/dx = 2xe-x - (x2 +1)e-x = -e-x (x -1)2 (ii) dy/dx = 0 at stationary points, thus: 0 = 2xe-x - (x2 +1)e-x This gives stationary point at (1, 2e-1) (iii) dy/dx (i. e. the gradient) remains the same sign either side of the stationary point. Therefore the stationary point must be a point of inflection. Mathematically shown below: dy/dx is less than zero before the stationary point (i. e. at x less than one) dy/dx equals zero at the stationary point dy/dx is less than zero after the stationary point (i. e. at x more than one) (5 marks) b) (i) Using the product rule on dy/dx gives: d2y/dx2 = 2e-x - 2xe-x - 2xe-x + (x2 +1)e-x = e-x (x -1)(x -3) Therefore a = 1 and b = 3. (4 marks) The gradient is always less than zero, so the curve slopes downwards. From the above there are two points of inflections at x = 1 and x = 3. The equation never becomes negative, therefore the curve does not cross the x axis. Therefore the curve looks like: (2 marks) (Marks available: 11) 2. A piece of wire, of length 20cm, is to be cut into two parts. One of the parts, of length x cm, is to be formed into a circle and the other part into a square. a) Show that the sum, A cm2, of the areas of the circle and the square is given by b) Show that A has a stationary value when (Marks available: 8) Answer Answer outline and marking scheme for question: 2 Give yourself marks for mentioning any of the points below: a) Area of circle Side of square Entering these to find a total area, gives: (3 marks) b) Differentiating the area equation above, gives: Solving this equation when dA/dx = 0, gives: (5 marks) (Marks available: 8) 3. The variables x and y are related by y = 4x. a) Find the value of x when y = 12, giving your answer to two decimal places b) Show that y = ekx, where k = In 4. c) Hence find dy/dx. d) Given that x is a function of a third variable t and that dy/dx = x, deduce that dy/dx = 12 ln 12, when y = 12. (Marks available: 7) Answer Answer outline and marking scheme for question: 3 Give yourself marks for mentioning any of the points below: a) 4x = 12, using logs on either side, gives: x log 4 = log 12 Solving this for x gives, x = 1.79 to 2d. p. (2 marks) b) We know that 4 = eln4 therefore, multiplying both side by the power of x, gives: 4x = (eln4)x From this you can deduce: y = ekx where k = ln 4. (1 mark) c) dy/dx = kekx = (ln 4)exln4 = (ln 4).4x = yln4. (1 mark) d) dy/dt = dy/dx . dx/dt = (y ln 4). x = (y ln 4).(ln 12/ ln 4) When y = 12, from (a) = 12 ln 12 (3 marks) (Marks available: 7)