and hence show that the only stationary point on the curve is the point for which x = 0. [4]

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1 C3 Differentiation June 00 qu Find in each of the following cases: = 3 e, [] = ln(3 + ), [] (iii) = + [] Jan 00 qu5 The equation of a curve is = ( +) 8 Find an epression for and hence show that the onl stationar point on the curve is the point for which = 0 d d Find an epression for and hence find the value of at the stationar point [5] 3 Jan 00 qu 7 (a) Leaking oil is forming a circular patch on the surface of the sea The area of the patch is increasing at a rate of 50 square metres per hour Find the rate at which the radius of the patch is increasing at the instant when the area of the patch is 900 square metres Give our answer correct to significant figures (b) The mass of a substance is decreasing eponentiall Its mass now is 50 grams and its mass, m grams, at a time t ears from now is given b m = 50e kt, where k is a positive constant Find, in terms of k, the number of ears from now at which the mass will be decreasing at a rate of 3 grams per ear 4 June 009 qu 6 The diagram shows the curve with equation = ( Find an epression for in terms of [] Hence find the equation of the tangent to the curve at the point (7, 3), giving our answer in the form = m + c [5] )

2 5 June 009 qu 9 k (a) Show that, for all non-zero values of the constant k, the curve = k + has eactl one stationar point [5] (b) Show that, for all non-zero values of the constant m, the curve = e m ( + m) has eactl two stationar points [7] 6 Jan 009 qu 4 For each of the following curves, find and determine the eact -coordinate of the stationar point: = (4 + ) 5, = ln 7 Jan 009 qu 5 The mass, M grams, of a certain substance is increasing eponentiall so that, at time t hours, the mass is given b M = 40e kt, where k is a constant The following table shows certain values of t and M t 0 63 M 80 In either order, (a) find the values missing from the table, (b) determine the value of k [] Find the rate at which the mass is increasing when t = 8 Jan 009 qu 8 The diagram shows the curve with equation 6 = 3 The point P has coordinates (0, p) The shaded region is bounded b the curve and the lines = 0, = 0 and = p The shaded region is rotated completel about the -ais to form a solid of volume V 7 Show that V = 6π 3 ( p + 3) [6] It is given that P is moving along the -ais in such a wa that, at time t, the variables p and t are related b d p = p 3 + dv Find the value of at the instant when p = 9 9 June 008 qu 3 Find, in the form = m + c, the equation of the tangent to the curve = ln at the point with -coordinate e [6]

3 0 Jan 008 qu 4 Earth is being added to a pile so that, when the height of the pile is h metres, its volume is V cubic metres, where V 6 ( + 6) 4 = h dv Find the value of when h = dh The volume of the pile is increasing at a constant rate of 8 cubic metres per hour Find the rate, in metres per hour, at which the height of the pile is increasing at the instant when h = Give our answer correct to significant figures Jan 008 qu 7 A curve has equation, e = + k where k is a non-zero constant Differentiate e e ( + k + k), and show that = ( + k) [5] Given that the curve has eactl one stationar point, find the value of k, and determine the eact coordinates of the stationar point [5] June 007 qu Differentiate each of the following with respect to 3 ( + ) 5 [] June 007 qu 8 4ln 3 4 Given that =, show that = 4ln + 3 (4ln + 3) 4ln 3 Find the eact value of the gradient of the curve = 4ln + 3 at the point where it crosses the -ais (iii) O e The diagram shows part of the curve with equation = (4ln + 3) The region shaded in the diagram is bounded b the curve and the lines =, = e and = 0 Find the eact volume of the solid produced when this shaded region is rotated completel about the -ais

4 4 Jan 007 qu + 3 Find the equation of the tangent to the curve = at the point,, giving our answer 3 in the form a + b + c = 0, where a, b and c are integers [5] 5 Jan 007 qu 4 + Given that = (4t + 9) and = 6e, find epressions for and Hence find the value of when t = 4, giving our answer correct to 3 significant figures 6 Jan 007 qu 8 8 The diagram shows the curve with equation = e The curve has maimum points at P and Q The shaded region A is bounded b the curve, the line = 0 and the line through Q parallel to the -ais The shaded region B is bounded b the curve and the line PQ Show b differentiation that the -coordinate of Q is [5] Use Simpson s rule with 4 strips to find an approimation to the area of region A Give our answer correct to 3 decimal places (iii) Deduce an approimation to the area of region B [] 7 June 006 qu Find the equation of the tangent to the curve = 4 at the point (, 3) [5] 8 June 006 qu 9 = ln( ) P O The diagram shows the curve with equation = ln( ) The point P has coordinates (0, p) The region R, shaded in the diagram, is bounded b the curve and the lines = 0, = 0 and = p The units on the aes are centimetres The region R is rotated completel about the -ais to form a solid Show that the volume, V cm 3, of the solid is given b V = π(e P + 4e + p 5) [8] It is given that the point P is moving in the positive direction along the -ais at a constant P

5 rate of 0 cm min Find the rate at which the volume of the solid is increasing at the instant when p = 4, giving our answer correct to significant figures [5] 9 Jan 006 qu 3 (a) Differentiate ( + ) 6 with respect to (b) Find the gradient of the curve = + 3 at the point where = 3 0 Jan 006 qu8 The diagram shows part of the curve = ln(5 ) which meets the -ais at the point P with coordinates (, 0) The tangent to the curve at P meets the -ais at the point Q The region A is bounded b the curve and the lines = 0 and = 0 The region B is bounded b the curve and the lines PQ and = 0 Find the equation of the tangent to the curve at P [5] Use Simpson s Rule with four strips to find an approimation to the area of the region A, giving our answer correct to 3 significant figures (iii) Deduce an approimation to the area of the region B [] June 005 qu6 (a) Find the eact value of the -coordinate of the stationar point of the curve = ln (b) The equation of a curve is = 4 + c, where c is a non-zero constant Show b 4 c differentiation that this curve has no stationar points

2. Jan 2010 qu June 2009 qu.8

2. Jan 2010 qu June 2009 qu.8 C3 Functions. June 200 qu.9 The functions f and g are defined for all real values of b f() = 4 2 2 and g() = a + b, where a and b are non-zero constants. (i) Find the range of f. [3] Eplain wh the function