C1 1 Worksheet G DIFFERENTIATION A curve has the equation y = 2 + 3x + kx2 − x3 where k is a constant. Given that the gradient of the curve is −6 at the point P where x = −1, a find the value of k. (4) Given also that the tangent to the curve at the point Q is parallel to the tangent at P, b find the length PQ, giving your answer in the form k 5 . 2 3 The point A lies on the curve y = 12 x2 (5) and the x-coordinate of A is 2. a Find an equation of the tangent to the curve at A. Give your answer in the form ax + by + c = 0, where a, b and c are integers. (5) b Verify that the point where the tangent at A intersects the curve again has the coordinates (−1, 12). (3) Given that y = x2 − 6 x − 3 3x 1 2 , show that dy ( x + a) 2 can be expressed in the form , where 3 dx bx 2 a and b are integers to be found. 4 (6) The curve with equation y = x3 + ax2 − 24x + b, where a and b are constants, passes through the point P (−2, 30). a Show that 4a + b + 10 = 0. (2) Given also that P is a stationary point of the curve, 5 b find the values of a and b, (4) c find the coordinates of the other stationary point on the curve. (3) y 1 y = x 2 − 4 + 3x − 12 O A B x C 1 −1 The diagram shows the curve with equation y = x 2 − 4 + 3x 2 . The curve crosses the x-axis at the points A and B and has a minimum point at C. 6 a Find the coordinates of A and B. (5) b Find the coordinates of C, giving its y-coordinate in the form a 3 + b, where a and b are integers. (5) f(x) ≡ x3 + 4x2 + kx + 1. a Find the set of values of the constant k for which the curve y = f(x) has two stationary points. (5) Given that k = −3, b find the coordinates of the stationary points of the curve y = f(x). Solomon Press (4) C1 7 DIFFERENTIATION Worksheet G continued f(x) ≡ 2x3 + 5x2 − 1. a Find the set of values of x for which f(x) is increasing. (5) b Find an equation of the normal to the curve y = f(x) at the point A (−1, 2). (3) The normal to the curve y = f(x) at the point B is parallel to the normal at the point A. c Find the exact coordinates of the point B. 8 (4) y l y = x3 − 3x2 − 8x + 4 P m O x Q The diagram shows the curve with equation y = x3 − 3x2 − 8x + 4. The straight line l is the tangent to the curve at the point P (−1, 8). a Find an equation of line l. (4) The straight line m is parallel to l and is the tangent to the curve at the point Q. 9 b Find an equation of line m. (4) c Find an equation of the normal to the curve at P. (2) d Hence, or otherwise, show that the distance between lines l and m is 16 2 . (4) A curve has the equation y = x (k − x), where k is a constant. Given that the gradient of the curve is 2 at the point P where x = 2, a find the value of k, (5) b show that the normal to the curve at P has the equation x + 2 y = c, where c is an integer to be found. 10 11 (5) f(x) = x3 − 3x2 + 4 a Show that (x + 1)(x − 2)2 ≡ x3 − 3x2 + 4. (2) b Hence state, with a reason, the coordinates of one of the stationary points of the curve with equation y = f(x). (2) c Using differentiation, find the coordinates of the other stationary point of the curve. (5) The curve C has the equation 3 y = 2x − x 2 , x ≥ 0. a Find the coordinates of any points where C meets the x-axis. (3) b Find the x-coordinate of the stationary point on C and determine whether it is a maximum or a minimum point. (6) c Sketch the curve C. (2) Solomon Press

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