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C4 Differentiation - Implicit differentiation PhysicsAndMathsTutor.com 1. A curve C has equation x 2 2 + y = 2xy Find the exact value of dy at the point on C with coordinates (3, 2). dx (Total 7 marks) 2. The curve C has the equation cos2x + cos3y = 1, −π ≤ x ≤ π , 0 ≤ y ≤ π 4 4 6 (a) Find dy in terms of x and y. dx (3) The point P lies on C where x = π . 6 (b) Find the value of y at P. (3) (c) Find the equation of the tangent to C at P, giving your answer in the form ax + by + cπ = 0, where a, b and c are integers. (3) (Total 9 marks) –2x 2 3. The curve C has the equation ye = 2x + y . (a) Find dy in terms of x and y. dx (5) Edexcel Internal Review 1 C4 Differentiation - Implicit differentiation PhysicsAndMathsTutor.com The point P on C has coordinates (0, 1). (b) Find the equation of the normal to C at P, giving your answer in the form ax + by + c = 0, where a, b and c are integers. (4) (Total 9 marks) 2 3 4. A curve C has the equation y – 3y = x + 8. (a) Find dy in terms of x and y. dx (4) (b) Hence find the gradient of C at the point where y = 3. (3) (Total 7 marks) 2 2 5. A curve has equation 3x – y + xy = 4. The points P and Q lie on the curve. The gradient of the tangent to the curve is 8 at P and at Q. 3 (a) Use implicit differentiation to show that y – 2x = 0 at P and at Q. (6) (b) Find the coordinates of P and Q. (3) (Total 9 marks) 6. A curve is described by the equation 3 2 x – 4y = 12xy. (a) Find the coordinates of the two points on the curve where x = –8. (3) Edexcel Internal Review 2 C4 Differentiation - Implicit differentiation PhysicsAndMathsTutor.com (b) Find the gradient of the curve at each of these points. (6) (Total 9 marks) 7. A set of curves is given by the equation sin x + cos y = 0.5. (a) Use implicit differentiation to find an expression for dy . dx (2) For –π < x < π and –π < y < π, (b) find the coordinates of the points where dy = 0. dx (5) (Total 7 marks) 8. A curve C is described by the equation 2 2 3x – 2y + 2x – 3y + 5 = 0. Find an equation of the normal to C at the point (0, 1), giving your answer in the form ax + by + c = 0, where a, b and c are integers. (Total 7 marks) 9. A curve C is described by the equation 2 2 3x + 4y – 2x + 6xy – 5 = 0. Find an equation of the tangent to C at the point (1, –2), giving your answer in the form ax + by + c = 0, where a, b and c are integers. (Total 7 marks) Edexcel Internal Review 3 C4 Differentiation - Implicit differentiation PhysicsAndMathsTutor.com 10. The value £V of a car t years after the 1st January 2001 is given by the formula –t V = 10 000 × (1.5) . (a) Find the value of the car on 1st January 2005. (2) (b) Find the value of dV when t = 4. dt (3) (c) Explain what the answer to part (b) represents. (1) (Total 6 marks) 11. A curve has equation 2 2 x + 2xy – 3y + 16 = 0. Find the coordinates of the points on the curve where dy = 0. dx (Total 7 marks) 2 2 12. The curve C has equation 5x + 2xy – 3y + 3 = 0. The point P on the curve C has coordinates (1, 2). (a) Find the gradient of the curve at P. (5) (b) Find the equation of the normal to the curve C at P, in the form y = ax + b, where a and b are constants. (3) (Total 8 marks) Edexcel Internal Review 4
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