The document provides examples of using differentiation to find the tangents and normals to curves defined parametrically. It gives several examples of finding the equations of tangents and normals to curves at given points. It also considers a parametric curve defined by equations y = at^2 + bt + 1 and x = t + c, and shows that given values for any two of the parameters a, b, c, d, and e, the remaining parameters can be determined, resulting in two possible solutions for c.

1. Starter a) Find the tangent to the curve y = e 2x at the point (3 , e 6 ). y = e 6 (2x – 5) b) Find the normal to the curve y = 5 ln x at the point where x = e. 5y = –ex + e 2 + 25

2. Objectives: to use differentiation to find the tangents and normals to curves defined parametrically

3. Differentiation of parametric equations Find expressions for in terms of t: a) b) c) d)

4. Tangents and normals a) A curve has parametric equations x = t ², y = 2 t . Find the equation of the tangent to the curve at the point where y = 4. 2 y = x + 4 b) Find the tangent to the curve x = t ², y = 4 t at the point (9 , 12). 3 y = 2 x + 18 c) Find the equation of the normal at (–8 , 4) to the curve x = t ³, y = t ². y = 3 x + 28

5. Core 3 & 4 Textbook Exercise 3C Page 247

6. Homework C4 Differentiation Worksheet C

7. A parametric curve is defined as follows: y = at 2 + bt + 1 , x = t + c. When t = 1, dy/dx = d. When t = 2, dy/dx = e. The curve goes through (1, 1). If a = 2 and b = 3, find c, d and e. If c = 2 and d = 3, find a, b and e. Given two of a, b, c, d and e, can we always find the others? How many solutions will there be?

8. Solution dy/dx = 2at + b. So 2a + b = d, and 4a + b = e. 1 = t + c, so t = 1 – c so 1 = a(1 - c) 2 + b(1 - c) + 1. This gives (1 - c)(a + b - ac) = 0 so c = 1 or (a + b)/a. If a = 2 and b = 3, c = 1 or 2.5, d = 7, e = 11 (2 solutions). If d = 2 and e = 3, a = 0.5, b = 4, c = 1 or 9 (2 solutions). Given two of a, b, d and e, we can always find c and the others, with c taking two possible values, one being 1 (2 solutions). If we are given c (not 1) and one other from a, b, d, and e, then we can always find the others uniquely. But if we are given c = 1 and b = 2, say, we do not have enough information to determine the other constants so an infinite number of solutions exist.