An industrial laser cutter follows a trajectory C C\,C defined by the parametric equations
x=2p2,y=23p3+4p2−14p+k x = 2p^2, \quad y = \frac{2}{3}p^3 + 4p^2 - 14p + k x=2p2,y=32p3+4p2−14p+kwhere k k\,k is a constant and p≠0p \neq 0p=0.
Find dydx\displaystyle \frac{dy}{dx}dxdy in terms of ppp.
The line l l\,l is the normal to the curve C C\,C at the point A A\,A where p=1p = 1p=1.
Given that l l\,l is also a tangent to the curve C C\,C at the point BBB,
show that the parameter p p\,p at point B B\,B is a solution of the equation
p2+2p−7=0 p^2 + 2p - 7 = 0 p2+2p−7=0Hence find the value of p p\,p at BBB, justifying your choice given that the xxx-coordinate of B B\,B is greater than 10.
Given that the yyy-intercept of l l\,l is 23\displaystyle \frac{2}{3}32,
determine the value of kkk.
Practise Edexcel A Level Maths Differentiation with exam-style questions for A Level Maths. 311 questions covering 9.1 Differentiating sin x and cos x, 9.2 Differentiating exponentials and logarithms, 9.3 The Chain Rule, 9.4 The Product Rule, 9.5 The Quotient Rule, 9.6 Differentiating Trigonometric Functions, 9.7 Parametric Differentiation, 9.8 Implicit Differentiation, 9.9 Using Second Derivatives, and 9.10 Rates of Change, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
