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 AQA A Level Maths 1.10 G: Differentiation with exam-style questions for A Level Maths. 321 questions covering 1.10.1 The derivative and second derivative, 1.10.2 Differentiating standard functions, 1.10.3 Applications of differentiation, 1.10.4 Product, quotient and chain rules (A-level only), 1.10.5 Implicit and parametric differentiation (A-level only), and 1.10.6 Constructing differential equations (A-level only), matched to the AQA A Level Maths (7357) 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.
