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 the tangent to C C\,C at the point B B\,B is parallel to lll,
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.
717 exam-style questions on Edexcel A Level Maths Differentiation, 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. Each one has a worked solution and a mark scheme showing where the marks go.