The profile of a suspended track for a high-speed transport system is modelled by the curve CCC. The path of the track starts at point P P\,P and ends at point QQQ, as defined by the parametric equations
x=12t2+2t+10 x = \frac{1}{2}t^2 + \frac{2}{t} + 10 x=21t2+t2+10 h=3t+12t h = 3t + \frac{12}{t} h=3t+t12where 1≤t≤61 \le t \le 61≤t≤6.
The horizontal distance from a sensor at the origin is x x\,x metres, and h h\,h is the height of the track above the ground in metres.
P P\,P is the point on the track where t=1t = 1t=1 and Q Q\,Q is the point where t=6t = 6t=6.
Safety regulations require that the difference in height between the start point P P\,P and the end point Q Q\,Q must be less than 6 metres. Show that the track meets this requirement.
Find an expression for dhdx\displaystyle \frac{dh}{dx}dxdh in terms of ttt.
A vertical reinforcement pillar is placed between the ground and the lowest point R R\,R on the track. Find the height of this pillar.
Calculate the acute angle the track makes with the horizontal at the end point QQQ. Give your answer to the nearest degree.
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.