The vertical displacement, hhh metres, of a specialised weather drone relative to its launch platform is modelled by the function
h(x)=2(x2−24)(4x+26)12,x≥−6.5 h(x) = 2(x^2 - 24)(4x + 26)^{\frac{1}{2}}, \quad x \ge -6.5 h(x)=2(x2−24)(4x+26)21,x≥−6.5where xxx is the horizontal distance in kilometres from the platform.
Show that
h′(x)=k(5x2+26x−24)(4x+26)12 h'(x) = \frac{k(5x^2 + 26x - 24)}{(4x + 26)^{\frac{1}{2}}} h′(x)=(4x+26)21k(5x2+26x−24)where kkk is an integer to be found.
Hence, find the values of xxx for which the drone is moving perfectly horizontally.
The path of the drone has a local maximum at the point PPP.
Find the exact coordinates of PPP.
A second drone's altitude is tracked by the function ggg, defined by
g(x)=2h(x)+15,−6.5≤x≤0 g(x) = 2h(x) + 15, \quad -6.5 \le x \le 0 g(x)=2h(x)+15,−6.5≤x≤0Determine the range of ggg, giving your answer in exact form.
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.
