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 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.
