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.
425 exam-style questions on OCR A Level Maths 1.7 Differentiation, covering 1.7.1 Derivative as gradient of the tangent, 1.7.2 Gradient of the tangent at a point, 1.7.3 Sketching the gradient function, 1.7.4 Second derivatives, 1.7.5 Second derivative as rate of change of gradient, 1.7.6 Convex, concave and points of inflection (A-level only), 1.7.7 Differentiation from first principles for powers of x, 1.7.8 Differentiation from first principles for sin x and cos x (A-level only), 1.7.9 Differentiating x^n, 1.7.10 Differentiating e^(kx) and a^(kx) (A-level only), 1.7.11 Differentiating trigonometric functions (A-level only), 1.7.12 Derivative of ln x (A-level only), 1.7.13 Tangents and normals, 1.7.14 Stationary points, 1.7.15 Increasing and decreasing functions, 1.7.16 Points of inflection (A-level only), 1.7.17 Product and quotient rules (A-level only), 1.7.18 Chain rule (A-level only), 1.7.19 Parametric and implicit differentiation (A-level only), 1.7.20 Constructing differential equations (A-level only), and 1.7 Differentiation. Each one has a worked solution and a mark scheme showing where the marks go.