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.
825 exam-style questions on OCR (MEI) A Level Maths 1.9 Calculus, covering 1.9.1 Gradient of a curve at a point, 1.9.2 Gradient as limit of chord gradient, 1.9.3 Derivative as gradient of tangent, 1.9.4 Sketch the gradient function, 1.9.5 Differentiate y = kx^n, 1.9.6 Second derivative as rate of change of gradient, 1.9.7 Stationary points: maxima and minima, 1.9.8 Increasing and decreasing functions, 1.9.9 Tangent and normal at a point, 1.9.10 Differentiate e^kx, a^kx and ln x (A-level only), 1.9.11 Differentiate trigonometric functions (A-level only), 1.9.12 Product rule (A-level only), 1.9.13 Quotient rule (A-level only), 1.9.14 Chain rule (A-level only), 1.9.15 Rates of change with the chain rule (A-level only), 1.9.16 Implicit differentiation (A-level only), 1.9.17 Concavity and the second derivative (A-level only), 1.9.18 Points of inflection (A-level only), 1.9.19 Integration as reverse of differentiation, 1.9.20 Integrate kx^n, 1.9.21 Constant of integration, 1.9.22 Indefinite and definite integrals, 1.9.23 Area between a graph and the x-axis, 1.9.24 Integrate e^kx, 1/x, sin kx, cos kx (A-level only), 1.9.25 Integration as the limit of a sum (A-level only), 1.9.26 Area between two curves (A-level only), 1.9.27 Integration by substitution (reverse chain rule) (A-level only), 1.9.28 Integration by substitution (other cases) (A-level only), 1.9.29 Integration by parts (A-level only), 1.9.30 Integration using partial fractions (A-level only), 1.9.31 Formulate first order differential equations (A-level only), 1.9.32 Solve first order differential equations (A-level only), and 1.9.33 Interpret solutions of differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.