(i) The altitude of a research drone HHH, measured in decameters, is modeled by the function
H(t)=e−0.4tsec0.5t,0≤t<π H(t) = \text{e}^{-0.4t} \sec 0.5t, \quad 0 \le t < \pi H(t)=e−0.4tsec0.5t,0≤t<πwhere ttt is the time in minutes after deployment.
Find H′(t)H'(t)H′(t).
Hence determine the time ttt at which the altitude of the drone is stationary.
A separate flight path is defined by the implicit relationship
x=ln(2siny),0<y<π2 x = \ln(2\sin y), \quad 0 < y < \frac{\pi}{2} x=ln(2siny),0<y<2πShow that
dydx=exf(x) \frac{\text{d}y}{\text{d}x} = \frac{\text{e}^x}{\text{f}(x)} dxdy=f(x)exwhere f(x)\text{f}(x)f(x) is a function of ex\text{e}^xex that should be found.
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.