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1.9 Calculus

1.9 Calculus

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Question 456

(i) The altitude of a research drone HHH, measured in decameters, is modeled by the function

H(t)=e−0.4tsec⁡0.5t,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<π

where ttt is the time in minutes after deployment.

ia.

Find H′(t)H'(t)H′(t).

[4]
ib.

Hence determine the time ttt at which the altitude of the drone is stationary.

[3]
ii.

A separate flight path is defined by the implicit relationship

x=ln⁡(2sin⁡y),0<y<π2 x = \ln(2\sin y), \quad 0 < y < \frac{\pi}{2} x=ln(2siny),0<y<2π​

Show that

dydx=exf(x) \frac{\text{d}y}{\text{d}x} = \frac{\text{e}^x}{\text{f}(x)} dxdy​=f(x)ex​

where f(x)\text{f}(x)f(x) is a function of ex\text{e}^xex that should be found.

[5]
Markscheme

1.9 Calculus Questions

  1. A Level
  2. /Maths
  3. /1.9 Calculus

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.

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