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1.9 Calculus

1.9 Calculus

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Question 642

An autonomous underwater vehicle (AUV) is released from the surface of the ocean and descends vertically towards the seabed.

a.

Using a simple model of constant acceleration 0.25 m s-2, a researcher predicts that the velocity of the AUV exactly 80 seconds after being released from rest is 20 m s-1. Show how the researcher has obtained this prediction.

[1]
b.

Using a refined model that accounts for fluid resistance, it is assumed that the AUV's acceleration, a m s−2a \text{ m s}^{-2}a m s−2, at time t t\,t seconds after release is given by

a=0.25−0.01v a = 0.25 - 0.01v a=0.25−0.01v

where v m s−1v \text{ m s}^{-1}v m s−1 is the velocity of the AUV at time ttt. Find an expression for v v\,v in terms of ttt.

[6]
c.

Compare the behavior of v v\,v as t→∞ t \to \infty\,t→∞ for both models.

[2]
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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