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1.9 Calculus

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Question 48

The altitude of a specialized research probe, H(t)H(t)H(t) in kilometres, is modelled for time t>0.5 t > 0.5\,t>0.5 seconds using its vertical acceleration.

It is given that:

  • the acceleration is H′′(t)=12t2+2t2H''(t) = 12t^2 + \dfrac{2}{t^2}H′′(t)=12t2+t22​
  • the point P P\,P on the graph of altitude against time has a ttt-coordinate of 1
  • the tangent to the graph of H(t)H(t)H(t) at P P\,P has the equation H=8t−3H = 8t - 3H=8t−3
a.

Determine the equation of the normal to the graph of altitude against time at the point PPP. Give your answer in the form at+bH+c=0at + bH + c = 0at+bH+c=0, where a,b, a, b,\,a,b, and c c\,c are integers.

[3]
b.

Find an expression for H(t)H(t)H(t) in terms of ttt.

[7]
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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