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1.9 Calculus

1.9 Calculus

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Question 558
a.

Use the substitution u=1+tu = 1 + \sqrt{t}u=1+t​ to show that the integral

∫12t1+t dt \int \frac{12\sqrt{t}}{1+\sqrt{t}} \, dt ∫1+t​12t​​dt

can be written in the form

∫(24u−48+24u) du \int \left( 24u - 48 + \frac{24}{u} \right) \, du ∫(24u−48+u24​)du
[4]
b.

The mass of a fungal colony, mmm grams, grows at a rate modelled by the equation

dmdt=12t1+t \frac{dm}{dt} = \frac{12\sqrt{t}}{1+\sqrt{t}} dtdm​=1+t​12t​​

where ttt is the number of days since the colony was first observed, for 1≤t≤91 \le t \le 91≤t≤9.

Determine the total increase in the mass of the colony from the end of day 1 to the end of day 9. Show each stage of your working and give your answer to one decimal place.

[4]
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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