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1.9 Calculus

1.9 Calculus

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Question 518
i.

Find

∫(3w+2)24w dw,w>0 \int \frac{(3w+2)^2}{4\sqrt{w}} \, dw, \quad w > 0 ∫4w​(3w+2)2​dw,w>0

giving your answer in simplest form.

[4]
ii.

The concentration C C\,C of a chemical in a solution, measured in mol/L, changes over time t t\,t seconds according to the model C=f(t)C = f(t)C=f(t).

It is given that:

  • f′(t)=t2+kt+mf'(t) = t^2 + kt + mf′(t)=t2+kt+m, where k k\,k and m m\,m are constants.
  • The initial concentration of the chemical is 12 mol/L.
  • At t=3t = 3t=3, the concentration is 18 mol/L.
  • At t=3t = 3t=3, the rate of change of the concentration is 2 mol/L/s.

Find f(t)f(t)f(t) in simplest form.

[6]
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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