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1.9 Calculus

1.9 Calculus

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Question 437

The rate of biomass accumulation in a soil sample, BBB (measured in mg per day), is modeled as a function of the nitrogen concentration nnn (in parts per million) by the equation:

B(n)=5n−3n2−n,n>1 B(n) = \frac{5n - 3}{n^2 - n}, \quad n > 1 B(n)=n2−n5n−3​,n>1
a.

Express B(n)B(n)B(n) in partial fractions.

[3]
b.

Hence find ∫B(n) dn\displaystyle \int B(n) \, dn∫B(n)dn.

[3]
c.

Use your answer to part (b) to find the value of kkk for which the total biomass accumulation between concentration levels n=kn = kn=k and n=2kn = 2kn=2k is exactly ln⁡72\ln 72ln72, where k>1k > 1k>1.

∫k2kB(n) dn=ln⁡72 \int_k^{2k} B(n) \, dn = \ln 72 ∫k2k​B(n)dn=ln72
[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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