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1.9 Calculus

1.9 Calculus

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Question 531

A chemical bioreactor releases thermal energy at a rate R(t)R(t)R(t) MJ per hour, where t t\,t is the time in hours since the start of the reaction, given by

R(t)=50t+25e2t+1,0≤t≤4 R(t) = \sqrt{50t+25} e^{\sqrt{2t+1}}, \quad 0 \le t \le 4 R(t)=50t+25​e2t+1​,0≤t≤4
a.

Using the substitution u=2t+1u = \sqrt{2t+1}u=2t+1​, show that the total energy released, given by ∫04R(t) dt\int_{0}^{4} R(t) \, dt∫04​R(t)dt, can be expressed in the form

∫abku2eu du \int_{a}^{b} k u^2 e^u \, du ∫ab​ku2eudu

where aaa, b b\,b and k k\,k are constants to be found.

[5]
b.

Hence find, by algebraic integration, the exact value of the total energy released in the first 4 hours, giving your answer in the form Ae3+BeAe^3 + BeAe3+Be where A A\,A and B B\,B are integers.

[5]
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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