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1.9 Calculus

1.9 Calculus

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Question 570
i.

Find

∫e2xex+1 dx \int \frac{e^{2x}}{\sqrt{e^x + 1}} \, dx ∫ex+1​e2x​dx
[5]
ii.

During a chemical reaction, the rate of change of the mass mmm of a byproduct, in grams per hour, is modeled by the equation

dmdt=27t3t+4,t≥0 \frac{dm}{dt} = \frac{27t}{\sqrt{3t + 4}}, \quad t \ge 0 dtdm​=3t+4​27t​,t≥0

Use the substitution u=3t+4u = \sqrt{3t + 4}u=3t+4​ to show that

∫27t3t+4 dt=2(3t+4)12(At+B)+k \int \frac{27t}{\sqrt{3t + 4}} \, dt = 2(3t + 4)^{\frac{1}{2}}(At + B) + k ∫3t+4​27t​dt=2(3t+4)21​(At+B)+k

where AAA and BBB are integers to be found and kkk is a constant of integration.

[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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