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1.9 Calculus

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Question 159

The mass, m m\,m grams, of a certain chemical during a reaction is modeled by a differential equation involving time ttt, where 0≤t<π4\displaystyle 0 \le t < \frac{\pi}{4}0≤t<4π​.

a.

Find the derivative with respect to m m\,m of

1(1+2ln⁡m)2 \frac{1}{(1 + 2\ln m)^2} (1+2lnm)21​
[2]
b.

Hence find the general solution to the differential equation

4sec⁡(2t)dmdt=m(1+2ln⁡m)3tan⁡(2t) 4\sec(2t) \frac{\text{d}m}{\text{d}t} = m(1 + 2\ln m)^3 \tan(2t) 4sec(2t)dtdm​=m(1+2lnm)3tan(2t)

for m>e−1/2m > e^{-1/2}m>e−1/2.

[4]
c.

Show that the particular solution of this differential equation for which the initial mass is 1 g (so m=1m = 1m=1 when t=0t = 0t=0) is given by

m=eAsec⁡t−12 m = e^{A\sec t - \frac{1}{2}} m=eAsect−21​

where A A\,A is a constant to be found.

[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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