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1.7 Differentiation

1.7 Differentiation

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Question 149

The temperature, θ∘C\theta^\circ\text{C}θ∘C, of a liquid in a cooling tank, ttt minutes after cooling begins, is modeled by the equation

θ=B+150e−kt \theta = B + 150e^{-kt} θ=B+150e−kt

where BBB and kkk are positive constants. Given that the initial temperature of the liquid is 175∘C175^\circ\text{C}175∘C,

a.

find the value of BBB.

[2]
b.

The temperature of the liquid 4 minutes after cooling begins is 67∘C67^\circ\text{C}67∘C.

Show that k=pln⁡qk = p \ln qk=plnq where ppp and qqq are rational numbers to be found.

[3]
c.

Hence find

the temperature of the liquid 8 minutes after cooling begins, giving your answer to 3 significant figures,

[3]
d.

the rate of decrease of the temperature of the liquid 8 minutes after cooling begins. Give your answer in ∘C min−1^\circ\text{C min}^{-1}∘C min−1 to 3 significant figures.

[3]
Markscheme

1.7 Differentiation Questions

  1. A Level
  2. /Maths
  3. /1.7 Differentiation

425 exam-style questions on OCR A Level Maths 1.7 Differentiation, covering 1.7.1 Derivative as gradient of the tangent, 1.7.2 Gradient of the tangent at a point, 1.7.3 Sketching the gradient function, 1.7.4 Second derivatives, 1.7.5 Second derivative as rate of change of gradient, 1.7.6 Convex, concave and points of inflection (A-level only), 1.7.7 Differentiation from first principles for powers of x, 1.7.8 Differentiation from first principles for sin x and cos x (A-level only), 1.7.9 Differentiating x^n, 1.7.10 Differentiating e^(kx) and a^(kx) (A-level only), 1.7.11 Differentiating trigonometric functions (A-level only), 1.7.12 Derivative of ln x (A-level only), 1.7.13 Tangents and normals, 1.7.14 Stationary points, 1.7.15 Increasing and decreasing functions, 1.7.16 Points of inflection (A-level only), 1.7.17 Product and quotient rules (A-level only), 1.7.18 Chain rule (A-level only), 1.7.19 Parametric and implicit differentiation (A-level only), 1.7.20 Constructing differential equations (A-level only), and 1.7 Differentiation. Each one has a worked solution and a mark scheme showing where the marks go.

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