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1.9 Calculus

1.9 Calculus

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Question 421

The thermal profile of a synthetic crystal is modeled by the equation

5t−3tT+T2=23,T≥0 5^t - 3tT + T^2 = 23, \quad T \ge 0 5t−3tT+T2=23,T≥0

where TTT is the temperature in degrees Celsius and ttt is the time in hours. A specific observation point PPP on the profile has t=1t = 1t=1.

a.

Find the temperature TTT at point PPP.

[2]
b.

Find an expression for dTdt\frac{dT}{dt}dtdT​ in terms of TTT and ttt.

[3]
c.

The tangent to the profile at point PPP intersects the horizontal ttt-axis at point QQQ.

Determine the ttt-coordinate of QQQ, giving your answer in the form aln⁡5+bcln⁡5+d\frac{a \ln 5 + b}{c \ln 5 + d}cln5+daln5+b​ where a,b,c,a, b, c,a,b,c, and ddd are integers 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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