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1.9.10 Differentiate e^kx, a^kx and ln x (A-level only)

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Question 10

The temperature, TTT Kelvin, of a superconducting material being cooled in a cryogenic chamber after sss seconds is modeled by the equation:

log⁡10T=1.94−0.12s \log_{10} T = 1.94 - 0.12s log10​T=1.94−0.12s
a.

Show that this equation can be written in the form T=ab−sT = ab^{-s}T=ab−s, where aaa and bbb are constants. Give the value of aaa to the nearest whole number and the value of bbb to 2 significant figures.

[3]
b.

With reference to the equation in part (a), interpret the value of the constant aaa.

[1]
c.

When the cooling system is adjusted to a different cycle, the temperature TTT after sss seconds satisfies the equation:

T=180×1.15−s T = 180 \times 1.15^{-s} T=180×1.15−s

Use calculus to find, to 2 significant figures, the value of dTds\frac{dT}{ds}dsdT​ when s=3s = 3s=3.

[4]

1.9.10 Differentiate e^kx, a^kx and ln x (A-level only) Questions

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