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1.9.16 Implicit differentiation (A-level only)

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Question 33

The intensity, III, of a specialized acoustic wave at a distance rrr from the source is modeled by the equation:

I=rsin⁡(2r),r>0,I>0 I = r^{\sin(2r)}, \quad r > 0, \quad I > 0 I=rsin(2r),r>0,I>0
a.

By first taking natural logarithms, find an expression for dIdr\frac{dI}{dr}drdI​ in terms of III and rrr.

[5]
b.

Hence show that at any stationary point of the intensity, the distance rrr must satisfy the equation:

tan⁡(2r)+2rln⁡r=0 \tan(2r) + 2r \ln r = 0 tan(2r)+2rlnr=0
[3]

1.9.16 Implicit differentiation (A-level only) Questions

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