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>0By first taking natural logarithms, find an expression for dIdr\frac{dI}{dr}drdI in terms of III and rrr.
Hence show that at any stationary point of the intensity, the distance rrr must satisfy the equation:
tan(2r)+2rlnr=0 \tan(2r) + 2r \ln r = 0 tan(2r)+2rlnr=0425 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.