Show that the first two terms of the binomial expansion of f(x)=16−8x2f(x) = \sqrt{16 - 8x^2}f(x)=16−8x2 are
4−x2 4 - x^2 4−x2State the range of values of x x\,x for which the expansion found in part (a) is valid.
The signal intensity S(θ)S(\theta)S(θ) of a specialized sensor is modeled by the function S(θ)=16cosθS(\theta) = \sqrt{16 \cos \theta}S(θ)=16cosθ for small angles θ\thetaθ (in radians).
Using the result from part (a) and a suitable small angle approximation for cosθ\cos \thetacosθ, find an approximation for the total energy E E\,E detected across a narrow sweep:
E=∫00.416cosθ dθ E = \int_{0}^{0.4} \sqrt{16 \cos \theta} \, d\theta E=∫00.416cosθdθgiving your answer to four decimal places. Fully justify your answer.
A student attempts to estimate the total energy across a wider sweep using the same method:
Ewide=∫01.516cosθ dθ E_{wide} = \int_{0}^{1.5} \sqrt{16 \cos \theta} \, d\theta Ewide=∫01.516cosθdθExplain why this method is not suitable for this range.
