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−x24 - x^24−x2
State 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 dθE = \int_{0}^{0.4} \sqrt{16 \cos \theta} \, d\thetaE=∫00.416cosθdθ giving 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 dθE_{wide} = \int_{0}^{1.5} \sqrt{16 \cos \theta} \, d\thetaEwide=∫01.516cosθdθ Explain why this method is not suitable for this range.
Practise Edexcel A Level Maths 4.2 Expanding (a + bx)^n with exam-style questions for A Level Maths. 99 questions, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.