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.
308 exam-style questions on OCR (MEI) A Level Maths 1.6 Sequences and Series, covering 1.6.1 Binomial expansion for positive integer n, 1.6.2 Factorial and combinations notation, 1.6.3 Binomial expansion for rational n (A-level only), 1.6.4 Binomial expansion of (a + bx)^n via factoring (A-level only), 1.6.5 Binomial approximating polynomials (A-level only), 1.6.6 What a sequence is; finite and infinite (A-level only), 1.6.7 Generate a sequence by formula or recurrence (A-level only), 1.6.8 A series as a sum of terms (A-level only), 1.6.9 Sigma notation (A-level only), 1.6.10 Increasing, decreasing and periodic sequences (A-level only), 1.6.11 Convergent and divergent sequences (A-level only), 1.6.12 Arithmetic sequences and series (A-level only), 1.6.13 Standard formulae for arithmetic series (A-level only), 1.6.14 Geometric sequences and series (A-level only), 1.6.15 Standard formulae for geometric series (A-level only), 1.6.16 Sum to infinity of a geometric series (A-level only), and 1.6.17 Sequences and series in modelling (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.