A laser sensor monitors the vertical profile of a specialized optical lens. The height h h\,h of the lens surface, measured in millimeters, at a horizontal distance x x\,x mm from its center is modeled by the equation
h=1+4ex2/2,x≥0h = \sqrt{1 + 4e^{x^2/2}}, \quad x \ge 0h=1+4ex2/2,x≥0
Find dhdx\displaystyle \frac{dh}{dx}dxdh, giving your answer in its simplest form.
A specific point P P\,P on the lens surface has xxx-coordinate β\betaβ. The tangent to the lens profile at P P\,P passes through the origin (0,0)(0, 0)(0,0).
Show that β \beta\,β is a root of the equation
2ex2/2(x2−2)−1=02e^{x^2/2}(x^2 - 2) - 1 = 02ex2/2(x2−2)−1=0
Show that β \beta\,β lies between 1.4 and 1.5.
Show that applying the Newton-Raphson method to the equation f(x)=2ex2/2(x2−2)−1=0f(x) = 2e^{x^2/2}(x^2 - 2) - 1 = 0f(x)=2ex2/2(x2−2)−1=0 leads to the iteration formula
xn+1=xn−xn2−2xn3+e−xn2/22xn3x_{n+1} = x_n - \frac{x_n^2 - 2}{x_n^3} + \frac{e^{-x_n^2/2}}{2x_n^3}xn+1=xn−xn3xn2−2+2xn3e−xn2/2
Use the iteration formula from part (d) with x1=1.5x_1 = 1.5x1=1.5 to find, to 4 decimal places, the value of (i) x3x_3x3 (ii) β\betaβ
Practise Edexcel A Level Maths 10.3 The Newton-Raphson Method with exam-style questions for A Level Maths. 23 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.