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≥0 h = \sqrt{1 + 4e^{x^2/2}}, \quad x \ge 0 h=1+4ex2/2,x≥0Find 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=0 2e^{x^2/2}(x^2 - 2) - 1 = 0 2ex2/2(x2−2)−1=0Show 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−xn2/22xn3 x_{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−xn2/2Use 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
137 exam-style questions on OCR (MEI) A Level Maths 1.10 Numerical Methods (A-level only), covering 1.10.1 Locate roots by change of sign (A-level only), 1.10.2 When change of sign methods fail (A-level only), 1.10.3 Fixed point iteration (A-level only), 1.10.4 Newton-Raphson method (A-level only), 1.10.5 Convergence of iterations (A-level only), 1.10.6 Trapezium rule (A-level only), 1.10.7 Upper and lower bounds using rectangles (A-level only), and 1.10.8 Numerical methods to solve problems (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.