In a precision optics experiment, the refractive index μ \mu\,μ of a specific polymer varies with the distance x x\,x from the optical center of a lens according to the model
μ=7+2ex2,x≥0 \mu = \sqrt{7 + 2e^{x^2}}, \quad x \ge 0 μ=7+2ex2,x≥0Find dμdx\displaystyle \frac{d\mu}{dx}dxdμ, giving your answer in its simplest form.
A light ray intersects the curve at a point P P\,P with xxx-coordinate β\betaβ. It is determined that the tangent to the curve at P P\,P passes through the origin.
Show that x=βx = \betax=β is a root of the equation
(2x2−2)ex2−7=0 (2x^2 - 2)e^{x^2} - 7 = 0 (2x2−2)ex2−7=0Hence show that β \beta\,β lies between 1.2 and 1.3.
Show that the Newton-Raphson formula for the equation in part (b) can be written as
xn+1=4xn4−2xn2+2+7e−xn24xn3 x_{n+1} = \frac{4x_n^4 - 2x_n^2 + 2 + 7e^{-x_n^2}}{4x_n^3} xn+1=4xn34xn4−2xn2+2+7e−xn2Using the Newton-Raphson formula with x1=1.3x_1 = 1.3x1=1.3, find to 4 decimal places the value of (i) x3x_3x3 (ii) β\betaβ
