Task 1 Calculate the first three approximation for a root of f(x)s Ï2-7, use xo-3 as the first approximation. Find the equation of the tangent line to f(x) x2-7 at xo 3, Find where the tangent line intersects the x-axis. This is x1 Give your answer to 5 decimal places. Compute the successive approximations: Repeat the process using x, to find equation of the tangent line atx x Find where the this new tangent line intersects the x-axis (again 5 decimal places). This is your x2 Use the formula (1): xn+1n-(xn)/f'n) to compute x3 to 5 decimal places. Xy- Check how this result is differ from the value given by your calculator for Error-lv7-x3 Calculate the percentage error error * 100%
The figure shows how a function f(x) and its linear approximation (i.e., its tangent line) change value when x changes from x0 to x0+dx. Suppose f(x)=x2+4x, x0=2 and dx=0.04. Your answers below need to be very precise, so use many decimal places.
(a) Find the change Îf=f(x0+dx)âf(x0)). Îf =
(b) Find the estimate (i.e., the differential) df=fâ²(x0)dx. df =
(c) Find the approximation error |Îfâdf| Error =