HOMEWORK MATH101 SHEET 6 - 2008 Hand in your solutions to your MATH101 tutor by 5pm on Monday 10 November. (1) A small rectangular plot of farmland with an area of 216 m2 is to be enclosed by a fence and divided into two equal equal plots by another fence parallel to one of the sides. What dimensions for the outer plot will require the smallest total length of fencing? How much fencing will be needed? (2) The function f is defined by f (x) = 2x3 - 3x + 2 . Draw the graph of y = f (x) carefully, showing all relevant mathematical analysis in detail. Hence verify that the equation f (x) = 0 has only one real root x = xR , where -2 < xR < -1 . Use the NewtonRaphson method to evaluate xR correct to six decimal places: starting with x0 = -1.5, compute x3 . Quote the value of f (x3 to ) illustrate the accuracy of your answer. (3) A rectangle is to be inscribed (symmetrically about the y-axis) between the arch of the curve y = 4 cos(x/2) from x = - to x = and the x-axis. Show that the area A of the rectangle with largest area is given by x A = 8x cos( ) 2 where x satisfies the equation x 2 = tan( ) . x 2 By sketching the graphs of y = 1/x and y = tan x in the interval 0 < x < /2, demonstrate that the equation x tan x = 1 has only one solut...