Question

6. (a) Newton's method for approximating a root of an equation f(x) 0 (see Section 3.8) can be adapted to approximating a solution of a system of equations f(x, y) 0 and gx, y) 0. The surfaces z f(x, y) and z g(x, y) intersect in a curve that intersects the xy-plane at the point (r, s), which is the solution of the system. If an initial approxi- mation (xi, yı) is close to this point, then the tangent planes to the surfaces at (xi, y) intersect in a straight line that intersects the xy-plane in a point (x2, y2), which should be closer to (r, s). (Compare with Figure 3.8.2.) Show that where f, g, and their partial derivatives are evaluated at (x,y). If we continue this pro- cedure, we obtain successive approximations (x, yn) (b) It was Thomas Simpson (1710-1761) who formulated Newton's method as we know it today and who extended it to functions of two variables as in part (a). (See the biogra- 1025 phy of Simpson on page 560.) The example that he gave to illustrate the method was to solve the system of equations +y-100 In other words, he found the points of intersection of the curves in the figure. Use the method of part (a) to find the coordinates of the points of intersection correct to six decimal places. ry1000 4 xy 100 0 4 age tho airle y2 + , what values of a

Answer 1

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