
Concept explainers
(a) Newton's method for approximating a root of an equation f(x) = 0 (see Section 4.8) can be adapted lo approximating a solution of a system of equations f(x, y) = 0 and g(x, 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 approximation (x1, y1) is close to this point, then the tangent planes to the surfaces at (x1, y1) 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 4.8.2.) Show that
where f, g, and their partial derivatives are evaluated at (x1, y1). If we continue this procedure, we obtain successive approximations (xn, 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 biography of Simpson on page 520.) The example that he gave to illustrate the method was to solve the system of equations
xx + yy= 1000 xy + yx= 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.

Want to see the full answer?
Check out a sample textbook solution
Chapter 14 Solutions
Bundle: Calculus: Early Transcendentals, 8th + WebAssign Printed Access Card for Stewart's Calculus: Early Transcendentals, 8th Edition, Multi-Term
- Explain the relationship between 12.3.6, (case A of 12.3.6) and 12.3.7arrow_forwardExplain the key points and reasons for the establishment of 12.3.2(integral Test)arrow_forwardUse 12.4.2 to determine whether the infinite series on the right side of equation 12.6.5, 12.6.6 and 12.6.7 converges for every real number x.arrow_forward
- use Corollary 12.6.2 and 12.6.3 to derive 12.6.4,12.6.5, 12.6.6 and 12.6.7arrow_forwardExplain the focus and reasons for establishment of 12.5.1(lim(n->infinite) and sigma of k=0 to n)arrow_forwardExplain the focus and reasons for establishment of 12.5.3 about alternating series. and explain the reason why (sigma k=1 to infinite)(-1)k+1/k = 1/1 - 1/2 + 1/3 - 1/4 + .... converges.arrow_forward
- Explain the key points and reasons for the establishment of 12.3.2(integral Test)arrow_forwardUse identity (1+x+x2+...+xn)*(1-x)=1-xn+1 to derive the result of 12.2.2. Please notice that identity doesn't work when x=1.arrow_forwardExplain the key points and reasons for the establishment of 11.3.2(integral Test)arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageFunctions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning
