CALC 1501BNewton’s Method2020Newton’s MethodNewton’s Methodor theNewton–Raphson Methodis a method for finding arootof theequationf(x) = 0. We start with a guessx0for the rootr. Draw a tangent lineLto the graphy=f(x) at the pointx0. Letx1be the point at which the lineLintersects thex-axis.If the initial guessx0is “close” to an actual rootr, thenx1is a better approximation tor.The process can be repeated to produce a sequence of numbersx2, x3, . . .converging to therootr. The numberxn+1is thex-intercept of the tangent line toy=f(x) at (xn, f(xn)).Newton’s method for finding roots.We can find the equation forxn+1by noticing that the lineLpasses through the point (xn+1,0)and (xn, f(xn)) and hence has the slope0-f(xn)xn+1-xn. But it is also tangent toy=f(x) atx=xnand hence slope equalsf0(xn), giving usf0(xn) =0-f(xn)xn+1-xn=⇒xn+1=xn-f(xn)f0(xn)(1)Remark 1.A formula like (1) that givesxn+1in terms of the previous values is called arecursiveformulafor the sequencex0,x1,x2,. . . .1
CALC 1501BNewton’s Method2020Example 2.Approximate√5 using Newton’s method.Example 3.Find an approximate root off(x) =x3+x-1 using Newton’s method.2
CALC 1501BNewton’s Method2020Thus the numberx5= 0.682327803828019 is an approximate root ofx3+x-1.