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Math 237. Calculus II Solutions to the HW on Newton's Method (3.7) Assigned: 5, 9, 11, 14, and find the zeros of y = ex – 3x. Selected for Grading: 9, 14 Solutions. 5. Find the largest root of x3 + 6x2 + 9x + 1 = 0. Let f (x) = x3 + 6x2 + 9x + 1 = 0. Then Newton's formula for this setting is A quick peek at its graph shows that this function has its largest zero somewhere between –1 and zero. y x Starting off with x0 = –0.5, I got the following approximations. x0 = –0.5 x1 = 0.0666666667 x2 = –0.0991243961 x3 = –0.1202800956 x4 = –0.1206146753 x5 = –0.1206147584 At this point I started getting the same value over and over again. So the largest root of x3 + 6x2 + 9x + 1 = 0 is approximately –0.1206147584. 9. For this one I will use Newton's method and the function f (x) = cos(x) – 2x. Here is this function's graph. y x There is a zero between x = 0 and x = 1. I'll start with x1 = 0.5. First, though, I need to build the formula to be used. f '(x) = –sin(x) – 2 and So the formula to use is Here are the successive approximations that I got using this formula with my calculator. x1 = 0.5 x2 = 0.4506266931 x3 = 0.4501836476 x4 = 0.4501836113 x5 = 0.4501836113 The solution to the equation cos x = 2x is approximately x = 0.4501836113. 11. To find all real zeros of f (x) = x4 – 8x3 + 22x2 – 24x + 8, I'll use Newton's method as many times as necessary, once for each zero. I know that since this function is a polynomial function of degree four it has at most four zeros. A peek at its graph . . . y x shows that there are only three. It looks like the middle one is exactly 2. A quick check . . . f (2) = 16 – 64 + 88 – 48 + 8 = 112 – 112 = 0 . . . shows that this is true. So I need to use Newton's Method only twice. For each use, I will use the formula: First, with x1 = 0.5: x2 = 0.575 x3 = 0.5855859612 x4 = 0.5857863666 x5 = 0.5857864376, and all the subsequent approximations were equal to this fifth one. And using x1 = 3.5: x2 = 3.425 x3 = 3.414414039 x4 = 3.414213633 x5 = 3.414212562, and all the rest were this same value. Summary: The roots of x4 – 8x3 + 22x2 – 24x + 8 = 0 are x ≈ 0.5857864376, x = 2, and x ≈ 3.414212562. 14. We seek the smallest positive root of 2 cot(x) = x. Let f (x) = 2 cot(x) – x. In order to begin the process, I need a first guess. A look at the graph of f . . . y x shows that it lies very close to x = 1. So I'll start off with the guess x1 = 1. I'll use the formula Here are the estimates that I got. x2 = 1.074305226 x3 = 1.076871416 x4 = 1.076873986, and all successive approximations had this same value. The smallest positive root of 2 cot(x) = x is approximately 1.076873986. The Additional Question I'll use the function f (x) = ex – 3x. Here is its graph. y x As you can see, it has zeros between 0 and 1 and between 1 and 2. To find the zero between 0 and 1 I started off with x0 = 0.5. And I'll use the recursive formula Here are the successive values that I got. x0 = 0.5 x1 = 0.610059655 x2 = 0.6189967797 x3 = 0.6190612834 x4 = 0.6190612867 At this point I started getting the same value over and over again. So the first zero for f (x) = ex – 3x is approximately 0.6190612867. To find the zero between 1 and 2 I'll use the same recursive formula as above, but I'll start off with x0 = 1.5. Here are the succissive values that I got. x0 = 1.5 x1 = 1.512358146 x2 = 1.512134625 x3 = 1.512134552 At this point I started getting the same value over and over again. So the first zero for f (x) = ex – 3x is approximately 1.512134552.