Use Newton's method to approximate all the intersection points of the pair of curves. Some preliminary graphing or analysis may help in choosing good initial approximations. Round to six decimal places.

y = ex and y = x2 + 3

Question

Use Newton's method to approximate all the intersection points of the pair of curves. Some preliminary graphing or analysis may help in choosing good initial approximations. Round to six decimal places.

y = ex and y = x2 + 3

BlackTom AI · Accepted Answer

We are asked to use Newton's method to approximate all intersection points of the curves y = e^x and y = x^2 + 3, and round to six decimal places.

First, define the function whose zeros correspond to intersections: f(x) = e^x − (x^2 + 3). The derivative is f'(x) = e^x − 2x. Newton's method updates x_{n+1} = x_n − f(x_n)/f'(x_n). The goal is to find roots of f(x).

Now evaluate each candidate option relative to the true intersection(s):

- Option A: x ≈ 1.873123. This value is very close to the known root of e^x = x^2 + 3. Substituting roughly, e^{1.873123} ≈ 6.517, and 1.873123^2 + 3 ≈ 3.508 + 3 = 6.508; the difference is about 0.009, which is consistent with a high-precision Newton refinement when rounded to six decimals. The sequence of Newton iterations starting from a reasonable initial guess near this region would converge toward a value in this vicinity, and the six-decimal placement matches the typical refinement accuracy.

- Option B: x ≈ 1.373123. If we plug this into f(x), e^{1.373123} ≈ 3.947, while x^2 + 3 ≈ 1.885 + 3 = 4.885; the difference f(x) ≈ −0.938, which is not near zero, indicating this is not a root. Newton's method would require several iterations to move toward a root from such a point, but the final six-decimal value would be around the actual root near 1.87, not 1.373.

- Option C: x ≈ 1.762023. Here e^{1.762023} ≈ 5.82, and x^2 + 3 ≈ (1.762023)^2 + 3 ≈ 3.106 + 3 = 6.106; the difference f(x) ≈ −0.286, which is noticeably away from zero. While closer than option B, this value would still require several Newton steps to reach the root near 1.87, and a six-decimal rounding at the end would land near the actual root, not at 1.762023.

- Option D: x ≈ 1.873233. This is extremely close to Option A, differing by 0.000110. Since Newton's method tends to converge smoothly with small local changes once near the root, a value that close to the actual root would arise from the same basin of attraction and rounding to six decimals could yield either 1.873123 or 1.873233 depending on iteration count and numerical rounding. However, given the consistency with a well-converged root in the 1.8731... range, this nearby candidate is also plausible as a high-precision outcome of the same root.

Overall, the values that lie near the actual intersection point satisfy e^x ≈ x^2 + 3, and the numbers around 1.87 are the ones approaching f(x) = 0 under Newton's iteration. The more distant candidates around 1.37 or 1.76 yield larger residuals f(x) and would not be the stabilized six-decimal results of a Newton refinement starting from a reasonable initial guess. The two digits shown in the close options reflect a tiny variation that can occur due to rounding in iterative processes.

In summary, the options clustered near x ≈ 1.873... are the plausible Newton refinements to the intersection, while the others fail to minimize f(x) to near zero at six-decimal precision.

Use Newton's method to approximate all the intersection points of the pair of curves. Some preliminary graphing or analysis may help in choosing good initial approximations. Round to six decimal places. y = ex and y = x2 + 3 单项选择题

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