Question at position 3 By using differentials, an approximation of 1233\sqrt[3]{123} is497100.4\frac{97}{100}.41315.4\frac{13}{15}.42325.4\frac{23}{25}.52125\frac{2}{12}.473754\frac{73}{75}.单项选择题
A
4 97 100 .
B
4 13 15 .
C
4 23 25 .
D
5 2 12 .
E
4 73 75 .

Question

Question at position 3 By using differentials, an approximation of 1233\sqrt[3]{123} is497100.4\frac{97}{100}.41315.4\frac{13}{15}.42325.4\frac{23}{25}.52125\frac{2}{12}.473754\frac{73}{75}.

BlackTom AI · Accepted Answer

We start by restating the problem in our own terms: using differentials to approximate the quantity 1233 times the cube root of 123. The task is to compare several formatted numeric options and determine which one matches the differential approximation approach.

Option 1: "4
97
100
." This option, when interpreted as a decimal grouping, suggests a value around 4.97 with additional trailing digits 100. If we consider the differential approach around a nearby cube root anchor, the core step is to approximate ∛123 by a nearby known value and then scale by 1233. A common anchor is ∛125 = 5, since 125 is a nearby perfect cube. The differential f(x) = x^{1/3} has derivative f'(x) = (1/3)x^{-2/3}, and at x = 125 we get f'(125) = 1/75 ≈ 0.013333. With Δx = 123 - 125 = -2, the linear approximation gives ∛123 ≈ ∛125 + f'(125)Δx ≈ 5 + (1/75)(-2) = 5 - 2/75 ≈ 4.973333. Multiplying by 1233 yields 1233 × 4.973333 ≈ 6132.12. None of the digits in option 1 clearly reflect 6132.12, but if the option’s intended interpretation is a decimal chunked into 4, 97, and 100, the 4.97 component aligns with the approximate value of ∛123 (4.9733…), which is the critical piece before the multiplication by 1233. The slight mismatch in trailing digits suggests option 1 captures the principal ∛123 approximation (around 4.97) rather than the final product, but it is not a faithful full representation of 1233×∛123. The learning point is that the dominant factor in the differential step is getting ∛123 ≈ 4.973..., and option 1’s 4.97 component reflects that.

Option 2: "4
13
15
." Interpreted as 4.13…15, this value deviates notably from the true ∛123 ≈ 4.973, which means it underestimates the cube root by a substantial margin. Since the differential method hinges on a close approximation of ∛123, this option misrepresents the key quantity, making it an incorrect choice.

Option 3: "4
23
25
." Here, 4.23… is even further from 4.973, providing an even poorer approximation to ∛123. The error in the cube-root estimate would propagate directly into the final product, leading to a result far from the actual differential approximation. Thus this option is not suitable.

Option 4: "5
2
12
." This reads as a number around 5.2, which overshoots ∛123 (which is just under 5). Since the anchor ∛125 is 5 and the true ∛123 is slightly less than 5, using 5.2 as the cube-root approximation would systematically overestimate the product 1233×∛123, contradicting the differential-based estimate. Therefore this option is incorrect.

Option 5: "4
73
75
." Interpreting as 4.73… suggests another under- or over-estimate relative to the true ∛123 ≈ 4.973. This value is closer to the correct cube-root than some other options but still deviates enough that, when scaled by 1233, it would not match the differential-approximation result as closely as the primary approximation 4.9733 would. Consequently, this option does not align with the differential-based calculation as accurately as the principal form around 4.973.

Putting these observations together, the central quantity to match via the differential method is the cube-root approximation ∛123 ≈ 4.9733, which is best reflected in the pattern 4.97… (option 1’s leading 4, 97). The remainder digits (100, 15, 25, 12, 75) represent trailing digits that are not critical to the initial differential step, but they diverge in ways that make the full final product deviate from the true differential result. Therefore, option 1 most faithfully captures the essential magnitude of ∛123 and thus aligns best with the differential-based approximation strategy, while the other options introduce errors that are inconsistent with the method.

Question at position 3 By using differentials, an approximation of 1233\sqrt[3]{123} is497100.4\frac{97}{100}.41315.4\frac{13}{15}.42325.4\frac{23}{25}.52125\frac{2}{12}.473754\frac{73}{75}.单项选择题

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Question at position 3 By using differentials, an approximation of 1233\sqrt[3]{123} is473754\frac{73}{75}.52125\frac{2}{12}.42325.4\frac{23}{25}.497100.4\frac{97}{100}.41315.4\frac{13}{15}.

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