Problem Analysis #calculus_28

Motif 3 · Run 1

Python ● execution succeeded

Incorrect

53.45s

13,095 toks

Infron · Motif 3 · 46문항 × 3회 · Python TAR. 131,072 output tokens · 기본 1,800초 · 6+1 turns. ACC는 전체 138회, Pass@1은 각 문항의 첫 유효 시도, Pass@3는 3회 내 정답으로 계산합니다.

Problem Statement

함수 $f(x)=\frac12x^2-x+\ln(1+x)$ 와 양수 t에 대하여 점 $(s,f(s))$ ( $s>0$ )에서 y축에 내린 수선의 발과 곡선 $y=f(x)$ 위의 점 $(s,f(s))$ 에서의 접선이 y축과 만나는 점 사이의 거리가 t가 되도록 하는 s의 값을 $g(t)$ 라 하자. $\int_{1/2}^{27/4}g(t)\,dt$ 의 값은?

① $\frac{161}{12}+\ln3$ ② $\frac{40}3+\ln3$ ③ $\frac{53}4+\ln2$ ④ $\frac{79}6+\ln2$ ⑤ $\frac{157}{12}+\ln2$

최종 답은 선택지 번호 1~5 중 하나를 $\boxed{}$ 안에 쓰세요.

Ground Truth

Recorded model response / solution

Reasoning Process

Recorded Python call:

import sympy as sp
s=sp.symbols('s', positive=True)
h=s**3/(1+s)
# endpoints
a=sp.solve(sp.Eq(h, sp.Rational(1,2)), s)
b=sp.solve(sp.Eq(h, sp.Rational(27,4)), s)
print("roots 1/2:", a)
print("roots 27/4:", b)
I=sp.integrate(s*sp.diff(h,s), (s,1,3))
sp.simplify(I)

Tool Output

Execution succeeded roots 1/2: [1] roots 27/4: [3]

Reasoning Process

Recorded Python call:

import sympy as sp
s=sp.symbols('s')
h=s**3/(1+s)
I=sp.integrate(s*sp.diff(h,s), (s,1,3))
sp.simplify(I)

Tool Output

Execution succeeded

Reasoning Process

Recorded Python call:

import sympy as sp
s=sp.symbols('s')
h=s**3/(1+s)
I=sp.integrate(s*sp.diff(h,s), (s,1,3))
print(I)
print(sp.simplify(I))
print(float(I))

Tool Output

Execution succeeded -log(2) + log(4) + 157/12 log(2) + 157/12 13.77648051389328

Final Model Output

Answer channel integration error

The provider returned reasoning data, but no scorable content field.

Comparative Analysis (Pcalculus_28) (Average across runs)

Averages use reported measurements only. Missing usage is not treated as zero.

Avg Time Taken

Motif 3

114.6

Union Alpha

91.1

Solar Pro 4

69.8

Grok 4.7

56.3

MiMo V2.6 Pro

32.3

Muse Spark 1.3

19.4

Solar Mini 4 Preview

14.8

DeepSeek V4.1 Flash

13.8

Avg Token Usage

Grok 4.7

17,431

Motif 3

16,209

Solar Mini 4 Preview

14,146

Muse Spark 1.3

7,383

DeepSeek V4.1 Flash

7,373

Union Alpha

7,217

Solar Pro 4

6,260

MiMo V2.6 Pro

4,916