2026 HAFS 미적분학 I — 1학기 중간고사 예상문제 완전 분석

유형	문항 번호	문항 수	비율
Limits	1, 2, 3, 4, 5, 6, 7	7	32%
Derivatives	8, 9, 10, 11, 12	5	23%
Tangent line	13, 14, 15, 16	4	18%
Inverse trig	17, 18, 19, 20	4	18%
Logarithmic diff	21	1	4.5%
Orthogonal trajectories	22	1	4.5%

Q1. Limits 4.2점

\(\displaystyle\lim_{h \to 0} \frac{(2+h)^4 - 16}{h}\)

① 16
② 24
③ 32
④ 48
⑤ 64

풀이

\(f(x)=x^4\)의 \(x=2\)에서 도함수 정의.
\(f'(x)=4x^3\), \(f'(2)=4\cdot 8=32\).

정답: ③ 32

Q2. Limits 4.2점

\(\displaystyle\lim_{x \to 0} \frac{e^{3x} - 1}{x}\)

① 1
② 2
③ 3
④ \(e^3\)
⑤ \(3e\)

풀이

\(f(x)=e^{3x}\)의 \(x=0\)에서 도함수.
\(f'(x)=3e^{3x}\), \(f'(0)=3\).

정답: ③ 3

Q3. Limits 4.3점

\(\displaystyle\lim_{h \to 0} \frac{\sqrt{9+h} - 3}{h}\)

① 1/3
② 1/6
③ 1/9
④ 3
⑤ 1/2

풀이

\(f(x)=\sqrt{x}\)의 \(x=9\)에서 도함수.
\(f'(x)=\dfrac{1}{2\sqrt{x}}\), \(f'(9)=\dfrac{1}{6}\).

정답: ② 1/6

Q4. Limits 4.4점

\(\displaystyle\lim_{x \to \infty}\left(\sqrt{x^2 + 5x} - \sqrt{x^2 + x}\right)\)

① 0
② 1
③ 2
④ 5/2
⑤ ∞

풀이

유리화: \(\dfrac{(x^2+5x)-(x^2+x)}{\sqrt{x^2+5x}+\sqrt{x^2+x}}=\dfrac{4x}{\sqrt{x^2+5x}+\sqrt{x^2+x}}\).
\(x\)로 나누면 \(\dfrac{4}{\sqrt{1+5/x}+\sqrt{1+1/x}}\to\dfrac{4}{1+1}=2\).

정답: ③ 2

Q5. Limits 4.4점

\(\displaystyle\lim_{x \to \infty} \frac{3x^3 - 2x + 1}{5x^3 + 4x^2 - 7}\)

① 0
② 3/5
③ 3/4
④ 1
⑤ ∞

풀이

최고차항 계수 비: \(\dfrac{3}{5}\).

정답: ② 3/5

Q6. Limits 4.5점

Find largest \(\delta\) such that \(|x-2|<\delta \implies |x^3-8|<0.1\).

① 0.001
② 0.003
③ 0.007
④ 0.008
⑤ 0.01

풀이

\(|x^3-8|=|x-2|\cdot|x^2+2x+4|\). \(x=2+\delta\): \((2+\delta)^3-8=12\delta+6\delta^2+\delta^3\).
\(\delta=0.008\): \(12(0.008)+6(0.000064)+\cdots\approx 0.0964<0.1\) ✓
\(\delta=0.01\): \(\approx 0.1206>0.1\) ✗

정답: ④ 0.008

Q7. Limits 4.5점

Choose every true statement:
Ⓐ If \(\lim_{x\to a}f(x)\) exists and \(\lim_{x\to a}g(x)\) DNE, then \(\lim_{x\to a}[f(x)+g(x)]\) DNE.
Ⓑ If \(\lim_{x\to a}[f(x)\cdot g(x)]\) exists and \(\lim_{x\to a}f(x)\) exists, then \(\lim_{x\to a}g(x)\) must exist.
Ⓒ If \(\lim_{x\to a}f(x)=0\) and \(|g(x)|\le M\), then \(\lim_{x\to a}f(x)\cdot g(x)=0\).

① Ⓐ only
② Ⓒ only
③ Ⓐ and Ⓒ only
④ Ⓑ and Ⓒ only
⑤ All

풀이

Ⓐ TRUE — 귀류법: \(f+g\) 존재하면 \(g=(f+g)-f\)도 존재해야 하므로 모순.
Ⓑ FALSE — 반례: \(f(x)=0\), \(g(x)=\sin(1/x)\). \(fg=0\) 존재, \(f=0\) 존재, \(g\) 불존재.
Ⓒ TRUE — Squeeze theorem.

정답: ③ Ⓐ and Ⓒ only

Q8. Derivatives 4.5점

Find \(f'(1)\) when \(f(x)=x^5-3x^3+6\sqrt{x}-\dfrac{2}{x^2}\).

① 1
② 3
③ 5
④ 7
⑤ 9

풀이

\(f'(x)=5x^4-9x^2+3x^{-1/2}+4x^{-3}\).
\(f'(1)=5-9+3+4=3\).

정답: ② 3

Q9. Derivatives 4.6점

Find \(f'(0)\) when \(f(x)=(x^2+1)e^x\).

① 1
② 2
③ 3
④ \(e\)
⑤ 0

풀이

\(f'(x)=e^x(x^2+2x+1)=e^x(x+1)^2\).
\(f'(0)=1\cdot 1=1\).

정답: ① 1

Q10. Derivatives 4.6점

Find the derivative of \(y=\ln\!\left(\sqrt{x^2+4x+1}\right)\).

① \(\dfrac{2x+4}{x^2+4x+1}\)
② \(\dfrac{x+2}{x^2+4x+1}\)
③ \(\dfrac{2x+4}{2(x^2+4x+1)}\)
④ \(\dfrac{1}{2\sqrt{x^2+4x+1}}\)
⑤ \(\dfrac{x+2}{2(x^2+4x+1)}\)

풀이

\(y=\dfrac{1}{2}\ln(x^2+4x+1)\).
\(y'=\dfrac{2x+4}{2(x^2+4x+1)}=\dfrac{x+2}{x^2+4x+1}\).

정답: ②

Q11. Derivatives 4.7점

Find \(f'(1)\) when \(f(x)=x^x\) for \(x>0\).

① 0
② 1
③ \(e\)
④ 2
⑤ \(\ln 2\)

풀이

\(\ln f=x\ln x\). \(\dfrac{f'}{f}=\ln x+1\).
\(f'(1)=1^1\cdot(0+1)=1\).

정답: ② 1

Q12. Derivatives 4.7점

Choose every true statement:
Ⓐ Differentiable ⇒ continuous.
Ⓑ Continuous ⇒ differentiable.
Ⓒ \(f,g\) not differentiable ⇒ \(f+g\) not differentiable.

① Ⓐ only
② Ⓑ only
③ Ⓐ, Ⓑ
④ Ⓐ, Ⓒ
⑤ All

풀이

Ⓐ TRUE — 정리.
Ⓑ FALSE — 반례: \(|x|\) at \(x=0\).
Ⓒ FALSE — 반례: \(f=|x|, g=-|x|\), \(f+g=0\).

정답: ① Ⓐ only

Q13. Tangent line 4.7점

Find y-intercept of tangent to \(f(x)=x^3-2x+5\) at \(x=1\).

① 2
② 3
③ 4
④ 5
⑤ 6

풀이

\(f(1)=4\), \(f'(1)=1\). 접선: \(y=x+3\). y절편 \(=3\).

정답: ② 3

Q14. Tangent line 4.8점

Find equation of normal line to \(y=\dfrac{1}{x+1}\) at \(x=1\).

① \(y=4x-\dfrac{7}{2}\)
② \(y=4x-3\)
③ \(y=4x-\dfrac{15}{4}\)
④ \(y=-4x+\dfrac{9}{2}\)
⑤ \(y=\dfrac{1}{4}x+\dfrac{1}{4}\)

풀이

\(y(1)=\dfrac{1}{2}\), \(y'(1)=-\dfrac{1}{4}\). 법선 기울기 \(=4\).
\(y-\dfrac{1}{2}=4(x-1)\to y=4x-\dfrac{7}{2}\).

정답: ①

Q15. Tangent line 4.8점

Slope of tangent to \(x^2+xy+y^2=7\) at \((1,2)\).

① \(-4/5\)
② \(-1\)
③ \(-2/5\)
④ \(2/5\)
⑤ \(4/5\)

풀이

\(2x+y+xy'+2yy'=0\). At \((1,2)\): \(4+5y'=0\to y'=-\dfrac{4}{5}\).

정답: ① −4/5

Q16. Tangent line 4.8점

x-coordinate where tangent to \(y=e^{2x}\) passes through origin.

① 1/4
② 1/2
③ 1
④ \(\ln 2\)
⑤ \(1/e\)

풀이

점 \((a, e^{2a})\)에서 접선: 기울기 \(=2e^{2a}\). 원점 통과:
\(-e^{2a}=2e^{2a}(-a)\to 1=2a\to a=\dfrac{1}{2}\).

정답: ② 1/2

Q17. Inverse trig 4.9점

\(\dfrac{d}{dx}\arcsin(2x)\) at \(x=\dfrac{1}{4}\).

① \(\dfrac{4}{\sqrt{3}}\)
② \(\dfrac{2}{\sqrt{3}}\)
③ \(\dfrac{4\sqrt{3}}{3}\)
④ 2
⑤ \(\dfrac{8}{3}\)

풀이

\(f'(x)=\dfrac{2}{\sqrt{1-4x^2}}\).
\(f'\!\left(\dfrac{1}{4}\right)=\dfrac{2}{\sqrt{1-\frac{1}{4}}}=\dfrac{2}{\sqrt{3/4}}=\dfrac{2}{\frac{\sqrt{3}}{2}}=\dfrac{4}{\sqrt{3}}=\dfrac{4\sqrt{3}}{3}\).

정답: ③

Q18. Inverse trig 4.9점

Derivative of \(y=\arctan\!\left(\dfrac{x-1}{x+1}\right)\) for \(x>-1\).

① \(\dfrac{1}{1+x^2}\)
② \(\dfrac{2}{1+x^2}\)
③ \(\dfrac{1}{(x+1)^2}\)
④ \(\dfrac{1}{x^2+1}+\dfrac{1}{(x+1)^2}\)
⑤ \(\dfrac{2}{(x+1)^2+(x-1)^2}\)

풀이

\(u=\dfrac{x-1}{x+1}\), \(u'=\dfrac{2}{(x+1)^2}\).
\(y'=\dfrac{u'}{1+u^2}=\dfrac{2}{(x+1)^2+(x-1)^2}=\dfrac{2}{2x^2+2}=\dfrac{1}{x^2+1}\).

정답: ①

Q19. Inverse trig 5.0점

\(\theta=\arctan\!\left(\dfrac{5}{12}\right)\). Find \(\sin(2\theta)\).

① 60/169
② 120/169
③ 5/13
④ 10/13
⑤ 12/13

풀이

\(\tan\theta=\dfrac{5}{12}\to\) 직각삼각형 \(5\text{-}12\text{-}13\).
\(\sin\theta=\dfrac{5}{13}\), \(\cos\theta=\dfrac{12}{13}\).
\(\sin(2\theta)=2\cdot\dfrac{5}{13}\cdot\dfrac{12}{13}=\dfrac{120}{169}\).

정답: ② 120/169

Q20. Inverse trig 5.0점

Derivation of \(\dfrac{d}{dx}(\arcsin x)\). Choose correct pair for Ⓐ (\(\cos y\)) and Ⓑ (\(dy/dx\)).

① \(\cos y=1-x^2\), \(\dfrac{dy}{dx}=\dfrac{1}{1-x^2}\)
② \(\cos y=\sqrt{1-x^2}\), \(\dfrac{dy}{dx}=\dfrac{1}{\sqrt{1-x^2}}\)
③ \(\cos y=\sqrt{1+x^2}\), \(\dfrac{dy}{dx}=\dfrac{1}{\sqrt{1+x^2}}\)
④ \(\cos y=\sqrt{x^2-1}\), \(\dfrac{dy}{dx}=\dfrac{1}{\sqrt{x^2-1}}\)
⑤ \(\cos y=(1-x^2)^2\), \(\dfrac{dy}{dx}=\dfrac{1}{(1-x^2)^2}\)

풀이

\(\sin^2 y+\cos^2 y=1\to\cos y=\sqrt{1-x^2}\) (\(-\frac{\pi}{2}\le y\le\frac{\pi}{2}\)에서 양수).
\(\dfrac{dy}{dx}=\dfrac{1}{\sqrt{1-x^2}}\).

정답: ②

Q21. Logarithmic diff 5.0점

Find \(f'(1)\) when \(f(x)=x^{x+1}\) for \(x>0\).

① 1
② 2
③ 3
④ \(e\)
⑤ \(1+\ln 2\)

풀이

\(\ln f=(x+1)\ln x\). \(\dfrac{f'}{f}=\ln x+\dfrac{x+1}{x}=\ln x+1+\dfrac{1}{x}\).
\(f'(1)=1^2\cdot(0+1+1)=2\).

정답: ② 2

Q22. Orthogonal trajectories 5.1점

Find orthogonal trajectories of \(y=Cx^2\).

① \(x^2+2y^2=K\)
② \(x^2-2y^2=K\)
③ \(2x^2+y^2=K\)
④ \(x^2+y^2=K\)
⑤ \(y^2-2x^2=K\)

풀이

\(y=Cx^2\to C=y/x^2\). 미분: \(y'=2Cx=2(y/x^2)\cdot x=\dfrac{2y}{x}\).
직교: \(\dfrac{dy}{dx}=-\dfrac{x}{2y}\). 분리변수: \(2y\,dy=-x\,dx\to y^2=-\dfrac{x^2}{2}+C_1\to x^2+2y^2=K\).

정답: ①

우선순위	영역	핵심 포인트
★★★	극한 (Limits)	미분 정의 극한 3문제 (Q1-Q3), 무한대 극한 유리화 (Q4-Q5)
★★★	Chain Rule	\(e^{x^2}\), \(\ln(\sqrt{\cdots})\), \(\arcsin/\arctan\) 합성 (Q10, Q17, Q18)
★★☆	접선 / 법선	y절편 (Q13), 원점 통과 (Q16), implicit diff (Q15)
★★☆	역삼각함수	\(\arcsin/\arctan\) 미분 공식 + 삼각비 대입 (Q19-Q20)
★☆☆	고난이도	ε-δ (Q6), logarithmic diff (Q11, Q21), orthogonal trajectories (Q22)

2026 HAFS 미적분학 I — 1학기 중간고사 예상문제 완전 분석

1. 유형별 분포

2. 22문항 전체 분석

3. 정답표

4. 학습 전략 요약