2026학년도 1학기 1차 지필평가 (예상)

limit of polynomial

1. [4.2점]

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

① 16 ② 24 ③ 32 ④ 48 ⑤ 64

limit of exponential

2. [4.2점]

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

① 1 ② 2 ③ 3 ④ \(e^3\) ⑤ \(3e\)

limit of irrational

3. [4.3점]

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

① \(\dfrac{1}{3}\) ② \(\dfrac{1}{6}\) ③ \(\dfrac{1}{9}\) ④ 3 ⑤ \(\dfrac{1}{2}\)

limit at infinity

4. [4.4점]

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

① 0 ② 1 ③ 2 ④ \(\dfrac{5}{2}\) ⑤ \(\infty\)

limit at infinity

5. [4.4점]

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

① 0 ② \(\dfrac{3}{5}\) ③ \(\dfrac{3}{4}\) ④ 1 ⑤ \(\infty\)

precise definition of limit

6. [4.5점]

Find the largest value of \(\delta\) 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

limits — true/false

7. [4.5점]

Choose every true statement in the box below.

Ⓐ If \(\lim_{x\to a}f(x)\) exists and \(\lim_{x\to a}g(x)\) does not exist, then \(\lim_{x\to a}[f(x)+g(x)]\) does not exist.

Ⓑ 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.

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

derivative — power rule

8. [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

derivative — product rule

9. [4.6점]

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

① 1 ② 2 ③ 3 ④ \(e\) ⑤ 0

derivative — chain rule

10. [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 \(\dfrac{x+2}{2(x^2+4x+1)}\)

derivative — logarithmic

11. [4.7점]

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

① 0 ② 1 ③ \(e\) ④ 2 ⑤ \(\ln 2\)

derivatives — true/false

12. [4.7점]

Choose every true statement in the box below.

Ⓐ If \(f\) is differentiable at \(x = a\), then \(f\) is continuous at \(x = a\).

Ⓑ If \(f\) is continuous at \(x = a\), then \(f\) is differentiable at \(x = a\).

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

tangent line

13. [4.7점]

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

① 2 ② 3 ③ 4 ④ 5 ⑤ 6

normal line

14. [4.8점]

Find the equation of the 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 \(y = \dfrac{1}{4}x + \dfrac{1}{4}\)

implicit differentiation

15. [4.8점]

Find the slope of the tangent line to the curve \(x^2 + xy + y^2 = 7\) at the point \((1, 2)\).

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

tangent through origin

16. [4.8점]

Find the \(x\)-coordinate of the point on the curve \(y = e^{2x}\) where the tangent line passes through the origin.

① \(\dfrac{1}{4}\) ② \(\dfrac{1}{2}\) ③ 1 ④ \(\ln 2\) ⑤ \(\dfrac{1}{e}\)

inverse trig — arcsin

17. [4.9점]

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

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

inverse trig — arctan composition

18. [4.9점]

Find the 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 \(\dfrac{2}{(x+1)^2+(x-1)^2}\)

inverse trig — triangle

19. [5.0점]

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

① \(\dfrac{60}{169}\) ② \(\dfrac{120}{169}\) ③ \(\dfrac{5}{13}\) ④ \(\dfrac{10}{13}\) ⑤ \(\dfrac{12}{13}\)

inverse trig — proof completion

20. [5.0점]

The following is a derivation of the derivative of \(y = \arcsin(x)\).

Step 1: \(y = \arcsin(x)\) implies \(\sin(y) = x\).

Step 2: Differentiating: \(\cos(y)\cdot\dfrac{dy}{dx} = 1\), so \(\dfrac{dy}{dx} = \dfrac{1}{\cos(y)}\).

Step 3: \(\cos(y) = \underline{\quad\text{Ⓐ}\quad}\) (positive root since \(-\frac{\pi}{2}\le y\le\frac{\pi}{2}\))

Step 4: \(\dfrac{dy}{dx} = \underline{\quad\text{Ⓑ}\quad}\)

Choose the correct pair for Ⓐ and Ⓑ.

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 \(\cos(y) = (1-x^2)^2,\;\dfrac{dy}{dx}=\dfrac{1}{(1-x^2)^2}\)

logarithmic differentiation

21. [5.0점]

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

① 1 ② 2 ③ 3 ④ \(e\) ⑤ \(1 + \ln 2\)

orthogonal trajectories

22. [5.1점]

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

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 \(y^2 - 2x^2 = K\)

배점표

문항	1	2	3	4	5	6	7	8	9	10	11
배점	4.2	4.2	4.3	4.4	4.4	4.5	4.5	4.5	4.6	4.6	4.7
문항	12	13	14	15	16	17	18	19	20	21	22
배점	4.7	4.7	4.8	4.8	4.8	4.9	4.9	5.0	5.0	5.0	5.1

2026학년도 1학기 1차 지필평가 (예상)

2학년 미적분학 I

배점표