Đề thi giữa học kỳ I môn Giải tích 1 (Calculus I) - 10 câu tự luận

Instructions:
- Calculators and reference materials are not permitted.
- Candidates must show all relevant steps and intermediate calculations to receive full credit.

Câu 1 (1,0 điểm). Evaluate the following limits without using L'Hôpital's Rule:
a) $\lim_{x \to 2} \dfrac{x^2 + x - 6}{x^2 - 4}$
b) $\lim_{x \to 0} \dfrac{\sqrt{1 + 4x} - 1}{x}$

Câu 2 (1,0 điểm). Determine the value of the constant $k$ such that the function $f(x)$ is continuous at $x = 0$:

$$f(x) = \begin{cases} \dfrac{\sin(3x)}{x} + k & \text{if } x < 0 \\ 5x + 2k^2 - 1 & \text{if } x \ge 0 \end{cases}$$

Câu 3 (1,0 điểm). Compute the derivative $f'(x)$ for each of the following functions:
a) $f(x) = \dfrac{x^2 + 1}{2x - 3}$
b) $f(x) = \ln\left(x + \sqrt{x^2 + 1}\right)$

Câu 4 (1,0 điểm). Consider the curve defined implicitly by the equation:

$$x^3 + y^3 - 6xy = 0$$

a) Find $\dfrac{dy}{dx}$ in terms of $x$ and $y$.
b) Write the equation of the tangent line to the curve at the point $(3, 3)$.

Câu 5 (1,0 điểm). A ladder $10\text{ m}$ long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a constant rate of $1\text{ m/s}$, how fast is the top of the ladder sliding down the wall when the bottom is $6\text{ m}$ from the wall?

Câu 6 (1,0 điểm). Let $f(x) = x^3 - 3x^2 - 9x + 5$ on the interval $[-2, 4]$.
a) Find the absolute maximum and absolute minimum values of $f(x)$ on $[-2, 4]$.
b) Determine all intervals on which the graph of $f(x)$ is concave upward or concave downward, and find the inflection point(s).

Câu 7 (1,0 điểm). An open-top rectangular storage box with a square base and vertical sides is to be constructed. The material for the base costs $\$4$ per square meter, while the material for the sides costs $\$2$ per square meter. If the total volume of the box must be $32\text{ m}^3$, find the dimensions of the box that will minimize the total cost of construction.

Câu 8 (1,0 điểm). Evaluate the following indefinite integrals:
a) $\int x^2 \sqrt{x^3 + 5}\,dx$
b) $\int \dfrac{e^x}{1 + e^{2x}}\,dx$

Câu 9 (1,0 điểm).
a) Use the Fundamental Theorem of Calculus and the Chain Rule to determine $g'(x)$, where:

$$g(x) = \int_{x^2}^{\sin x} \sqrt{1 + t^4}\,dt$$

b) Evaluate the definite integral:

$$\int_{0}^{1} \dfrac{x}{\sqrt{1 + 3x^2}}\,dx$$

Câu 10 (1,0 điểm).
a) Find the area of the finite region bounded by the parabolas $y = x^2$ and $y = 2x - x^2$.
b) Evaluate the limit by expressing it as a definite integral on $[0, 1]$:

$$L = \lim_{n \to \infty} \sum_{i=1}^{n} \dfrac{n}{n^2 + i^2}$$

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