C958 MATH 2100
Calculus I
Comprehensive Objective
Assessment
2024
,1. Multiple Choice
- Question: Which of the following definitions correctly describes the
limit of a function \(f(x)\) as \(x\) approaches \(a\)?
- A) \( \lim_{{x \to a}} f(x) = L \) means that as \(x\) gets arbitrarily close
to \(a\), \(f(x)\) gets arbitrarily close to \(L\).
- B) \( \lim_{{x \to a}} f(x) = L \) if and only if \(f(a) = L\).
- C) \( \lim_{{x \to a}} f(x) = L \) means that \(f(x)\) is discontinuous at
\(a\).
- D) \( \lim_{{x \to a}} f(x) \) converges to \(L\) only when both \(x\) and
\(L\) are finite.
- Answer: A
- Rationale: Option A correctly describes the definition of the limit of a
function as \(x\) approaches \(a\).
2. Fill-in-the-Blank
- Question: The ______________ test is used to determine whether an
infinite series converges or diverges by comparing the ratio of successive
terms.
- Answer: Ratio
- Rationale: The Ratio Test compares the ratio of successive terms to
assess series convergence.
, 3. True/False
- Question: The Mean Value Theorem states that for a function
continuous on \([a, b]\) and differentiable on \((a, b)\), there exists some \(c
\in (a, b)\) such that \(f'(c) = \frac{f(b) - f(a)}{b - a}\).
- Answer: True
- Rationale: This is the precise statement of the Mean Value Theorem.
4. Multiple Choice
- Question: When finding the area under the curve of \(f(x)\) from \(a\)
to \(b\), which of the following integrals would you use?
- A) \( \int_{b}^{a} f(x) \, dx \)
- B) \( \int_{a}^{b} f(x) \, dx \)
- C) \( \int_{a}^{b} f(x) \, \partial x \)
- D) \( \int f(x) \, dx \)
- Answer: B
- Rationale: The integral from \(a\) to \(b\) of \(f(x)\) is used to find the
area under the curve.
5. Fill-in-the-Blank
- Question: The function \(f(x) = \frac{1}{x}\) has a vertical asymptote at
\(x = ________\).
- Answer: 0
- Rationale: The function \(\frac{1}{x}\) becomes undefined and
approaches infinity as \(x\) approaches 0, indicating a vertical asymptote
there.
Calculus I
Comprehensive Objective
Assessment
2024
,1. Multiple Choice
- Question: Which of the following definitions correctly describes the
limit of a function \(f(x)\) as \(x\) approaches \(a\)?
- A) \( \lim_{{x \to a}} f(x) = L \) means that as \(x\) gets arbitrarily close
to \(a\), \(f(x)\) gets arbitrarily close to \(L\).
- B) \( \lim_{{x \to a}} f(x) = L \) if and only if \(f(a) = L\).
- C) \( \lim_{{x \to a}} f(x) = L \) means that \(f(x)\) is discontinuous at
\(a\).
- D) \( \lim_{{x \to a}} f(x) \) converges to \(L\) only when both \(x\) and
\(L\) are finite.
- Answer: A
- Rationale: Option A correctly describes the definition of the limit of a
function as \(x\) approaches \(a\).
2. Fill-in-the-Blank
- Question: The ______________ test is used to determine whether an
infinite series converges or diverges by comparing the ratio of successive
terms.
- Answer: Ratio
- Rationale: The Ratio Test compares the ratio of successive terms to
assess series convergence.
, 3. True/False
- Question: The Mean Value Theorem states that for a function
continuous on \([a, b]\) and differentiable on \((a, b)\), there exists some \(c
\in (a, b)\) such that \(f'(c) = \frac{f(b) - f(a)}{b - a}\).
- Answer: True
- Rationale: This is the precise statement of the Mean Value Theorem.
4. Multiple Choice
- Question: When finding the area under the curve of \(f(x)\) from \(a\)
to \(b\), which of the following integrals would you use?
- A) \( \int_{b}^{a} f(x) \, dx \)
- B) \( \int_{a}^{b} f(x) \, dx \)
- C) \( \int_{a}^{b} f(x) \, \partial x \)
- D) \( \int f(x) \, dx \)
- Answer: B
- Rationale: The integral from \(a\) to \(b\) of \(f(x)\) is used to find the
area under the curve.
5. Fill-in-the-Blank
- Question: The function \(f(x) = \frac{1}{x}\) has a vertical asymptote at
\(x = ________\).
- Answer: 0
- Rationale: The function \(\frac{1}{x}\) becomes undefined and
approaches infinity as \(x\) approaches 0, indicating a vertical asymptote
there.
