BMath Bachelor of Mathematics Calculus and Real Analysis 2 — Questions and Answers
Question 1: Using L'Hôpital's Rule, what is lim(x→0) (e^x − 1)/x?
- 0
- 1 (Correct answer)
- e
- ∞
Correct answer: 1
Since the limit is 0/0 form, L'Hôpital gives lim e^x / 1 = e^0 = 1.
Question 2: The Mean Value Theorem guarantees that for a differentiable function on [a, b], there exists c ∈ (a, b) such that:
- f(c) = 0
- f'(c) = 0
- f'(c) = (f(b) − f(a))/(b − a) (Correct answer)
- f(c) = (f(a) + f(b))/2
Correct answer: f'(c) = (f(b) − f(a))/(b − a)
The MVT states that the instantaneous rate of change equals the average rate of change at some interior point c.
Question 3: Which of the following is NOT an improper integral?
- ∫[0 to ∞] e^(−x) dx
- ∫[0 to 1] 1/√x dx
- ∫[−1 to 1] x² dx (Correct answer)
- ∫[1 to ∞] 1/x dx
Correct answer: ∫[−1 to 1] x² dx
∫[−1 to 1] x² dx has a continuous integrand on a closed bounded interval, so it is a proper integral.
Question 4: What is ∫ x e^x dx?
- x e^x + C
- (x − 1)e^x + C (Correct answer)
- x² e^x / 2 + C
- e^x / x + C
Correct answer: (x − 1)e^x + C
Integration by parts with u = x, dv = e^x dx yields xe^x − ∫e^x dx = xe^x − e^x + C = (x−1)e^x + C.
Question 5: The Taylor series of ln(1 + x) centered at x = 0 converges for which values of x?
- |x| < 1 only
- −1 < x ≤ 1 (Correct answer)
- −1 ≤ x ≤ 1
- All real x
Correct answer: −1 < x ≤ 1
The Taylor series Σ (−1)^(n+1) x^n / n converges for −1 < x ≤ 1 by the alternating series and comparison tests at the endpoints.
Question 6: If f is continuous on [a, b], which theorem guarantees f attains both its maximum and minimum values?
- Intermediate Value Theorem
- Mean Value Theorem
- Extreme Value Theorem (Correct answer)
- Bolzano's Theorem
Correct answer: Extreme Value Theorem
The Extreme Value Theorem states that a continuous function on a closed bounded interval attains its maximum and minimum.
BMath Bachelor of Mathematics Calculus and Real Analysis 2
Using L'Hôpital's Rule, what is lim(x→0) (e^x − 1)/x?