BMath Bachelor of Mathematics Calculus and Real Analysis 1 — Questions and Answers
Question 1: What is the limit of (sin x)/x as x approaches 0?
- 0
- 1 (Correct answer)
- ∞
- Undefined
Correct answer: 1
The standard limit lim(x→0) (sin x)/x = 1 is a fundamental result in calculus.
Question 2: Which condition must hold for a function f to be differentiable at a point x = a?
- f must be continuous at a (Correct answer)
- f must be bounded near a
- f must be periodic
- f must be monotone near a
Correct answer: f must be continuous at a
Differentiability at a point implies continuity there, so continuity is a necessary condition.
Question 3: What does the Fundamental Theorem of Calculus, Part 1, state about F(x) = ∫[a to x] f(t) dt?
- F(x) = f(x)
- F'(x) = f(x) (Correct answer)
- F(x) = F(a) + f(x)
- F'(x) = f(a)
Correct answer: F'(x) = f(x)
Part 1 of the FTC states that if f is continuous, then F'(x) = f(x), linking differentiation and integration.
Question 4: Which series convergence test is best suited when the general term involves a factorial?
- Integral test
- Ratio test (Correct answer)
- Root test
- Comparison test
Correct answer: Ratio test
The Ratio test is particularly effective for series with factorials because the ratio of consecutive terms simplifies cleanly.
Question 5: What is the radius of convergence R of the power series Σ (x^n / n!) for n = 0 to ∞?
- 0
- 1
- e
- ∞ (Correct answer)
Correct answer: ∞
Applying the ratio test gives lim |x|/(n+1) = 0 for all x, so R = ∞ and the series converges everywhere.
Question 6: A function f: ℝ → ℝ is said to be uniformly continuous on ℝ if:
- For every ε > 0, there exists δ > 0 such that |f(x) − f(y)| < ε whenever |x − y| < δ, with δ independent of x and y (Correct answer)
- For every x, there exists δ > 0 such that |f(x) − f(y)| < ε for |x − y| < δ
- f is differentiable everywhere on ℝ
- f is bounded on every closed interval
Correct answer: For every ε > 0, there exists δ > 0 such that |f(x) − f(y)| < ε whenever |x − y| < δ, with δ independent of x and y
Uniform continuity requires a single δ that works for all pairs of points, unlike pointwise continuity where δ may depend on x.
BMath Bachelor of Mathematics Calculus and Real Analysis 1
What is the limit of (sin x)/x as x approaches 0?