BMath Bachelor of Mathematics Calculus and Real Analysis

Question 1

What is the limit of (sin x)/x as x approaches 0?

Accepted Answer

1

Answer

0

Answer

∞

Answer

Undefined

Question 2

Which condition must hold for a function f to be differentiable at a point x = a?

Accepted Answer

f must be continuous at a

Answer

f must be bounded near a

Answer

f must be periodic

Answer

f must be monotone near a

Question 3

What does the Fundamental Theorem of Calculus, Part 1, state about F(x) = ∫[a to x] f(t) dt?

Accepted Answer

F'(x) = f(x)

Answer

F(x) = f(x)

Answer

F(x) = F(a) + f(x)

Answer

F'(x) = f(a)

Question 4

Which series convergence test is best suited when the general term involves a factorial?

Accepted Answer

Ratio test

Answer

Integral test

Answer

Root test

Answer

Comparison test

Question 5

What is the radius of convergence R of the power series Σ (x^n / n!) for n = 0 to ∞?

Accepted Answer

∞

Answer

0

Answer

1

Answer

e

Question 6

A function f: ℝ → ℝ is said to be uniformly continuous on ℝ if:

Accepted 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

Answer

For every x, there exists δ > 0 such that |f(x) − f(y)| < ε for |x − y| < δ

Answer

f is differentiable everywhere on ℝ

Answer

f is bounded on every closed interval