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

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?

Accepted Answer

f must be continuous at a

Answer

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?

Accepted Answer

F'(x) = f(x)

Answer

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?

Accepted Answer

Ratio test

Answer

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 ∞?

Accepted Answer

∞

Answer

Applying the ratio test gives lim |x|/(n+1) = 0 for all x, so R = ∞ and the series converges everywhere.

BMath Bachelor of Mathematics Practice Test