BMath Bachelor of Mathematics Calculus and Real Analysis 3

Question 1

What is the derivative of arctan(x)?

Accepted Answer

1/(1 + x²)

Answer

The derivative of arctan(x) is 1/(1 + x²), derived by implicit differentiation of tan(y) = x.

Question 2

Which of the following correctly defines a Cauchy sequence in ℝ?

Accepted Answer

A sequence where for every ε > 0 there exists N such that |a_m − a_n| < ε for all m, n > N

Answer

A Cauchy sequence requires that terms become arbitrarily close to each other, not necessarily to a fixed limit a priori.

Question 3

For the function f(x) = x³ − 3x, what are the local extrema?

Accepted Answer

Local max at x = −1, local min at x = 1

Answer

f'(x) = 3x² − 3 = 0 gives x = ±1; f''(−1) < 0 (max) and f''(1) > 0 (min).

Question 4

What does it mean for a sequence {a_n} to diverge to +∞?

Accepted Answer

For every M > 0, there exists N such that a_n > M for all n > N

Answer

Divergence to +∞ means the sequence eventually exceeds any fixed bound M, which is the formal definition.

Question 5

What is the integral ∫[0 to π] sin(x) dx?

Accepted Answer

2

Answer

∫sin(x) dx = −cos(x) + C, so [−cos(x)] from 0 to π = −cos(π) + cos(0) = 1 + 1 = 2.

BMath Bachelor of Mathematics Practice Test