BMath Bachelor of Mathematics Calculus and Real Analysis 3

Question 1

What is the derivative of arctan(x)?

Accepted Answer

1/(1 + x²)

Answer

1/√(1 − x²)

Answer

−1/(1 + x²)

Answer

1/(x² − 1)

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 sequence where each term is smaller than the previous

Answer

A sequence that converges to zero

Answer

A sequence bounded above by some M

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

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

Answer

No local extrema

Answer

Local max at x = 0

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

The sequence has no upper bound

Answer

The sequence oscillates without bound

Answer

The sequence has no convergent subsequence

Question 5

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

Accepted Answer

2

Answer

0

Answer

1

Answer

π

Question 6

Which rule allows differentiation of a composite function f(g(x))?

Accepted Answer

Chain rule

Answer

Product rule

Answer

Quotient rule

Answer

Power rule