BMath Bachelor of Mathematics Calculus and Real Analysis 2

Question 1

Using L'Hôpital's Rule, what is lim(x→0) (e^x − 1)/x?

Accepted Answer

1

Answer

0

Answer

e

Answer

∞

Question 2

The Mean Value Theorem guarantees that for a differentiable function on [a, b], there exists c ∈ (a, b) such that:

Accepted Answer

f'(c) = (f(b) − f(a))/(b − a)

Answer

f(c) = 0

Answer

f'(c) = 0

Answer

f(c) = (f(a) + f(b))/2

Question 3

Which of the following is NOT an improper integral?

Accepted Answer

∫[−1 to 1] x² dx

Answer

∫[0 to ∞] e^(−x) dx

Answer

∫[0 to 1] 1/√x dx

Answer

∫[1 to ∞] 1/x dx

Question 4

What is ∫ x e^x dx?

Accepted Answer

(x − 1)e^x + C

Answer

x e^x + C

Answer

x² e^x / 2 + C

Answer

e^x / x + C

Question 5

The Taylor series of ln(1 + x) centered at x = 0 converges for which values of x?

Accepted Answer

−1 < x ≤ 1

Answer

|x| < 1 only

Answer

−1 ≤ x ≤ 1

Answer

All real x

Question 6

If f is continuous on [a, b], which theorem guarantees f attains both its maximum and minimum values?

Accepted Answer

Extreme Value Theorem

Answer

Intermediate Value Theorem

Answer

Mean Value Theorem

Answer

Bolzano's Theorem