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

Since the limit is 0/0 form, L'Hôpital gives lim e^x / 1 = e^0 = 1.

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

The MVT states that the instantaneous rate of change equals the average rate of change at some interior point c.

Question 3

Which of the following is NOT an improper integral?

Accepted Answer

∫[−1 to 1] x² dx

Answer

∫[−1 to 1] x² dx has a continuous integrand on a closed bounded interval, so it is a proper integral.

Question 4

What is ∫ x e^x dx?

Accepted Answer

(x − 1)e^x + C

Answer

Integration by parts with u = x, dv = e^x dx yields xe^x − ∫e^x dx = xe^x − e^x + C = (x−1)e^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

The Taylor series Σ (−1)^(n+1) x^n / n converges for −1 < x ≤ 1 by the alternating series and comparison tests at the endpoints.

BMath Bachelor of Mathematics Practice Test