Department of Mathematics

Midterm # 2, MATH-UA.0325 - Fall 2024

Exercise 1. (6 pts) True or false. Justify your answer.

a) The function g(x) = sin x is uniformly continuous on R.

b) Let f(x) be a continuous function on [0, 1], then

c)

Exercise 2. (10 pts) Compute the following limits:

a)

b)

Exercise 3. (4 pts) Find the antiderivative of the function

if any.

Exercise 4. (10 pts) Find the Taylor series centered at x0 and find the interval on which the expansion is valid.

a) where x0 = 1.

b) where x0 = 0.

Exercise 5. (20 pts) Answer to the following questions:

Determine whether or not the sequence where

converges by answering to the following questions:

a) Is the function f(x) obtained from the n−term of the sequence continuous?

b) Is the function f(x) positive?

c) Is the function f(x) decreasing?

d) Does the improper integral f(x)dx converge? Justify your answer.