UP Academic League of Chemical Engineering Students (UP ALCHEMES) Academic Affairs Committee - Reviews and Tutorials Series, AY 2014-2015 Math 53 First Exam (SAMPLE EXAM)

 

 

I. Given f(x) = ln (x + 6), find f-1 (x) and the range of f.

II. Evaluate the following limits:

1. lim!→!!!!!!

!! !!!!!

2. lim!→!!!!! !!!|!!!|

3. lim!→!! 𝑥 + 𝑥 − 𝑥 4. lim!→!

!!""!!!!!

5. lim!→!!!!!"# !!

!

III. Given f(x) defined as:

𝑓 𝑥 =

𝑥!

1 + sin 𝑥          𝑖𝑓 − 2 < 𝑥 < −

𝜋2

5𝑥 − 6          𝑖𝑓   −𝜋2≤ 𝑥 < 1

𝑥−|𝑥|

         𝑖𝑓  1 ≤ 𝑥 < 3

42 − 𝑥 − 𝑥!

6 − 𝑥          𝑖𝑓  𝑥 ≥ 3

a. Find and classify all points of discontinuity of the following functions. b. Find the union of intervals at

which f(x) is continuous.

IV. Use the Squeeze Theorem to find the value of

lim!→!!

2 sin(𝑥) + 9𝑥! − 13