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MIAMI DADE COLLEGE PROFESSOR; RAYSURI A. ZAITER-CICCONE MAC 2233 SPRING 2021

MAC 2233 Homework 2 Spring 2021

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❖ Continuity • For each of the following problems [from problem 21) to problem 26) ]

Find (if any) the points of discontinuity of 𝑓(𝑥) (points 𝒙’𝒔 where 𝑓(𝑥) is NOT Continuous ) :

21) 𝑓(𝑥) = 𝑥2−𝑥−2

𝑥−2

22) 𝑓(𝑥) = { 1

𝑥2 , 𝑖𝑓 𝑥 ≠ 0

1 , 𝑖𝑓 𝑥 = 0

MIAMI DADE COLLEGE PROFESSOR; RAYSURI A. ZAITER-CICCONE MAC 2233 SPRING 2021

23) 𝑓(𝑥) = { 𝑥2−𝑥−2

𝑥−2 , 𝑖𝑓 𝑥 ≠ 2

3 , 𝑖𝑓 𝑥 = 2

24) 𝑓(𝑥) = { 𝑥2−𝑥

𝑥2−1 , 𝑖𝑓 𝑥 ≠ 1

1 , 𝑖𝑓 𝑥 = 1

25) 𝑓(𝑥) = {

𝑥2 + 1 , 𝑖𝑓 𝑥 ≤ 0 2 − 𝑥 , 𝑖𝑓 0 < 𝑥 ≤ 2

(𝑥 − 2)2, 𝑖𝑓 𝑥 > 2

MIAMI DADE COLLEGE PROFESSOR; RAYSURI A. ZAITER-CICCONE MAC 2233 SPRING 2021

26) 𝑓(𝑥) = {

𝑥 + 1 , 𝑖𝑓 𝑥 ≤ 1

1

𝑥 , 𝑖𝑓 1 < 𝑥 < 3

√𝑥 − 3 , 𝑖𝑓 𝑥 ≥ 3

27) For what values of the constant 𝑐 is the function 𝑓(𝑥) continuous on (−∞, ∞) ?

𝑓(𝑥) = { 𝑐𝑥2 + 2𝑥 , 𝑖𝑓 𝑥 < 2

𝑥3 − 𝑐𝑥 , 𝑖𝑓 𝑥 ≥ 2

MIAMI DADE COLLEGE PROFESSOR; RAYSURI A. ZAITER-CICCONE MAC 2233 SPRING 2021

28) Find the values of the 𝑎 and 𝑏 that makes 𝑓(𝑥) continuous everywhere.

𝑓(𝑥) =

{

𝑥2 − 4

𝑥 − 2 , 𝑖𝑓 𝑥 < 2

𝑎𝑥2 − 𝑏𝑥 + 3, 𝑖𝑓 2 ≤ 𝑥 < 3 2𝑥 − 𝑎 + 𝑏 , 𝑖𝑓 𝑥 ≥ 3

29) Find all the values of 𝑎 such that 𝑓(𝑥) continuous everywhere (continuous for all real numbers x).

𝑓(𝑥) = { 𝑥 + 1 , 𝑖𝑓 𝑥 ≤ 𝑎

𝑥2 , 𝑖𝑓 𝑥 > 𝑎

MIAMI DADE COLLEGE PROFESSOR; RAYSURI A. ZAITER-CICCONE MAC 2233 SPRING 2021

❖ More Limits and its Applications

30) Compute 𝐥𝐢𝐦 𝒙→𝟏

√𝑥 3 − 1 √ 𝑥 − 1

[hint: use “limit substitution” 𝒕𝟔 = 𝒙 notice that as 𝒙 → 𝟏 then 𝒕𝟔 → 𝟏 and so 𝒕 → 𝟏]. 31) Find numbers 𝑎 and 𝑏 such that

𝐥𝐢𝐦 𝒙→𝟏

√ 𝑎𝑥 + 𝑏 − 2

𝑥 = 𝟏

MIAMI DADE COLLEGE PROFESSOR; RAYSURI A. ZAITER-CICCONE MAC 2233 SPRING 2021

32) Definition: Horizontal-Asymptotes of a function 𝑓(𝑥) are defined by the equations 𝒚 = 𝒍𝒊𝒎

𝒙→∞ 𝑓(𝑥) and

𝒚 = 𝐥𝐢𝐦 𝐱→ −∞

𝑓(𝑥) .

These horizontal-Asymptotes exists only if their limits exist.

Find all the horizontal-Asymptotes for the given function below:

𝑓(𝑥) = √2𝑥2 + 1

3𝑥 − 5

33) Compute the following limit:

𝒍𝒊𝒎 𝒙→∞

√𝟗𝒙𝟐 + 𝒙 − 𝟑𝒙

MIAMI DADE COLLEGE PROFESSOR; RAYSURI A. ZAITER-CICCONE MAC 2233 SPRING 2021

34) Compute the following limit:

𝒍𝒊𝒎 𝒙→ −∞

√𝒙𝟐 + 𝒙 + 𝟏 + 𝟑𝒙

35) Use the definition of the derivative to find 𝑓′(𝑥) given

𝑓(𝑥) = 3𝑥2 − 𝑥 + 1

MIAMI DADE COLLEGE PROFESSOR; RAYSURI A. ZAITER-CICCONE MAC 2233 SPRING 2021

36) Let 𝑠(𝑡) = √ 𝑡 .

a) Find the average rate of change of 𝑠(𝑡) with respect to

𝑡 as 𝑡 changes from 𝑡 = 1

9 to 𝑡 = 4 .

b) Use calculus to find the instantaneous rate of change of 𝑠(𝑡) at 𝑡 = 4.

MIAMI DADE COLLEGE PROFESSOR; RAYSURI A. ZAITER-CICCONE MAC 2233 SPRING 2021

37) The gross annual earnings of certain company were

𝐴(𝑡) = 0.1𝑡2 + 10𝑡 + 20 thousand dollars 𝑡 years after its formation in 2008.

a) At what rate were the gross annual earnings of the company growing with respect to time in 2012?

b) At what percentage rate were the gross annual earnings growing with respect to time in 2012?

MIAMI DADE COLLEGE PROFESSOR; RAYSURI A. ZAITER-CICCONE MAC 2233 SPRING 2021

38)Find the derivative of the given function.

a) 𝑦 = −𝑥2

2 + 3

𝑥 − 2𝑥

3

2 + 1

3𝑥3 + 𝑥

5

b) 𝑓(𝑥) = √3𝑥5 + 3

MIAMI DADE COLLEGE PROFESSOR; RAYSURI A. ZAITER-CICCONE MAC 2233 SPRING 2021

c) 𝑔(𝑥) = 2(3𝑥 + 1)4(5𝑥 − 3)2

39) The gross annual earnings of certain company are

𝑓(𝑡) = √2𝑥 + 6 thousand dollars 𝑡 years

after its formation in January 2010.

a) At what rate were the gross annual earnings of the

company be growing in January 2015?

MIAMI DADE COLLEGE PROFESSOR; RAYSURI A. ZAITER-CICCONE MAC 2233 SPRING 2021

b) At what percentage rate were the gross annual earnings be growing in January 2015?

40) Find the second derivative of the given function.

A. 𝑓(𝑥) = 2

5𝑥+1

B. 𝑓(𝑥) = (3𝑥 + 5)5