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

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MAC 2233 Homework 3 Spring 2021

Name___________________________________

Show all work in the space provided. If no work is shown, the problem

will receive at most half credit for a correct answer.

 Technique and rules to compute the derivative

ο‚· For each problem from problem 41) to 49), given 𝒇(𝒙) compute

it derivative function 𝒇 β€²(𝒙) .

41) 𝑓(π‘₯) = 2βˆ’π‘₯2

3π‘₯2+1

42) 𝑓(π‘₯) = (π‘₯3 + 2π‘₯ βˆ’ 7)(3 + π‘₯ βˆ’ π‘₯2 )

43) 𝑓(π‘₯) = √π‘₯2 + 1

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

44) 𝑓(π‘₯) = (π‘₯ + 1

π‘₯ )

2 βˆ’

5

√3π‘₯

45) 𝑓(π‘₯) = (5π‘₯ 4 βˆ’ 3π‘₯ 2 + 2π‘₯ + 1)10

46) 𝑓(π‘₯) = ( π‘₯+1

1βˆ’π‘₯ )

2

47) 𝑓(π‘₯) = (3π‘₯ + 1)√6π‘₯ + 5

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

48) 𝑓(π‘₯) = (3π‘₯+1)3

(1βˆ’3π‘₯)4

49) 𝑓(π‘₯) = √ 1βˆ’2π‘₯

3π‘₯+2

ο‚· From problem 50) to 52) find the equation of The Tangent-Line for given function 𝑓(π‘₯) at a given π‘₯ = 𝑐.

50) 𝑓(π‘₯) = π‘₯2 βˆ’ 3π‘₯ + 2 ; π‘Žπ‘‘ π‘₯ = 1

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

51) 𝑓(π‘₯) = π‘₯

x2+1 ; π‘Žπ‘‘ π‘₯ = 0

52) 𝑓(π‘₯) = √π‘₯2 + 5 ; π‘Žπ‘‘ π‘₯ = βˆ’2

ο‚· For problem 53) and 54) Find the rate of change of 𝒇(𝒕)

with respect to 𝒕 at the given value of 𝒕 .

53) 𝑓(𝑑) = 2𝑑 2βˆ’5

1βˆ’3t ; π‘Žπ‘‘ 𝑑 = βˆ’1

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54) 𝑓(𝑑) = ( t2 βˆ’ 3𝑑 + 6 ) 1

2 ; π‘Žπ‘‘ 𝑑 = 1

ο‚· For problem 55) and 56) Find the Percentage-Rate of

change of 𝒇(𝒕) with respect to 𝒕 at the given value

of 𝒕 .

55) 𝑓(𝑑) = 𝑑 2 βˆ’ 3𝑑 + βˆšπ‘‘ ; π‘Žπ‘‘ 𝑑 = 4

56) 𝑓(𝑑) = 2

1+𝑑 2 ; π‘Žπ‘‘ 𝑑 = 0

ο‚· For problem 57), 58) and 59)Find the 2nd –derivatives:

57) 𝑓(π‘₯) = 6π‘₯5 βˆ’ 4π‘₯3 + 5π‘₯2 βˆ’ 2π‘₯ + 1

π‘₯

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

58) 𝑓(π‘₯) = 2

1+π‘₯2

59) 𝑓(π‘₯) = (3π‘₯2 + 2)4

60) For the function 𝑓(π‘₯) given below,

Find the nth-derivative 𝑓(𝑛)(π‘₯) .

𝑓(π‘₯) = 1

1βˆ’π‘₯

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

MAC 2233 Homework 4 Spring 2021

Name___________________________________

Show all work in the space provided. If no work is shown, the problem

will receive at most half credit for a correct answer.

❖ Application of 𝒇 β€²(𝒙) to find the graph of 𝒇(𝒙)

β€’ For problems 61) through 64) Find the intervals of increase and

decrease for each given function 𝑓(π‘₯)

61) 𝑓(π‘₯) = 1

3 π‘₯ 3 βˆ’ 9π‘₯ + 2

62) 𝑓(π‘₯) = π‘₯2

π‘₯βˆ’3

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

63) 𝑓(π‘₯) = 1

16βˆ’π‘₯2

64) 𝑓(π‘₯) = π‘₯ 2 + 1

π‘₯2

β€’ For problems 65) through 69) determine the critical-numbers 𝒙 = 𝒄 of each of the given function 𝑓(π‘₯) and classify each

critical-point ( 𝑐 , 𝑓(𝑐) ) as a Local-Maximum, Local-Minimum, or neither:

65) 𝑓(π‘₯) = 3π‘₯ 4 βˆ’ 8π‘₯ 3 + 6π‘₯ 2 + 2

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

66) 𝑓(π‘₯) = 2π‘₯ 3 + 6π‘₯ 2 + 6π‘₯ + 5

67) 𝑓(π‘₯) = π‘₯2

π‘₯βˆ’1

68) 𝑓(π‘₯) = 3 + (π‘₯ 2 βˆ’ 2π‘₯ βˆ’ 3) 2

5

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

69) 𝑓(π‘₯) = π‘₯√9 βˆ’ π‘₯

70) Find constants π‘Ž , 𝑏 and 𝑐 so that the graph of the function 𝑓(π‘₯) = π‘Žπ‘₯ 2 + 𝑏π‘₯ + 𝑐 has a Local-Maximum at (5, 12) and crosses the y-axis at (0, 3).

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

β€’ For problems 71) through 73) determine the intervals of concavity

and classify each of these intervals as concave-up and concave-

down for each of the given function 𝑓(π‘₯). Also, find all the inflection-points ( if any).

71) 𝑓(π‘₯) = 2π‘₯ 6 βˆ’ 5π‘₯ 4 + 7π‘₯ βˆ’ 3

72) 𝑓(π‘₯) = π‘₯ 2 βˆ’ 1

π‘₯

73) 𝑓(π‘₯) = (π‘₯ βˆ’ 4) 7

3

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

β€’ [PUTTING PIECES TOGETHER…]

For problems 74) through 76) for each given function

𝑓(π‘₯) determine a) Find intervals on which 𝑓(π‘₯) increasing and decreasing. b) Find intervals on which the graph of 𝑓(π‘₯) is concave-up

and concave-down. c) Find all critical-numbers and classify them as

Local-maximum, Local-Minimum or neither.

d) Find all the inflection-points of 𝑓(π‘₯) ( if any). 74) 𝑓(π‘₯) = π‘₯ 4 βˆ’ 4π‘₯ 3 + 10

75) 𝑓(π‘₯) = (π‘₯ 2 βˆ’ 5)3

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

76) 𝑓(π‘₯) = π‘₯ 4 + 6π‘₯ 3 βˆ’ 24π‘₯ 2 + 24

β€’ For problems 77) through 80) for each given function 𝑓(π‘₯) Find the Critical-Points and use the Second-Derivative-Test

To find Local-Maximum and Local-Minimum of the given

𝑓(π‘₯) .

77) 𝑓(π‘₯) = π‘₯ 4 βˆ’ 2π‘₯ 3 + 3

78) 𝑓(π‘₯) = 2π‘₯ + 1 + 18

π‘₯

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

79) 𝑓(π‘₯) = ( π‘₯

π‘₯+1 )

2

80) 𝑓(π‘₯) = (π‘₯+3)3

(π‘₯βˆ’1)2

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

MAC 2233 Homework 5 Spring 2021

Name___________________________________

Show all work in the space provided. If no work is shown, the problem

will receive at most half credit for a correct answer.

❖ More Applications of 𝒇 β€²(𝒙) and 𝒇 β€²β€²(𝒙) to sketch the graph of 𝒇(𝒙) , also optimization problems for business

β€’ For problems 81) through 84), sketch the graph of a function

𝑓(π‘₯) that has all the given properties 81)

o 𝑓′(π‘₯) > 0 π‘€β„Žπ‘’π‘› π‘₯ < 0 π‘Žπ‘›π‘‘ π‘€β„Žπ‘’π‘› π‘₯ > 5 ,

o 𝑓′(π‘₯) < 0 π‘€β„Žπ‘’π‘› 0 < π‘₯ < 5,

o 𝑓 β€²β€²(π‘₯) > 0 π‘€β„Žπ‘’π‘› βˆ’ 6 < π‘₯ < βˆ’3 π‘Žπ‘›π‘‘ π‘€β„Žπ‘’π‘› π‘₯ > 2 ,

o 𝑓 β€²β€²(π‘₯) < 0 π‘€β„Žπ‘’π‘› π‘₯ < βˆ’6 π‘Žπ‘›π‘‘

π‘€β„Žπ‘’π‘› βˆ’ 3 < π‘₯ < 2 .

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

82)

o 𝑓 β€²(π‘₯) > 0 π‘€β„Žπ‘’π‘› π‘₯ < βˆ’2 π‘Žπ‘›π‘‘ π‘€β„Žπ‘’π‘› βˆ’ 2 < π‘₯ < 3,

o 𝑓 β€²(π‘₯) < 0 π‘€β„Žπ‘’π‘› π‘₯ > 3,

o 𝑓 β€²(βˆ’2) = 0 and 𝑓 β€²(3) = 0.

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

83)

o 𝑓′(π‘₯) > 0 π‘€β„Žπ‘’π‘› 1 < π‘₯ < 2,

o 𝑓 β€²(π‘₯) < 0 π‘€β„Žπ‘’π‘› π‘₯ < 1 π‘Žπ‘›π‘‘ π‘€β„Žπ‘’π‘› π‘₯ > 2 ,

o 𝑓 β€²β€²(π‘₯) > 0 π‘“π‘œπ‘Ÿ π‘₯ < 2 π‘Žπ‘›π‘‘ π‘“π‘œπ‘Ÿ π‘₯ > 2 ,

o 𝑓 β€²(1) = 0 and 𝑓 β€²(2) 𝑖𝑠 𝑒𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑

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

84)

o 𝑓′(π‘₯) > 0 π‘€β„Žπ‘’π‘› π‘₯ < 1,

o 𝑓 β€²(π‘₯) < 0 π‘€β„Žπ‘’π‘› π‘₯ > 1 ,

o 𝑓 β€²β€²(π‘₯) > 0 π‘€β„Žπ‘’π‘› π‘₯ < 1 π‘Žπ‘›π‘‘ π‘€β„Žπ‘’π‘› π‘₯ > 1 ,

o 𝑓 β€²(1) 𝑖𝑠 𝑒𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑.

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

β€’ For problems 85) through 88), find the Absolute-Maximum and

the Absolute-Minimum values (if any) of the function 𝑓(π‘₯) on the specified interval

85) 𝑓(π‘₯) = βˆ’2π‘₯ 3 + 3π‘₯ 2 + 12π‘₯ βˆ’ 5 ; π‘œπ‘› βˆ’ 3 ≀ π‘₯ ≀ 3

86) 𝑓(π‘₯) = βˆ’3π‘₯ 4 + 8π‘₯ 3 βˆ’ 10 ; π‘œπ‘› 0 ≀ π‘₯ ≀ 3

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

87) 𝑓(π‘₯) = π‘₯2

π‘₯+1 ; π‘œπ‘› βˆ’

1

2 ≀ π‘₯ ≀ 1

88) 𝑓(π‘₯) = 2π‘₯ + 8

π‘₯ + 2 ; π‘œπ‘› π‘₯ > 0

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

89) Find two positive numbers whose sum is 50 and whose product is as large as possible.

90) The are 320 yards of fencing available to enclose a rectangular field. How should this fencing be used so that the enclosed area is as large as possible?

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

91) Prove that of all rectangles with a given perimeter, the square has the largest-Area. 92) Prove that of all rectangles with a given area, the square has the Smallest-Perimeter.

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

93) A carpenter has been asked to build an open box with a square base. The sides of the box will cost $3 per square meter, and the base will cost $4 per square meter. What are the dimensions of the box of Greatest- Volume that can be constructed for $48 ? 94) Diana is a carpenter who has been hired to make a Closed-Box with a square-base and Volume of 250 cubic- meters. The material for the Top and the Bottom of the box costs $2 per square meter, and the material for the sides costs $1 per square meter. Can Diana construct the box for less than $300 ?

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

95) Use Logarithmic-Differentiation to compute 𝑓 β€²(π‘₯) given

𝑓(π‘₯) = (π‘₯ 2 + 𝑒 2π‘₯)3 𝑒 βˆ’2π‘₯

(1 + π‘₯ + π‘₯ 2)2/3

96) Use Logarithmic-Differentiation to compute 𝑓 β€²(π‘₯) given

𝑓(π‘₯) = 𝑒 βˆ’2π‘₯ (2 βˆ’ π‘₯ 3)3/2

√ 1+π‘₯ 2

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

97) Use Logarithmic-Differentiation to compute 𝑓 β€²(π‘₯) given

𝑓(π‘₯) = (π‘₯ + 1)3 (6 βˆ’ π‘₯)2 √2π‘₯ + 1 3

98) Use Logarithmic-Differentiation to

compute 𝑓 β€²(π‘₯) given

𝑓(π‘₯) = 5π‘₯ 3

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

99) Notice that 𝑦 = 𝑓(π‘₯) [π‘‘β„Žπ‘Žπ‘‘ 𝑖𝑠 𝑦 𝑖𝑠 π‘Ž π‘“π‘’π‘›π‘π‘‘π‘–π‘œπ‘› 𝑑𝑒𝑝𝑒𝑛𝑑𝑖𝑛𝑔 π‘œπ‘› π‘₯]. Use Implicit-Differentiation to find 𝑦′ [π‘€β„Žπ‘–π‘β„Ž 𝑖𝑠 𝑦′ = 𝑓 β€²(π‘₯) ] given

π‘₯ 𝑒 βˆ’π‘¦ + 𝑦 𝑒 βˆ’π‘₯ = 3

100) The Marginal-Profit of a certain commodity is

𝑃 β€²(π‘₯) = 100 βˆ’ 2π‘₯ When π‘₯ units are produced. When 10 units are produced, the Profit is $700.

a) Find the profit 𝑃(π‘₯) . b) What production level π‘₯ results in maximum

Profit ? What is the maximum Profit ?

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