Calculus Assignments

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

MAC 2233 Homework 5 Spring 2021

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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 ′(𝒙) and 𝒇 ′′(𝒙) to sketch the graph of 𝒇(𝒙) , also optimization problems for business

• For problems 81) through 84), sketch the graph of a function

𝑓(𝑥) that has all the given properties 81)

o 𝑓′(𝑥) > 0 𝑤ℎ𝑒𝑛 𝑥 < 0 𝑎𝑛𝑑 𝑤ℎ𝑒𝑛 𝑥 > 5 ,

o 𝑓′(𝑥) < 0 𝑤ℎ𝑒𝑛 0 < 𝑥 < 5,

o 𝑓 ′′(𝑥) > 0 𝑤ℎ𝑒𝑛 − 6 < 𝑥 < −3 𝑎𝑛𝑑 𝑤ℎ𝑒𝑛 𝑥 > 2 ,

o 𝑓 ′′(𝑥) < 0 𝑤ℎ𝑒𝑛 𝑥 < −6 𝑎𝑛𝑑

𝑤ℎ𝑒𝑛 − 3 < 𝑥 < 2 .

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

82)

o 𝑓 ′(𝑥) > 0 𝑤ℎ𝑒𝑛 𝑥 < −2 𝑎𝑛𝑑 𝑤ℎ𝑒𝑛 − 2 < 𝑥 < 3,

o 𝑓 ′(𝑥) < 0 𝑤ℎ𝑒𝑛 𝑥 > 3,

o 𝑓 ′(−2) = 0 and 𝑓 ′(3) = 0.

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

83)

o 𝑓′(𝑥) > 0 𝑤ℎ𝑒𝑛 1 < 𝑥 < 2,

o 𝑓 ′(𝑥) < 0 𝑤ℎ𝑒𝑛 𝑥 < 1 𝑎𝑛𝑑 𝑤ℎ𝑒𝑛 𝑥 > 2 ,

o 𝑓 ′′(𝑥) > 0 𝑓𝑜𝑟 𝑥 < 2 𝑎𝑛𝑑 𝑓𝑜𝑟 𝑥 > 2 ,

o 𝑓 ′(1) = 0 and 𝑓 ′(2) 𝑖𝑠 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑

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

84)

o 𝑓′(𝑥) > 0 𝑤ℎ𝑒𝑛 𝑥 < 1,

o 𝑓 ′(𝑥) < 0 𝑤ℎ𝑒𝑛 𝑥 > 1 ,

o 𝑓 ′′(𝑥) > 0 𝑤ℎ𝑒𝑛 𝑥 < 1 𝑎𝑛𝑑 𝑤ℎ𝑒𝑛 𝑥 > 1 ,

o 𝑓 ′(1) 𝑖𝑠 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑.

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

• For problems 85) through 88), find the Absolute-Maximum and

the Absolute-Minimum values (if any) of the function 𝑓(𝑥) on the specified interval

85) 𝑓(𝑥) = −2𝑥 3 + 3𝑥 2 + 12𝑥 − 5 ; 𝑜𝑛 − 3 ≤ 𝑥 ≤ 3

86) 𝑓(𝑥) = −3𝑥 4 + 8𝑥 3 − 10 ; 𝑜𝑛 0 ≤ 𝑥 ≤ 3

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

87) 𝑓(𝑥) = 𝑥2

𝑥+1 ; 𝑜𝑛 −

1

2 ≤ 𝑥 ≤ 1

88) 𝑓(𝑥) = 2𝑥 + 8

𝑥 + 2 ; 𝑜𝑛 𝑥 > 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 ′(𝑥) given

𝑓(𝑥) = (𝑥 2 + 𝑒 2𝑥)3 𝑒 −2𝑥

(1 + 𝑥 + 𝑥 2)2/3

96) Use Logarithmic-Differentiation to compute 𝑓 ′(𝑥) given

𝑓(𝑥) = 𝑒 −2𝑥 (2 − 𝑥 3)3/2

√ 1+𝑥 2

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

97) Use Logarithmic-Differentiation to compute 𝑓 ′(𝑥) given

𝑓(𝑥) = (𝑥 + 1)3 (6 − 𝑥)2 √2𝑥 + 1 3

98) Use Logarithmic-Differentiation to

compute 𝑓 ′(𝑥) given

𝑓(𝑥) = 5𝑥 3

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

99) Notice that 𝑦 = 𝑓(𝑥) [𝑡ℎ𝑎𝑡 𝑖𝑠 𝑦 𝑖𝑠 𝑎 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑑𝑒𝑝𝑒𝑛𝑑𝑖𝑛𝑔 𝑜𝑛 𝑥]. Use Implicit-Differentiation to find 𝑦′ [𝑤ℎ𝑖𝑐ℎ 𝑖𝑠 𝑦′ = 𝑓 ′(𝑥) ] given

𝑥 𝑒 −𝑦 + 𝑦 𝑒 −𝑥 = 3

100) The Marginal-Profit of a certain commodity is

𝑃 ′(𝑥) = 100 − 2𝑥 When 𝑥 units are produced. When 10 units are produced, the Profit is $700.

a) Find the profit 𝑃(𝑥) . b) What production level 𝑥 results in maximum

Profit ? What is the maximum Profit ?