vijk

P1 is the point left parenthesis negative 5 comma 6 comma negative 1 right parenthesis

(−5,6,−1) and Upper P 2

P2 is the point left parenthesis 0 comma negative 8 comma negative 8 right parenthesis

(0,−8,−8).

i

j

k

 

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Calculus problems: Definite Integrals (Please show work and solution on paper. Thank you).

• Attachment 1
• Attachment 2
• Attachment 3
• Attachment 4
• Attachment 5

Express the limit as a definite integral on the given interval .lim17 – 00X ; In ( 1 + X ; 2 ) Ax , [0 , 1]i = 1dx

 

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Express each function in the form uðx; yÞ þ ivðx; yÞ, where u and v are real:

(a) z3; ðbÞ 1=ð1 zÞ; ðcÞ e3z; ðdÞ ln z.

ðaÞ w ¼ z3 ¼ ðx þ iyÞ3 ¼ x3 þ 3×2ðiyÞ þ 3xðiyÞ2 þ ðiyÞ3 ¼ x3 þ 3ix2y 3xy2 iy2

¼ x3 3xy2 þ ið3x2y y3Þ

Then uðx; yÞ ¼ x3 3xy2; vðx; yÞ ¼ 3x2y y3.

ðbÞ w ¼

1

1 z ¼

1

1 ðx þ iyÞ ¼

1

1 x iy

1 x þ iy

1 x þ iy ¼

1 x þ iy

ð1 xÞ2 þ y2

Then uðx; yÞ ¼

1 x

ð1 xÞ2 þ y2

; vðx; yÞ ¼

y

ð1 xÞ2 þ y2

:

ðcÞ e3z ¼ e3ðxþiyÞ ¼ e3x e3iy ¼ e3xðcos 3y þ i sin 3yÞ and u ¼ e3x cos 3y; v ¼ e3x sin 3y

ðdÞ ln z ¼ lnðeiÞ ¼ ln þ i ¼ ln

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2 þ y2

q

þ i tan1 y=x

 

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Express Builders is a company which supplies building materials. Express Builders supplies a number of local building contractors. They clearly state in all their contracts that the building materials which they supply remain their property until paid for in full, with a right to reclaim the goods if the building contractor fails to pay for them within the stipulated time. Express Builders has supplied the goods set out below to some contractors and they want to know if they will be able to claim the return of the goods if they are not paid.

Advise Express Builders, with reference to case law, as to whether any of these items may be removed and returned to them, in the event of non-payment:

a) Express Builders supplied 50 000 bricks to Mpho Manana for use in the construction of a house. Construction of the house has not yet begun and all of the bricks are stacked up on the site.

b) Express Builders supplied 40 wooden trusses to Joe Martin for use in the construction of a factory. The factory has been completed and the roof trusses have been placed in position and covered with plastic and tiles. The roof can be dismantled without causing damage to the trusses or the factory.

c) Express Builders supplied 50 aluminium window frames to Kate Price for use in the renovation of a block of flats. The block of flats has been renovated and the windows have been screwed to the walls of the building and stuck in place with strong epoxy resin glue. The windows can only be removed by damaging them.

Your discussion must include:

• An explanation of the law as it relates to the transfer of movables as discussed in the facts above and the impact of original acquisition through accession; (10 marks)

• the requirements indicating permanency; and (15 marks)

• an application of the law and legal precedent to each of the contracts identified above. (10 marks)

 

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