Exam (elaborations) MAT1512 - Calculus A (MAT1512) Calculus: Early Transcendentals, ISBN: 9780538497909

Calculus A
MAT1512
Assignment 4

Question 1

2
(a) 𝑧 = 𝑙𝑛(𝑥 + 𝑡 )
∂𝑧 1
𝑧𝑥 = ∂𝑥
= 2
(𝑥+𝑡 )
∂𝑧 2𝑡
𝑧𝑡 = ∂𝑡
= 2
(𝑥+𝑡 )


𝑥
𝑡
(b) 𝐹(𝑥, 𝑦) = ∫ 𝑐𝑜𝑠(𝑒 )𝑑𝑡
𝑦
𝑥
∂𝐹 𝑡
∂𝑥
= ∫ 𝑐𝑜𝑠(𝑒 )𝑑𝑡
𝑦
∂𝐹 𝑥
∂𝑥
= 𝑐𝑜𝑠(𝑒 )
Now
𝑥
∂𝐹 𝑡
∂𝑦
= ∫ 𝑐𝑜𝑠(𝑒 )𝑑𝑡
𝑦
−𝑦
𝑡
= ∫ 𝑐𝑜𝑠(𝑒 )𝑑𝑡
𝑥
𝑦
=− 𝑐𝑜𝑠(𝑒 )


2 −𝑥𝑧
(c) 𝐹(𝑥, 𝑦, 𝑧) = 𝑥𝑦 𝑒

, ∂𝐹 2 −𝑥𝑧 2 −𝑥𝑧
∂𝑥
=𝑦 𝑒 − 𝑥𝑧𝑦 𝑒
2 −𝑥𝑧
=𝑦 𝑒 (1 − 𝑥𝑧)

∂𝐹 −𝑥𝑧
∂𝑦
= 2𝑥𝑦𝑒

∂𝐹 2 2 −𝑥𝑧
∂𝑧
=− 𝑥 𝑦 𝑒




Question 2

(a) 𝑈 = 𝑙𝑛(𝑥 + 2𝑦)
1
𝑈𝑥(𝑥, 𝑦) = 𝑥+2𝑦
2
𝑈𝑦(𝑥, 𝑦) = 𝑥+2𝑦


∂ 1 2
𝑈𝑥𝑦(𝑥, 𝑦) = ∂𝑦
( 𝑥+2𝑦 ) =− 2
(𝑥+2𝑦)


∂ 2 2
𝑈𝑦𝑥(𝑥, 𝑦) = ∂𝑥
( 𝑥+2𝑦 ) =− 2
(𝑥+2𝑦)


Therefore 𝑈𝑥𝑦 = 𝑈𝑦𝑥

𝑥𝑦
(b) 𝑈 = 𝑒 𝑠𝑖𝑛𝑦