EUM113/3_Sem1(2014/15) CALCULUS OF MULTIVARIABLE

EUM113/3_Sem1(2014/15)

CALCULUS OF MULTIVARIABLE

Tutorial 4 1.

Find the domain and range of (i) 𝑓𝑓(𝑥𝑥, 𝑦𝑦) = �𝑦𝑦 − 𝑥𝑥 2 (ii) 𝑓𝑓(𝑥𝑥, 𝑦𝑦) = �1 − 𝑥𝑥 2 − 𝑦𝑦 2 1 (iii) 𝑓𝑓(𝑥𝑥, 𝑦𝑦) = 𝑓𝑓(𝑥𝑥, 𝑦𝑦) =

(iv) 2.

√𝑥𝑥−𝑦𝑦 𝑥𝑥 �𝑦𝑦

Sketch the level curve of 𝑧𝑧 = 𝑘𝑘 for the specified values of 𝑘𝑘. 𝑧𝑧 = 𝑥𝑥 2 + 𝑦𝑦 2 , 𝑧𝑧 = 𝑥𝑥 2 + 𝑦𝑦,

(i) (ii)

𝑘𝑘 = 0, 1, 2, 3, 4 𝑘𝑘 = −2, −1, 0, 1, 2

3.

Sketch the graph of 𝑓𝑓(𝑥𝑥, 𝑦𝑦) = 4 − 𝑥𝑥 2 − 𝑦𝑦 2

4.

Use limit laws and continuity properties to evaluate the limit lim

7𝑥𝑥 − 8𝑦𝑦 =0 −8

(𝑥𝑥,𝑦𝑦)→(0,0) sin 𝑦𝑦

5.

Determine whether 𝑓𝑓 (𝑥𝑥, 𝑦𝑦) has a removable discontinuity at (0,0).

𝑥𝑥 2 + 4𝑦𝑦 2 𝑖𝑖𝑖𝑖 (𝑥𝑥, 𝑦𝑦) ≠ (0,0) 𝑓𝑓 (𝑥𝑥, 𝑦𝑦) = � −4 𝑖𝑖𝑖𝑖 (𝑥𝑥, 𝑦𝑦) = (0,0)

6.

Determine whether the following limit exists. If so, find its value. 𝑥𝑥 4 − 4𝑦𝑦 4 (𝑥𝑥,𝑦𝑦)→(0,0) 𝑥𝑥 2 + 2𝑦𝑦 2 lim

7.

A

function 𝑓𝑓 (𝑥𝑥, 𝑦𝑦) is said to have a removable discontinuity at (𝑥𝑥0 , 𝑦𝑦0 ) if 𝑓𝑓(𝑥𝑥, 𝑦𝑦) exists but 𝑓𝑓 is not continuous at (𝑥𝑥0 , 𝑦𝑦0 ) , either because 𝑓𝑓 is not defined at lim

(𝑥𝑥,𝑦𝑦)→(𝑥𝑥0 ,𝑦𝑦0 )

(𝑥𝑥0 , 𝑦𝑦0 ) or because 𝑓𝑓(𝑥𝑥0 , 𝑦𝑦0 ) differs from the value of the limit.

Determine whether 𝑓𝑓 (𝑥𝑥, 𝑦𝑦) has a removable discontinuity at (0,0). 6𝑦𝑦 2 𝑓𝑓 (𝑥𝑥, 𝑦𝑦) = 2 𝑥𝑥 − 𝑦𝑦 2

EUM113/3_Sem1(2014/15)

CALCULUS OF MULTIVARIABLE

lim

8.

Show that the following limit does not exist:

9.

Find all points where the function is continuous: 𝑥𝑥 2 𝑦𝑦 𝑖𝑖𝑖𝑖 (𝑥𝑥, 𝑦𝑦) ≠ (0,0) 𝑓𝑓 (𝑥𝑥, 𝑦𝑦) = �𝑥𝑥 2 +𝑦𝑦 2 0 𝑖𝑖𝑖𝑖 (𝑥𝑥, 𝑦𝑦) = (0,0)

𝑥𝑥𝑥𝑥

(𝑥𝑥,𝑦𝑦)→(0,0) 𝑥𝑥 2 +𝑦𝑦 2