Exercises before mid advanced calculus

Exercises:
1. Indicate domain definition of each function below graphically

(

x 2 + y 2 − 16

a.

ln

b.

x
x + y2 − 4

)

2

2. Investigate the value of A such that f continuous at ( 0,0)

x2 − y 2

f ( x, y ) =

x2 + y 2
A,

,

( x , y )≠ ( 0,0)
( x , y )= ( 0,0 )

dU
if x5 + y = t and x 2 + y 3 = t 2 .
dt
2
4. If z = f ( x, y ) = x y − 3 y , then

3. If U = x 3 y , find

a. find dz
b. determine dz if x = 4, y = 3, ∆x = −0, 01, ∆y = 0, 02
c. how might you determine f ( 5, 21;6,85 ) without direct computation?
2
3
5. Find the directional derivative of f ( x, y, z ) = x yz in the direction - i + 2 j + k at the point

(1,1,-1)
3

3

6. Find critical points of function f ( x, y ) = x + y − 3x − 12 y + 20 , then determine as max or
min point.
7. Find

the

maxima

and

minima

x + y + z = 6, x > 0, y > 0, z > 0

of

f ( x, y, z ) = xy 2 z 3

subject

to

the

conditions