MATH 54 5TH EXAM EXERCISES 1.

Determine the domain of the following functions.  a. ,  =



2.



c.

4 =  cos 

d.

, ,  =  cos    â&#x2C6;&#x2019; 1

d.

3=

1.05,2.1, approximate the change in . 12. Use differentials to estimate the amount of tin in a closed tin can with diameter 8cm and height 12cm if the tin is 0.04cm thick.

   , â&#x2020;&#x2019;!,!  % +  %

13. Use differentials to estimate the amount of metal in a closed

Determine the type of discontinuity at 0,0.

a. ,  =

d.

â&#x201E;&#x17D;,  = ,  = ),  =

(

* #

,

,  = (  +   ; 



d.

0, 1, 2, 3, 4, 5 = 30  12 + 12  4 â&#x2C6;&#x2019; 2234  â&#x2C6;&#x2019;

+ 4 + 9 ;/ 

Find the partial derivatives by holding all but one of the variables constant and applying ordinary differentiation techniques. a. 0, 9, : = 40  sin 9 + 5< = cos 9 sin : â&#x2C6;&#x2019; 2 cos : ;

c. d. e.

8. 9.

A3 tan  5 ; A5

0, 9 =

0  cos 9

â&#x2C6;&#x2019; 20 tan 9 ;B

%



3 = < ;  = tan  012 ;  = ln301 +

,  = 2 â&#x2C6;&#x2019; 3  +  ; ,  3

A 5 AAA

5 =    +    +   ;

g.

, ,  = ln cos3 â&#x2C6;&#x2019; 4 ;CC8 , 8C8

h.

), ,  = sin ;) , )% , )%

Find an equation of the tangent plane to the given surface at the specified point. a.  =   +  + 3  ;1,1,5  = ( â&#x2C6;&#x2019; ;5,1,2

Find the linearization (or equation of the linear approximation) of the following at the given point. a. ,  = (;1,4

b.

,  = ;6,3

c.

,  = tan   + 2 ;1,0

Approximate 1.95,1.08 given that ,  = (20 â&#x2C6;&#x2019;   â&#x2C6;&#x2019; 7  . Approximate 6.9,2.06 given that ,  = ln â&#x2C6;&#x2019; 3.

H

512 ;

A3 A3 A3 , , A0 A1 A2 A3 A3 , A2 A1

c.

3 = sin  ;  = 2< G ;  = 2  <  ;

d.

J = K  ;  = cos  sin 2 ;  =   < G ;

,L ,

15. Find the indicated (total or partial) derivative.

b. c. d.

3 = ln  +   ;  = < G ;  = < G ; 3=

G N O

N P

;  = 3 sin 2 ;  = ln 2 ; 



3 = ln +  + 2  % +  % = 8; %

%



MG

MC MG

;  = 2 sin 2 ;  = cos 2

M M

2  + 3 = 5;

f.

cos +  =  sin  ;  

MC

M

e.

M M

M

A A +    sin  ; ,

g.

=

h.

<  + 2< â&#x2C6;&#x2019; 4< = 3;



f.

b. 7.

, ,  = < sinh 2 â&#x2C6;&#x2019; < cosh 2 ; 3=

b.

a.

245 + 335  ;8

A3 A0

3 =   ;  = ;  = 0< H ;  = 0< H ;



, ,  =

=

a.

,,

c.

b.

metal in the top and bottom is 0.1cm thick and the metal in

14. Find the indicated partial derivatives.

 %

cylindrical can that is 10cm high and 4cm in diameter if the

the sides is 0.05cm thick.

Apply the definition to find the partial derivative.  ,a. ,  = ; b.

= H G

11. If  = 5  + ^2 and ,  changes from 1,2 to

lim

c.

6.

5 = ln (  +   +  

,  = ln4 â&#x2C6;&#x2019;   â&#x2C6;&#x2019;    + 

b.

5.

b.

c.



4.

3 = < G sin 9

, ,  =

 # ". lim  , â&#x2020;&#x2019;!,!  # +  #

3.

a.

b.

Determine the existence of limits.   . lim  , â&#x2020;&#x2019;!,!   +  

\$.

10. Find the total differential of the following functions.

A

A A A ,  A A

-EAArances

Math 54 exercises 5th le