MATH 54 5TH EXAM EXERCISES 1.
Determine the domain of the following functions. a. , =
2.
c.
4 = cos
d.
, , = cos − 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.
, → !,! % + %
13. Use differentials to estimate the amount of metal in a closed
Determine the type of discontinuity at 0,0 .
a. , =
d.
â„Ž , = , = ) , =
(
* #
,
, = ( + ;
d.
0, 1, 2, 3, 4, 5 = 30 12 + 12 4 − 2234 −
+ 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