미적 공수 차시험 기출문 제(2), (2) 201 7 1
1. Find the limit , if it exists, and pr ove your s tatemen t using
!"#
definition of limits, or
show that the limit does not exist.
(a) lim
'()*+→'-)-+
.
/(
0
1*
0
(*cos*
(b) lim
'()*+→'-)-+
.
/(
0
1*
0
(*sin*
2. Find the equa tion of the tangen t plane to the surface
0(
0
"("/*
0
167-
at the point
'0)"8)"/+
.
3. Find the linear approximation of the function
9'(
)*+7
.
81(
0
1*
0
(*sin'("*+
at
'8)8+
and use it
to app rox ima te
9'8:8)-:;+:
4. Le t
<
'(
)*
)6+
be a differentiable function of three variables for which
∇
<
'
.
0
>
.
/)"
.
0
8)
>
.
/
+
7?8)0)"0@
.
Use the Chain Ru le to ev aluate
.
∂A
∂
<
at
'B)A)C+7
'
0)
.
D
E)"
.
D
E
+
.
(Use
(7BsinAcosC)*7BsinAsinC)67BcosA
.)
5. Su ppose that you are c limbin g a hill whose shap e is given b y the equation
678---"-:-8(
0
"-:-0*
0
, where
(
,
*
, and
6
are measured in meters, and you are standing
at a point with coordinates
'F-)G-)GHI+
.
The positive x-axis po ints east and the positive y-axis poin ts north.
(a) If you walk due south, will you start to ascend or d escend? At what r ate?
(b) In wh ich d irection is the slope largest? What is the ra te of ascend in that d irection?
6. Determine whether the statement is true or false. If it is true, explain why. If it is
false, explain why or g ive an e xample that disproves the statement.
(a) There exists a function
9J
K
0
→
K
with continuous second-order partial derivatives
such that
.
L(
L9 7*
and
.
L*
L9 7(
0
.
(b) Let
9'(
)*+7
M
N
O
P
P
.
(
0
1*
0
/(
0
*'()*+≠'-)-+
-'()*+ 7'-)-+
:
9
(
is continuous a t
'-)-+:
