Добавил:
ivanov666
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:Mathematics (Математика). Учебное пособие
.pdf
201
1
cos
sin
x
xy2
cos
=
x
2
=
dx
dx
x
x
-
=
=
y
в)
1
=
г)
2
д)
е) xxtgy 7sin5
ж) xxy 3sin2cos
= ,
з)
и)
y
+
=
x
,
+
,
2
)( xarctgxy += ,
,
22
2
)arcsin1( xy += ,
xx
-
1010
,
к)
л) )2(ln xctgy
м)
2
= .
xxeycossin +
,
)4ln(
xxy -=
,
2. To find the derivative
3. To find the derivative
Variant 9.
1. To find the derivative:
3 712
43 +-+=
а)
xxy
б) xxy sin)4( -= ,
x
e
y
=
в)
arctgx
г) )53sin(
д) tgxey
= ,
ln
=
y
е)
x 32-
sin
,
xy ,
x
1
-
x
,
dy
x
: 0
2
yd
:
2
1
4
10
2
xyln
=
,
32
=+- yxtgye
.
.
202
1
dx
dx
2
x
3
sin
x
+
xarctgy
ж)
+=
x
x
e
x
=
з)
и)
к)
л)
м) )1ln(
tgey
sin xy = ,
y ,
)1ln(3+=x
y =
,
4
1
-
x
22
)arcsin(lnxx
,
)ln(arcsin
2
-+= xxy .
2. To find the derivative
3. To find the derivative
Variant 10.
1. To find the derivative:
1
3
7 +++= x
а)
б) xxy sin)1(
в)
г)
д)
е)
xy
cos
y
=
=
y
= ,
arccos
=
2
3
+= ,
x
,
x
tgx
,
+
xxysin2+
xy1
,
x
ж) )2ln(
2
xtgy = ,
,
x
dy
arcsin=
:
y
2
yd
:
2
3
5
,
xy
.
2
1 xxy += .
203
з)
dx
-
=
dx
x
x
+
4
=
= ,
и) )(
к)
л)
м)
xxey+-
earctgy ,
22
sin xy = ,
3 2
2
exy .
2+=x
2-= xarctgy ,
44
+=x
2. To find the derivative
3. To find the derivative
Variant 11.
1. To find the derivative:
a)
6 57
5 +-+=
1
xxy
dy
:
2
yd
:
2
3
3
3
,
.
7cosln
xyy
2
1arcsin xxxy --= .
b) arctgxxy )1(
c)
3
+= ,
arcsin
y
=
x
5
,
xx
-
d) )14(
e)
arcsin
=
f)
g) 23
y ,
h) )1ln(cos
sin2sin xxxy ++= ,
xy1
2
)1(
+xarctg
+=
2
+= xy ,
i) xctgarсxy sin= ,
j)
k)
l) )(cos xtgy
y
2= ,
=
xysin
-
2xxee
+
.
2
+= xarcctgy ,
,
,
33
204
2. To find the derivative
dx
dx
x
x
x
-
1
+
x
+
dx
=
dx
3. To find the derivative
Variant 12.
1. To find the derivative:
a)
b) arctgxey
c)
d)
4 -+-=
x
= ,
ctgx
y
=
sin(
=
y
e)
f)
2
log
=
y
g) )arcsin(
h) )24(log
2
i) )13ln(
6
xy 2
×= ,
2
exy 5+= .
j)
k)
l)
7 39
,
3
x
2
x
3
2
xctgy )7(7
x
tgx
1
xxy
)
,
2)(x
earctgxy += ,
1
,
2
xtgy = ,
-+- xxy ,
2
+-= xy ,
+= ,
2. To find the derivative
dy
:
2
yd
: xxy 3sin=.
2
7
2
4
x
dy
: xyy
,
ln2 .
2244
yxyx =+ .
3. To find the derivative
2
yd
:
2
2
31 xy -= .
205
Variant 13.
x
x
e
sin
-
2
-
x
=
dx
dx
=
x
x
arcsin
1. To find the derivative:
3 85
79
a)
xxy
b) xxy cos= ,
arctgx
y
=
c)
x
,
3
4
52
-+-=
8
,
d)
=
xy sinlog
7
,
e) 2sin1 +-= xy ,
x
log
f)
y
g)
h)
i)
j)
k)
y 2= ,
l) xxy 2cossin
2= ,
=
arctgy
arctgx
ey = ,
3
=
xctgx
3
3
x
,
-
2
xxy lgarcsin ×=
,
2
xy-
,
.
2. To find the derivative
3. To find the derivative
Variant 14.
1. To find the derivative:
5 33
27 -++-=
a)
b) ctgxey
c)
x
= ,
3
y
=
x
xxy
12
-
,
dy
4
:
2
yd
: )ln( xctgy
2
3
43
8
,
yxyx +=+
.
.
206
d)
x
9
+
dx
dx
x
+
=
e)
f)
g)
h)
i)
=
=
=
5
2
sin=
4
arccos
xy
3
)2(log
xxy +=
,
23 2
)(sin xxy +=
,
xy sinlog
,
xyln
,
32 xey-
,
12
x
,
j) )1)(2(
=
k)
1
arctgy2
3xx
eey +-= ,
,
x
l) xxxxy )1(arcsin2
2. To find the derivative
3. To find the derivative
Variant 15.
1. To find the derivative:
54
4
35 -+--=
2
+= xy ,
xxy
,
,
a)
b) arcctgxey
c)
x
= ,
y
=
arcctgx
d) )15sin(
e) )sin21ln( xy
x
e
33
--= .
dy
:
2
yd
:
2
5
7
5
6
x
4
xy 3
= .
,
yy
exey sin1 ++= .
207
f) xarctgy ln=,
x
x
2
dx
+
=
dx
13
+
+
x
x
arcsin
=
g)
xy31
-
,
h)
i) xy
= ,
tgy
=
j)
k)
l)
sin= .
x
-
ln
4
1sin xy += ,
arccos3
x
,
-
x
ey
2
3
)121( xy +-= ,
dy
2. To find the derivative
3. To find the derivative
: arctgyxyx
2
yd
y
=
:
2
)sin( .
x
2
12x
+
.
Topic 11. Using derivatives in curve tracing
Variant 1.
1. To find the largest and smallest value of the function on the interval:
6
x
=
y
2
[ ]
5;5;
-
.
2. Draw the graph of the function:
Variant 2.
1. To find the largest and smallest value of the function on the inter-
val:
x
y +=
2. Draw the graph of the function:
[ ]
p
;0;cos2x
.
y
( )
x
3
xy16
+
= .
.
2
1-
x
=
208
Variant 3.
16
+
-
x
41x
7
+
+
x
x
x
2
-
-
16
x
x
-
1. To find the largest and smallest value of the function on the inter-
3
val:
2. Draw the graph of the function:
Variant 4.
1. To find the largest and smallest value of the function on the inter-
val:
2. Draw the graph of the function:
Variant 5.
1.
val:
2. Draw the graph of the function:
Variant 6.
1. To find the largest and smallest value of the function on the inter-
val:
2. Draw the graph of the function:
x
=
y
2
x
=
y
2
To find the largest and smallest value of the function on the inter-
x
y
x
=
y
2
+
3
[ ]
-
[ ]
-
;sin2x
[ ]
-
.
10;5;
3
x
-
y
=
.
7;3;
y
= .
p
é
--=
ê
ë
ù
p
;
.
ú
2
û
=
y
.
7;3;
= .
2
x
2
3
x
x
2
xy1
+
.
1
.
2
)1(2 +
Variant 7.
1. To find the largest and smallest value of the function on the inter-
val:
x
y
[ ]
pp
;;cos
--= x
.
2
2. Draw the graph of the function:
y
( )
1
-
x
.
2
12
x
=
209
Variant 8.
-
1
x
2
12x
-
x
-
x
1. To find the largest and smallest value of the function on the inter-
4
val:
x
=
y
x
[ ]
2
6
+
.
6;4;
-
2. Draw the graph of the function:
2
4
x
=
y .
3
-
Variant 9.
1. To find the largest and smallest value of the function on the inter-
val:
2. Draw the graph of the function:
3
[ ]
--= xxy
.
pp
;;cos
x
3 x
+
.
2
y
=
Variant 10.
1. To find the largest and smallest value of the function on the inter-
x
val:
cos
xy
;
2
2. Draw the graph of the function:
pp
é
--=
ê
ë
ù
.
ú
2;2
û
x
y
=
.
2
+
Variant 11.
1. To find the largest and smallest value of the function on the inter-
val:
2. Draw the graph of the function:
23
313 xxxy --+= ; ]2;1[
.
3
xy3
+
=
.
Variant 12
1. To find the largest and smallest value of the function on the inter-
val: 3
2. Draw the graph of the function:
2+=- x
exy ; ]4;1[
.
2
xy4
+
= .
210
Variant 13.
352
1
x
1
x
2
5
+
x
2
4
-
1. To find the largest and smallest value of the function on the inter-
val:
1
35
xxy -+= ; ]2;0[ .
2. Draw the graph of the function:
2
=
y .
32
+-
xx
-
Variant 14.
1. To find the largest and smallest value of the function on the inter-
2
val: 2
x
=
y ; ]4;
-
1
[- .
+
2. Draw the graph of the function:
Variant 15.
1. To find the largest and smallest value of the function on the inter-
1
val:
3. Draw the graph of the function:
2
ln2
+-= xxy ; ]1;
1
[ .
y .
x
=
2
82
x
=
y
x
.
3
)3(
-
Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]