Math 21A
Vogler
1.) Find the slope and concavity of the graph of x y
Discussion
Sheet 6of
Sketch
2
+ theygraph
= this equation.
2
a
t
x
0
a
nthe slope
1.)
2.) Find
Evaluate
thed following
and concavity
limits by
of the
using
graph
one of the
x y three limit definitions for e.
t
Sketch
of
2a + theygraph
= this equation.
2
a
t
xx
0
a. ) lni cmc ( 1 n+ - ) 4n
+
o
o
n
- co ( 5 1n
1
a
2.) Evaluate
2 )n
thed following limits by using one of the three limit definitions for e.
n
bt .
)
a
l
i
n
)( n + 4 )
x
d.)
e.) l i m
. ) l Tii m
( nn- 5n) h - H
f.)- l 3
icom ( (51 1+n 3h) 4 n
n c- 4c00 ( 1- n+ n- +
o
o
1 a.
(
1
2 )
2 n
,
nb
. n)
)
3.) Solve
li for
n x.() n + 4 )
3
n
d.)
e.)
l
i
m
.
(2nn- 5) h - H
f.) l 3
i m ( 1 + 3h
Ti - 4 00
n
()a.) l n x
1 = -3
b.)
l
n
x
=
ln
3
c
.
)
/
ln(2x
+
1)
ln(x
+ 3) = 0
2 n
-C
n ,
)h
-.
3.) Solve
for
x.
d.)
l
n
(x
1)
+
ln(x
2)
=
0
e
.
)
l
n (x - 2) + ln(x + 2) - ln x = ln 3
)
3
n
2
l)
a.)
lnx = 3
b.) l n x = ln 3 c . ) / ln(2x + 1) - ln(x + 3) = 0
Ci
4.) L.Me t f (x) = x
h
each.
3 i d.)
)1n l n (x
x .- 1) + ln(x - 2) = 0 e . ) l n (x - 2) + ln(x + 2) - ln x = ln 3
+
S ol— l v e
i
f5.) LFind
4.)
M
e t f y(x)= =—x
dx
/3dy i 1a
each.
n s x .
(s io+—m
x l pv )l ey
S
•
4
=
a.)
y
=
ln(5x+7)
b
.
)
y
=
ln(ln(ln(sin
x)))
c
.
)
y
=
ln
s
i
n
(
3
x
)
•
(x - 2)
f5.)
a Find
s y = —
(x x +1)
dx
0
/dy
p oa ss s
5 •
(sa
x l ypne=)l xy.
i ib d.)
m
d
1 y = ln(5x+7) b . ) y = ln(ln(ln(sin x))) c . ) y = ln sci on •(s 3 x ) •4(x - 2)
=
a.)
D
af
( 2x +x1)
f.)
(x
ns" y = (2x)x+
0
o
p( o3
s
s
5) •
7 xyng= .x )
a
d.)
n
i) b ye l . e .
c• o s
6.)
for x.
d
1
o Solve
D
(t 2 ax
) " y = (2x)x+
fo=
n=
t f.)
b.) 7 • e
)n
yix ex
3xa.)
gY =. 2)
(n0
s 7
2
x
±
3
•1 3
=
ye2
)ofm Solve
6.)
=
d p . for x.
.
tx a
o =h
x.i
e
=
f
tlr )a.)
) exY = 2
n
b3. ) 7 • e
0
y xy3
)i
-x ± 3
s7.)
2
y
1 3
xy Find
•e y/ = —
2
fm
=
dx =x
d
p
x
dy
a h=
(D o .
o
eC
lno ox.2
ix t f
.
2'
+
4x
ryu )3
x y = 7'• e 3-)
a.)
d.) y
)
3X + 5X
i y•(-4 m y/p= l—
7.)
xysd
(
r Find
5
dx xe
Y
isa f e
1
y no
=
dy
a
(
D
C
•
x
+
1
oye 2s) o l
n
.x
2' + 4x
nus oxx4
t
e w
+
a.)
y
=
7'•
e
)
d.) y
ut i (-x5
rr p l
1
3X + 5X
sd
m
x
(
b
.
)
rae
4
5)•n
s
e)
isau
1
s f Y
=
yyy+n.
•
x
1
x(
w
p )s(0eo l r n
ye
e
=
s 4
e2
+
=. r w
utesa x5
1x
e
xlx r .o ) g
b
s
)
)au
•=X
.yn . s2
si
(5
+e
w
p y(5
r
9.) L e t f (x) = x
each.
2 • e
d
x
10.) Find —
S o l vdx ed x 2
dy
2
y
(
x
X = t
a
n
d
8.)
Find
all
horizontal
asymptotes for the function in 7.)c.).
a.)
)
w h t y2+ t
=
e nt f 2(x)
tx = sin t
9.)
= x
0 L eb.)
teach.
2a • e n y = cos 2t, f o r 0 < t < 2ir
t ,
-d1
f
d
xff11.) Differentiate.
"
10.)
Findo—
o
dx
d
x
2
S
(dy o xl r v e
r(
b.) y = x • arcsin(e
y
x y X=2 =arctan(3x)
)a n a.)
t
d
0
e a.)
2
arccos x - arctan x
)=
2 t
t
y
w
h
<
a
d.)
y
=
c
o
s
x)
c
.
)
+
arctan x - aresin x
=
0
ec n2 2t tx = sin t
y
0f
b.) -< t a n
t
cos 2t, f o r 0 =< t < 2ir
aoh (I ( antr yc, =
X
xf.)y=
rcsin+ 1 -1 a r c c o s
a
1s
o
x )
dr
fe )3+ fo2
( s i n
f11.)
"
xo Differentiate.
a
t(
) x r o. c+ +t +a+ +n + + + + + x+ + + + + + + + + + + + + + + + + + + + +
a
fie yr =. arctan(3x)
r
b.)
a.)
2 y = x • arcsin(e
)no
(
S
0
i
m
p
following
problem
is
for
eThe
2) recreational purposes
f
)
arccos x -only.
arctan x
=
d
lya yi < =f c yo s
ap d.)
x) c . ) arctan x - aresin x
0s
s pattern
w
c
2aa t na hidden
y determine the next number in the sequence
and
fer12.) Find
e
<
r
.
)
h ( (ea r c t a n
=
otm I0,1,3,7,14,25,
x41,63,•
rcsin+ 1 -1 •a• r c c o s
a
f.)y=
st
xr ) o X
ru ) + o2
e 3c
( s i n
xpi
+
+
+
+
+
+
+
+
+
+
+++++++++++++++++++++
a
r
c
t
a
n
.
te )
x
a
oq fie
.
2
nsThe
() S i problem
m p is for) recreational purposes only.
fu following
di
p lya i f y
sga a
s pattern
w
r12.) Find
ai na hidden
and determine the next number in the sequence
ent
e
r
.
)
m
e
tco 0,1,3,7,14,25, 41,63,• • •
tn
r
uh
i
c
pas
e
ar.
q
st
u
if
a
go
t
i
nr
o
c
n
2
h
s
a
.
r
t
f
o
r
2
Thanks to Dr. Kouba