Math 21A
Vogler
Worksheet 8
1.) Cons ider the func tion f (x) = x
3
Mean
Value Theorem ( MVT ) on the interval [0, 1], and determine all values of c guaranteed
by
the
/ 2 M+VT . X
1
1.)
/2.)Cons
ider
the func
Ex p la
in why
f (x)tion
= Xf (x) = x
32
Mean
Value
Theorem
(
MVT
) on the interval [0, 1], and determine all values of c guaranteed
3 —
the
intervXal [ - 1 ,
by
the
M
VT
.
/ .1 2 S + h X o
w
1t/
h
a
t
if 0 < x < 3
x
/2.)
f3 Ex d
p la in
o why
e sf (x) = X
3.) Cons
func
2
ao Xider
tal t [ -i the
fi tione f (x)
s = { 2X
3sn —
the
interv
1, s
,3
.1ts aS
ef
a.) Sketch
t hh i o sthewgraph
of f
,
t/ a
s as
tu
m
p
y hb.)
S h o w that f satisfies the cxonditions
Value Theorem ( M VT ) over
if 0o<f the
x < Mean
3
i
f3ta
i loo e nls s
d
the inter
v al [-1, 3], inc luding special2Xfattention a t x 0 , and determine all values o f c
3.)
Cons
sno
ao ider
t t fi thes func
fi tione f (x)
s = {
o
guaranteed
by the MVT .
,3—
tsft aa.) Sketch
ef
t hh i sthe graph
of f
,1
a
s
s
hu
m
p
yte b.)
S
h
o
w
that
f
satisfies
the
c
onditions
f the Mean
Value
Theorem
( M VTdecreas) over
i < tion ando where
Deter
of each func
each func
tion
is increasing,
ta4.)
e
i s l omine
n the sdomain
a inter
m luding special fattention
the
v als [-1,l u3], inc
a
t
x
0
,
and
determine
all
values
of c
x
ing, concave up, and concave down. Identify
all relative and absolute extrema, infl ec tion
o
o
p
t
i
f
o
n
guaranteed by the MVT .
—
<
x - and y-intercepts, and asymptotes
(vertical, horiz ontal, or tilted) . Sk etc h the
ft points,
s
h
1
0
tegraph.
h
o
< tion and where each func tion is increasing, decreas4.)
Deter
mine
the domain of each func
e
af
sa.) f s(x) =ux(5 —
m x)
x
ing, concave up, and concave down. Identify
all relative and absolute extrema, infl ec tion
pt
t4 i
o
n
<
points, x - and y-intercepts, and asymptotes
(vertical, horiz ontal, or tilted) . Sk etc h the
sh
0
graph.b.) y = x2 + 4
oe
f M a.) f (x) = x(5 — x )
t V 4c.) g( x ) =
hT
b.)
d.) y( =
x )x2= +x4
eo
2
Mn
e.)
f (xx) ) == O S i n X + COSX o n the interval [ 0 , 27r]
c.)
/ g(
V
3
+
T
x hemisphere
( x ) = x sits on top of a r ight circular cylinder
o5.) Ad.)
10 and radius r . Wr it e the volume V of the
2o
n
nof height
e.)
f
(
x
= aOfunc
S i n tion
X + COS
X ovolume
n the interval
, 27r]
hemisphere
of the
C of the[ 0cylinder.
/t
h ) as
3e +
5.) Axihemisphere
sits on top of a r ight circular cylinder
n t
of height
10
and
radius
r . Wr it e the volume V of the
oe rn v
hemisphere
t a hl as a func tion of the volume C of the cylinder.
e[
i1 n t
e, r v
a l
[
1
1
,
A
X 2
+
A
X 2
+
1
Thanks to Dr. Kouba