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Math 206-05 Final Review Ch4 (7365771)
Question
Cifi
1.
1 2 3 4 5 6
14 15:16 17 18 19 20 21
lkaJ VaLLied
W/2/2
.. A<-6)
A /(0--0 anDe _tdaVe-
Question Details
_
/LI*
SCalcET7 4.1.035.MI. [1640886]
Find the critical numbers of the function. (Enter your answers as a comma-separated list. If an answer does not exist,
„, 2
It , If
/ (1) —
1+ 3
g(y) :17
(
_y
Y -. 3,yi 3 - q-/)( y_L--_3)
3p 3)
(y -1 4yi- 3)
T / 1 3/ t 3 > 0 /pc/ &IR
y( ay)
enter DNE.)
-
,
2
-
2.
1/ 3
- -
0
.-_- 0
2
Question Details
SCalcET7 4.1.037.MI.SA. [2104375]
This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive
any points for the skipped part, and you will not be able to come back to the skipped part.
Find the critical numbers of the function.
h(t)
3.
-
0/4 — 2t1/4
h?b) =
C — 3 .6 c.
••■•
SCalcET7 4.1.049.MI.SA. [1886701]
Question Details
This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive
any points for the skipped part, and you will not be able to come back to the skipped part.
Find the absolute maximum and absolute minimum values of f on the given interval.
f(x) = 6x3 — 9x2 — 108x + 2,
f i( y )=- 1 8- x'-(kx
-(-
[-3, 4]
08- ---. (g( y 2:--x -6) =- (S-(K - 3)0(-1- ))
Crif--i,a_ ai,efAzt ,, 3 —2_
4.
(
Question
T3)—
Details S
3
(C- ) 7 3r f())_)41 (<)= H
[ 1640951]
0
' I
Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c
that satisfy the conclusion of Rolle's Theorem. (Enter your answers as a comma-separated list.)
f(x) n
c=
— -97x,
[0, 81]
Q) 6es o,,
to,ed
AIM yen tica 12. cn Lf9, 21)
Fitel C
9)
s,#-ft(0-0.
.fz)(
75- Yz=
x _L
x
Q
a
httnliwww wphasinn nptiv4rnivanniaplarkontrnlnl
1/7
5/30/2015
5.
Assignment Previewer
Question Details
SCalcET7 4.2.005. [1771966]
_
Consider the following function.
f(x)
16 —
x2/3
Find f(-64) and f(64).
r—
f( -64 ) =
0
f(64) =
Find all values c in (-64, 64) such that f '(c) = 0. (Enter your answers as a comma-separated list. If an answer does not
exist, enter DNE.)
E
c=
Based off of this information, what conclusions can be made about Rolle's Theorem?
This contradicts Rolle's Theorem, since f is differentiable, f(-64) = f(64), and f '(c) = 0 exists, but c is not in
(-64, 64).
This does not contradict Rolle's Theorem, since f'(0) = 0, and 0 is in the interval (-64, 64).
This contradicts Rolle's Theorem, since f(-64) = f(64), there should exist a number c in (-64, 64) such that
f '(c)
0.
This does not contradict Rolle's Theorem, since f'(0) does not exist, and so f is not differentiable on (-64, 64).
,.::> Nothing can be concluded.
6.
SCaIcET7 4.2.009. [1771985]
Question Details
_
Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval?
f(x) = 3x2 — 2x + 1,
[0, 2]
Yes, it does not matter if f is continuous or differentiable, every function satifies the Mean Value Theorem.
V
Yes, f is continuous on [0, 2] and differentiable on (0, 2) since polynomials are continuous and differentiable on
No, f is not continuous on [0, 2].
No, f is continuous on [0, 2] but not differentiable on (0, 2).
There is not enough information to verify if this function satifies the Mean Value Theorem.
If it satisfies the hypotheses, find all numbers c that satisfy the conclusion of the Mean Value Theorem. (Enter your
answers as a comma-separated list. If it does not satisify the hypotheses, enter DNE).
"h 1 fra)
c=
) --
b —0.
Cc _
f())-
AD)
-0
_ 44-f- =
1
7C6? ) =
Miry 1 hin•ou, taLmi,,nnirers rus+11 nnis
(71
. es1 nr../resretr.-.1 M
717
Assignment Previewer
5/30/2015
7.
SCalcET7 4.2.015. [1771956]
Question Details
Let f(x) = (x - 3) -2 . Find all values of c in (1, 7) such that f(7) - f(1) = f'(c)(7 - 1). (Enter your answers as a commaseparated list. If an answer does not exist, enter DNE.)
pme
c=
Based off of this information, what conclusions can be made about the Mean Value Theorem?
This contradicts the Mean Value Theorem since f satisfies the hypotheses on the given interval but there does
1)
not exist any c on (1, 7) such that
f'(c) = f(7
7 -
V
This does not contradict the Mean Value Theorem since f is not continuou
at x =
This does not contradict the Mean Value Theorem since f is continuous on (1, 7), and there exists a c on (1, 7)
such that
f'(c) - f(7 - 1)
7-1
This contradicts the Mean Value Theorem since there exists a c on (1, 7) such that
f'(c) =
7-1
but f is
not continuous at x = 3.
Nothing can be concluded.
(0
CK-3)=2 {Yx)= - .2 0(-1,) --- 3 ,
)6
8.
Question Details
4/4
6.1)
-L
(C ?:9
-fio =
C160
?*
(C-3 )
SCalcET7 4.2.504.XP. [1763199]
Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval?
f(x) = x 3 x - 5,
[0, 2]
No, f is continuous on [0, 2] but not differentiable on (0, 2).
0 No, f is not continuous on [0, 2].
Yes, it does not matter if f is continuous or differentiable; every function satisfies the Mean Value Theorem.
) Yes, f is continuous on [0, 2] and differentiable on (0, 2) since polynomials are continuous and differentiable on
R.
There is not enough information to verify if this function satisfies the Mean Value Theorem.
If it satisfies the hypotheses, find all numbers c that satisfy the conclusion of the Mean Value Theorem. (Enter your
answers as a comma-separated list. If it does not satisfy the hypotheses, enter DNE).
-s-F(0)
F(0)-f((c)= 3c2÷c,
c=
a -0
4
3c L
F6-107 VAIAA-t-
744Pce
AO= b) -1(o)
f
6-
07
3/7
5/30/2015
9.
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Question Details
SCalcET7 4.2.AE.005. [2045596]
Video Example4 •
EXAMPLE 5
Suppose that f(0) = —5 and f'(x) 5 9 for all values of x.
How large can f(3) possibly be?
SOLUTION We are given that f is differentiable (and therefore
continuous) everywhere. In particular, we can apply the Mean Value
Theorem on the interval [0, 3] . There exists a number c such that
f(3)
—
f(0) = f'(c)(11
—
0)
SO
f(3) = f(0) +
We are giv
that f'(x) 5 9 for all x, so in particular we know that
f'(c) 5
J. Multiplying both sides of this inequality by 3, we have
3f '(c) 5.
so
f(3) —5 +
'(c)
5
—5 +
The largest possible value for f(3) is
10.
'(c).
'(c) = —5 +
.
Question Details
SCalcET7 4.3.008. [1774014]
The graph of the first derivative f' of a function f is shown. (Assume the function is defined only for 0
x 5_ 9.)
y =fix)
(a) On what interval(s) is f increasing? (Enter your answer using interval notation.)
(0?.14111()( 6, q)
(b) At what value(s) of x does f have a local maximum? (Enter your answers as a comma-separated list.)
x
=
4
At what value(s) of x does f have a local minimum? (Enter your answers as a comma-separated list.)
/
x =' 0
(c) On what interval(s) is f concave upward? (Enter your answer using interval notation.)
(//32 UCC,?)u(,
On what interval(s) is f concave downward? (Enter your answer using interval notation.)
C o,
u( its)
(d) What are the x-coordinate(s) of the inflection point of f? (Enter your answers as a comma-separated list.)
Assignment Previewer
5/30/2015
11.
Question Details
SCaIcET7 4.3.056.MI. [1647513]
Z
f(x) =
.)C e
"fi )
Consider the function below.
x2e-x
(a) Find the exact value of the minimum of
o
f( x) = I
ze- z (2-z)
c
iv( x
f for x
e-tt loo-= 0
Find the exact value of the maximum of
f for x >
xe- z
0.
lumie z=0
(b) Find the exact value of
x
at whit
rov.1
j---
x
_
C ) 0 exx.pc
st rapidly.
cnti4.4-e
11
m ie., 0
fx./ -4
'( x )41"
c2
f
P
0 ,qz
e2
1
Ih
27I-a
)::-
X=
12.
Question Details
3‘f)
.7(
0?+4--2 corres
SCaIcET7 4.4.025. [1640900]
Find the limitbse ('Hospital's Rule if appropriate. f there is a more elementary method, consider using it.
Iim
x-+ 0
e3x
-
/—
1 - 3x
2̀—
,
x2
L
3 e,
)
e),(
1
Question Details
SCaIcET7 4.4.027. [1641048]
Find the limit. Use ('Hospital's Rule if appropriate. If there is a more elementary method, consider using it.
um tan x
x-)0 tanh x
14.
SCaIcET7 4.4.049. [1640949]
Question Details
Find the limit. Use ('Hospital's Rule if appropriate. If there is a more elementary method, consider using it.
lirn
x-rl
9x
x -1
In%)
SCaIcET7 4.4.056. [1888880]
Question Details
_
Find the limit Use ('Hospital's Rule if appropriate. If there is a more elementary method, consider using it.
lim (tan 64x
X-40+
ol
5/7
5130/2015
Assignment Previewer
Question Details
SCaIcET7 4.4.057.M1. [1641217]
Find the limit. Use ('Hospital's Rule if appropriate. If there is a more elementary method, consider using it.
I'm (1 — 2x) 1ix
x-0 0
Question Details
SCaIcET7 4.4.061. [1640888]
Find the limit. Use ('Hospital's Rule if appropriate. If there is a more elementary method, consider using it.
lim x 5/x
x -,
18.
SCaIcET7 4.7.007. MI. [1641693]
Question Details
Find the dimensions of a rectangle with perimeter 92 m whose
number, enter it into both blanks.)
23
(
3
4-‘›e
m (larger value)
22)
Question Details
possible. (If both values are the same
A= x(46-X)
7C
m (smaller value)
14'
19.
a large
fi.h4_
't"
rx4,
it
tcY)
mte.
— te4. 6
,(x)
,
_
f
160
A
T7
S;IcEx
Lie. 0 =
4.7.010 [1641739]
The rate (in mg carbon/m 3/h) at which photosynthesis takes place for a species of phytoplankton is modele
We eve
function
1001
p(r)z
0
P
•Z
Y the
prc
2
,) ,_
dz.+
100-er:2,-
4-)
where I is the light intensity (measured in thousands of foot-candles). For mat liyht intensity is P a maximum?
/
thousand foot-candles
iGro
I-0-
-1- t 4) 1
2
( L
SCaIcET7 4.8.007. M1. [1642315]
Question Details
Use Newton's method with the specified initial approximation x 1 to find x3 , the third approximation to the root: of the given
equation. (Round your answer to four decimal places.)
x 5 x - 1 = 0, x 1 = 1
X3 =
21.
1
SCaIcET7 4.9.003. [2559099]
Question Details
Find the most general antiderivative of the function. (Check your answer by differentiation. Use C for the constant of the
antiderivative.)
f(x)
5
+ 2 x2
—
3
A
±
a
x3
S•-•
6
1(x) =
C
I
Assignment Details
Name (AID): Math 206 - 05 Final Review Ch4 (7365771)
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