### Solutions Midterm 2 Spring 2014

```MATH
205 EXAM 2
7:15-8:15 pm,
Thursday
Variant A
Ma.rch, 13, 2014
Nam
Instructor:
Section
time:-
No books or notes a,re a,llowed. Simple calculators are OK, but no smartphones.
Circle your final answer.
Show all work to get full credit. Arguments must be clear.
K-state honor code: "On mg honor, as a student, I haue neither giuen nor receiued
uno,a,th,ori,zed, aid, on this acttd,emic work."
Equation of the tangent
line: g:
ft(a)(x
-
a) +
f(a)
(f(g("))'-- f'[email protected]'@)
U
@) g
("))'
:
ffi):
Good luck!
f' (") g(") + f (r) s' (r)
f'@)[email protected]) - f @)[email protected])
([email protected]))'
SCORE
Total:100
#L
#2 #3
Max
10
10
#4 #5 #6 #7 #8
10
16
18
T4
10
t2
1 (10 pts). Find the equation of the tangent line to the graph of the function
Jlr) : c" at r: -L
<'l -- (t/a)
4
fx- a ) + (ra'\
ct
=-l
{ to't = 5-' = L,
f- s)'
l'/a..' € '/-,) = e^ a'tt
€ '(xt
-
0^
u4nsoue-,
pts).
On the same axis sketch carefullg the graph of the derivative of the
function given by the following graph.
(1O
o
J
,7
/
' th 20-: J-
\
-v\
\
\
3 (16 pts). SHOW ALL THE WORK, GIVE FULL ARGUMENTS.
For the ftrnction f (x)
:
2ra
-
8x
a) Find a.ll critical points.
( ,= (2yt-gx)t= 8x3^8
x3- I =o
x7= I
F4
b) Identify each critical point as a local maximum, a local minimum or neither.
(*1'= zv xZ
|"U,)=e.V>o
BA zn"l .(o.-;ro#'uc t<a*, itW
|
"
c) Circle what test did you use:
d) Find all inflection
1st derivative test
points. N
'l (
,t"d dd":I't".tilt}-\
*l '= o
ZYxL=A
{ "r*)
- /iz\ mn\
x=o
c-laa'-'Z<
< r?.".
efaTlt^ poin*3
(10 pts). Suppose that
Estimate /(30).
/(l)
is afunction with /(25)
:4
and t'QS)
f,(ro)= €ces) + €'lzs)'(<o2s)
tt
-
o-l
.t =[A
=
:
-9.1.
(18
pts).
Find the derivatives of the following functions. Si,mphfu the answer
and ci.rcle
(5-1)
it!
(,): ,4 = x-{
f
Er=:;3
L------'-=)
(5-2) f(r):2'+In(r)
=W;{4
f'r"t
\---==-=--
(5-3) s(I): \F +r
_L
A'(+) =!(4'+t)'z'
(ts4)
rl\r )
:
312
3tz
=
e"" [email protected])
\n'lrt : 3 e"'l^
t-)
(12 +31
(5-5)
(ex':: sl - e-j (d+z)
{ i^\^ (e")z
r't
(5-6)
I
g(t):
,-4,
5'kt=#
-l-+ -'{
+-b
( ,{+{.)2
.)
(,t +t)2-
(ta pts). sHow ALL THE WORK, GIVE FULL ARGUMENTS.
a) Find the e - coordinate of the point of the global minimum for the {tnction
f(r) :3r f on the intervil 7f2 < r < 314
f
xa-l
ftj+i ca.-0 gls:
31 3. (-'), =o
x,2BoH^ poi't+1 a,.\9 Oqlsi& +-tq
{tv-i=
Lz
7 + 3.2-'-
6
,
=t
o,> 7.s <<- ^/o1 o-Q Va.Z.,r*q
I +v:2-91 v=6'svn
v
F;a
b) Find the z - coordinate of the point of the global minimum for the function
f(r) :2r - ln(r) on the interval z > 0
Gra1,u,.
o-{ { <>'1,
1'1
,.'
Glcr6^t
al-
\$'(*t=
^l
X
=O
?x-
x
I=
rzr,'r
i r"r irzr,r-r'r
oFe
czui
lt ca.! 3
vq is al)^l <
"ao\
oi vt*
pts).
For q units of a product, a manufacture's cost is C(q) dolars and
revenue is .B(q) dollars, with C(500) : 7200., n(500) : 9400, MC(500) : 15
and ME(500) :20.
a) What is the profit or loss at q : 500?
(1O
Tf
=
Q(soot- Q{soo)
= lYta-
Tzo-p
={ Lze-o
pto,fi+
b) If production is increased from 500 to 501 units, by approximately how much
does profit change?
_*tcCStto)
14 R /Soo>
=
f1 w&( 4 *2,
5
\
-
*l-f=
3'q'
4 G4)= 3 '9r
/.4 +
.3
g[^g*^r*
xL
\ q*<.rwa,.Q
x
9."
o
'= 5- d&ans.
8 (12 pts). Graph a function with
a,ll
of the following properties:
(1) The first derivative ls positive everywhere and
(2) The second derivative is positive for x > 2, is negative fot x < 2, and is zero
at r:2.
(r) tu) 7
(-u)
,1
I
I
```