12 10-90 Exam 2
Spring 2013
Name
Instructions.
Show all work and include appropriate explanations when necessary. A correct answer
full credit. Please try to do all work in the space provided. Please
unaccompanied by work may not receive
circle your final answers.
1. (24pts) For this problem, consider the function
f(x)
cctl?c
f’)
Fill
± 1.
f(x).
.=
xC3
t-p:
(c) (2pts)
3+
2x
=
(4pts) Find the critical point(s) of
0
—
f’(z).
(a) (4pts) Find
(b)
=
in the blanks:
f(x)
P ).
(
is decreasing on the interval
Note:
±00
are
acceptable answers.
(d)
(2pts) Which of the critical points you
r
51
—_—
0
(e) (4pts) Find
3
41?
0
lItR( s..u’.t.
f”(x).
c”(x)
(f)
ud above is a local minimum?
3x12-xi-’t
(4pts) Find the inflection point(s) of
f(x).
tee-I’.
I
(€-
Lé’
.±____-___.
(g) (2pts)
Fill
in the
blanks:
f(x)
(I))t(3,))
sLoff’
is concave down on the interval
).
Note:
±00
acceptable answers.
(h)
(2pts) What is the global minimum value of
1
f(x)?
If no
such
pts
minimum exists,
write
‘DNE’.
are
2. (l2pts) Consider the function
2 +8
x
1
(a) (6pts) Find the critical point(s) of f(x) in the interval [2,5].
-‘/x)
(2
)c ‘2- -2-x--X
2
-‘)()—L1(1
)c ‘t) 2-
3
-t -
(x -i )
O(
O=
(
7(i
j
4
aCz
c
1
9) /
C-
2
(b) (6pts) Find the extreme values
0_s
of f(x) on the interval [2,5]. Max=
Ce..c-( c!f
s
(- ) o-
iv
.€4-4t,OO LW3
Minz=
X
8’
2
rf(Lf
12-
f(s).-
‘ij
3. (l2pts) A box with a square base and no top is to be constructed out of 300 in
2 of cardboard. The
picture on the left shows the piece of cardboard (which has area 300 in
). and the picture on the right
2
shows the box obtained by folding along the dotted lines. What are the dimensions of the box (labeled
x and y in the pictures below) that will maximize the volume of the box? Note: You must use calculus
to get credit!!
(
3oc
2..
Y
A
UjeC4
b
LStYiuI6-4-
-
‘-‘3
so
0
Vcx)
-
(V’ ()
tj
X
7
J
3
-
=
÷
I
100 — ?C
xlo
-_----_
ID
2
lb.
3
4. (lOpts) Consider the function f(x)
=
2 + 1 on the interval [0. 2j.
3x
(a) (5pts) Find the value of c guaranteed by the Mean Value Theorem.
*
.
3 -1
b
(b) (5pts) Find the value of c guaranteed by the Mean Value Theorem for Integrals.
I
=
2O
J
(3x+i)ot,c)
-
(x3tx
I
—
ID
SO
5. (l6pts) Find the indicated antiderivatives:
(a) (4pts)
f(z
+9) dx
(
(b) (4pts)f/dx
(c) (4pts) f(x4 +x)(4x + 1) dx
i) .L?c
(d) (4pts)
f
tA-a
sin (5x
—
-I-c
3) dx
x—3
I
3
6.
(a) (5pts) Approximate fzx using a Riemann sum with 3 subintervais of equal length the
sample points being le left-endpoint of the subintervals.
Z.
A
o
-
i
(b) (5pts) Find the exact value of
I
=
3
—
2
x
dx. Hint: Use geometry and the area interpretation
J —3
of the definite integral. What do the graph of y
k O
ç
‘.3
L!
=
—
2 look like?
x
)c
J -K- -tLA)
f cw-te- o rt
‘.
3 .-(
3.
x
S
3
7. (lQpts) Evaluate the indicated definite integrals:
(a) (5pts)
2
r
I
Jo
3
(4x
—
2x) dx
2
(x_x’J
(b) (5pts)
xx2
(
± ldx
I
f
3
i
(z -i)
4t + 3. Assume that
8. (6pts) The velocity of a particle moving along the x-axis is given by v(t)
the units of the axis measure meters and the velocity is given in meters per second. If the particle is
located at x 5 at time t 0. where is the particle located at time t =37--=
=
—
=
of
‘
t4”r
j%
).J1I)
0
4
.