Slope Fields
\-
l.
@x't,+J-dy
@
e
e
^=2,
x--
Given
grid.
dyfr, :
|
-
€reate a slope field at the 20 given points on the
r
31- +
3, t2dx.}
/*,
/
i.
./
/,vPP
x=+, H= -J+
,-'rl
-.
The grid marks are every 1 unit.
Indicate rvhich differential equation is represented in the slope field
graph. Briefy explain your choice.
:::::+::,'lt:i#.
,1.--t'
--.--'-.--']{
qA
-S
irftrS r,l'r< 5-tr4:X
--1." i: t:{)!--t-r
..t--tt'
-*-.--.--/ i' I i' l-l:.-*-'
x [-6, 6] y: [*4,41
----
---'----l-.-] ..{ lri
(A)
gl,^' "af
dy
9*:,,
(3,'::i'"tf (B)
tn '.;;:"r,.Ii,[cl
fr:
dx
dy
-,
--=tan'xldxV
f
:logs
it^l t u.*.
(E)
dy
fr:
v\'-
r-i
Explanation:
w..j
/el t/y. | ^d-b
ha,n'' (-t) = -4
*'''
;-i.< 6ll
.1,4r" ' "
.r'
t-t
^qL
&'"
5
Y3X"'
n6L
' t''
L
'
e
"'n{.#J- n41"*1"'€
<r\ .g:{
'nct"
,, ii qf
1ir i"r
-t^r,r,rJ
t
5'
:I
'l)
V'" 3-'"y '
I]
t'6
3.
just y or a combinaA differential equation may also be a function of
tion of x and y. As an example, the;lope field belorv rvas created from
the differentiil equatio n dy]dx = *ly.Describe any patterns you notice
for regions of poiiti*,e, negative, ot\ero slope as they may be determined by values of x and Y.
ffiff
d+I
'1,
ITFRIIN
4.
x
2-
o
(,
-'
I i-f
011l
{
L3
create a slope field on the 12 points in the graph for the differential
equation dy,rtf : Yz(x - 1).
o ril I
o I z
---?
-f-
oo
/
/
-D"
,/
\:,
1
Additional Practice
l.
each slope field, draw the solution curve that satisfies the
2' (Marks on the axes are every I unit')
condition
or,
initial
f(-tl:
(A)
(B)
\=,
-r
Which of the following slope fields could be a solution to the differential
l/3
equarion clyidx : -t ? Briefly explain your choice. (Al1 rvindorvs are
[-4.7, 4.7] and [-3. l, 3. I ].)
(B)
.J
.J
tl
.t
,l t
.ri,
TI
:'aL
tl
I
l
{t
,l
(D)
(c)
?,,rt"4{1N.ffi
7tt
tl
0-'
-'HT
{fiilitli i"#
r;i*iilf
t{
lrt
t-
+
tl
I
I
.l .
t
J.
tl
I
I
l-t
-L
\
st
t"t
f
S 'i
#
t
)
\
t
'i\
rtJJJ
.t rf rf
With
C
.l
';c
.IL
t
I
See
'
;;0.
nerds?
atl
ar
<
trI
lrr-(
L.
t
I
,.1.
'i\i:!
h--.ru\*.
Explanation:
Need More HelP
+
r,
I
,T
I,
l+
'nft'nn't'
-a-
_t
.r!
--*\--\.H
gh'ar-f
.\ *t .\
t
f -f -./ .tr
,1.!fi_-
trt--n
(E)
ff-rf.
It
I,
.
,.
Calculus, Section 6.1
2004 AP" Calculus AB Exam,
Problem 6
.;i,
:,
i,t'
. ttr
E
[l
2OOO AP@
ti
CALCULUS BC FREE-RESPONSE QUESTTONS
Consider the differential equation given by
i
*
=
*b - t)' .
(a) On the axes provided, sketch a slope field for the given differential equation at the eleven points indicated.
(Note: Use the axes provlded in the pink test booklet)
I
I
Use the slope field for the given differentiai iiqu4tion to explain why a solution could not have the graph
shown below.
O)
+€/0
3=l
y = f (x)
(c) Find
the particular solution
(d) Find
the range of the solution found in part (c).
m
to the given differential equation with the initial condition
*2-- r (y-D'
c) dr(
!s-
-
/(0) = -1.
Xdr
tu-')"
END OF EXAMINATION
x"rF
Q-0-'aY=
r \_, Copyright O 2000 College Entrance Examination Board and Educational Testing Service. All rights reserved.
APisaregisteredtrademarkof theCollegeEntranceExaminationBoard,
q}'(r)+C=
r,
t
a.
ur-l = 4+C
J/
L
-a
=1__
-l-l
I
a=L
t,c
L
)
-t
u-l
-!-
g-t
t*
Z
L
-5-
L
l]J
L
v -ffi -" lY= *6 *tl
L
*l
|
=_1-=:^-=--\
-l-
I
,uli^'t'"r-
Slope Field Card Match
NancY StePhenson
Clements High School
Sugar Land, Texas
group
Studentswillworkingrouploftrvgorthreetomatchthethreeffiesofcards:
and conclusion cards' Each
;#f;;;ti"i"q"uu*
cards,
slope field
"Js'
their answers' on the
to
ililr*"i
*t
,"
and
cards
of
'""otd
will receive,one set
unit'
one
is
mark
tick
ri"p" naas, each
Slope Fields
Differential Equations
Conelusions
s
lo
q
8
6
I
Z
L
1 of3
3
lo
I
1