Particle Motion
Rectilinear motion: motion of a particle on a line
S(t)=x(t)= position at time t
S(t)=v(t)= instantaneous velocity
S(t)= a(t)= acceleration
v(t) = speed tells how fast but not direction
Vinstantaneous= slope of tangent line
Vavg= slope o
AP Calculus
31 to 34 Review 2
Verify that the hypothesis of Rolles Theorem are satisfied on the given interval and find all values of c that satisfy the
conclusion of the theorem.
1. f(x) = x2 6x + 8; [2, 4]
Ans: c = 3
3
2. f(x) = cos x ; ,
2 2
Ans:
3. 41n (x  1)  V2ln y + 3 In x
:in
4. 51n (x2
=1n
x~
(XIi'!

2)
+ V4ln y  31nx  tIn z
(j2._;) r"e
\) ~_
X 3 273
~
Find dy for each.
dx
5. Y = x2 In (x2

cfw_)x. 1n.lXll)+
,Ii:
6. Y = cos (In x)
1)
3
ij I =
J.x
x"/
J
Y=
x)
lj : hi 3
L,fi=:
J
I'"~
7.1
Simple Differential Equations
Lab 11
Find the particular solution to the differential equation given an initial
condition. Sketch the particular solution over the given slope field.
Run the program SLPFLD. Press IENTERI.
Select 1:DERIV OF F(X). E
Below you are given a differential equation, initial condition and slope field.
a) Sketch the solution of the differential equation that passes through the given point.
b) Then use integratiol! to find the particular solution .of the DE.
55#y'
~:
~
;
I
Draw a slope field for each of the following differential equations.
1. dy =x+l
2. dy =2y
~
~
/I
II
IIf
tII
II I
II/
,
II
4. dy =2x
3. dy =x+y
~
~
.~  /
I
\
II
/I
I
II
/
dy
5. =yl
~
/
/'/
/
~/

Permission to use granted by Nancy Stephenson
Available
!YOM:
Drawing Slope Fields
Name'
Date
AP Calculus
j(;f.l
0
Consider the differential equation dy = 2x.
~ ttLlcucfw_;.b and Qtf.vhI,
dx
Mch ~u
nt.
.
1. FInd the s ope 0f t he tan~ent.me at eac h of th e or"oe d pairs oncfw_he"table below.
r
.I
ere
dy
dy
dy