Transcript Polar and Parametric Powerpoint

CALCULUS BC
EXTRA TOPICS
Here come some questions on the extra
topics not covered in the AB course.
They will tend to be a little harder to
remember!!
When you think you know the answer,
(or if you give up
) click to get to
the next slide to see if you were correct.
Ready?
Explain:
dP
 kP ( M  P )
dt
Calculus rocks!
This is the Logistic Equation, where
k= Growth rate
M= Carrying Capacity
0.5M = the function value when it is growing the fastest.
Logistic Solution
P(t) = ?
M
P (t ) 
 kt
1  Ce
M  Po
Where: C  P
o
You’re right, I never showed you this. Haha!
Table for Euler’s
Method = ?
“Oiler”
( x, y )
dy
|( x , y )
dx
x
 dy 
y=   x
 dx 
 x  x, y  y 
Estimated “Change”
L?
Length of a curve defined by f(x)…i.e. arc length…

1  ( f ( x)) dx
2
L?
Length of a parametric curve…
b

a
2
2
 dx   dy 

dt
   
 dt   dt 
Formula for Speed in
Parametric equation?
Speedy
the
lightning
bolt
2
 dx   dy 

   
 dt   dt 
2
That is, speed is the rate of change along
the curve…the derivative of the integral
for arc length, i.e. the integrand by itself.
Formula for Speed
in Motion Problems?
| v (t ) |
L?
Length of a polar curve…
(Polar)

dr 2
r  ( ) d
d
2
A?
area of a
region “inside”
a polar
graph...
1 2
r
d

2
Master polar of
equations
dy
?
dx
(Parametric)
More
change
dy
(
)
dt
dx
(
)
dt
2
d y
(Parametric)

?
2
dx
The change
of the
change
 dy 
d 
 dx 
dt
dx
dt
dy
 ? (Polar)
dx
Polar Bear
dr
r cos  
sin 
d
dr
 r sin  
cos 
d
♪
if you forget the formula for the polar
derivative,
you can always derive it using:
x = r·cosӨ and y = r·sinӨ
along with the product rule and
dy

dx
dy
d
dx
d
S.A.  ?
(Parametric)
Surface
Area
About Y-axis
dx 2 dy 2
2

x
(
)

(
)
dt

dt
dt
About X-axis
dx 2 dy 2
2

y
(
)

(
)
dt

dt
dt
S.A.  ?
(Reg. Function)
About X-axis
dy 2
2

y
1

(
)
dx

dx
About Y-axis
dy 2
2

x
1

(
)
dx

dx

r (t )  ?
x(t ), y(t )
Where x, and y are treated
the same as parametric
equations
Another notation for a
vector function is (if in
three dimensions):
x(t ), y(t ), z(t )
What is the formula for the
velocity and acceleration vectors?
Velocity vector:
v (t )  x(t ), y(t )
Acceleration vector:
a(t )  x(t ), y(t )
AP vector problems will be
2-dimensional
Average Value of J
b
J
dx

a
ba
You’re
done!
Class of ’11 rocks all.
Stay in school, kids.
Be sure to study the power points for :
1) Integrals
2) Derivatives
3) Sequences and Series
4) Miscellaneous Topics