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CALCULUS BC
EXTRA TOPICS
Developed by
Susan Cantey and her students
at
Walnut Hills High School
2006
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:
dy
y
 ky(1  )
dx
K
Calculus rocks!
This is the Logistic Equation, where
k= Growth rate
K= Carrying Capacity
Logistic Solution
P(t) = ?
K
P (t ) 
 kt
1  Ae
Where:
K  Po
A
Po
Next term using
Euler’s Method = ?
“Oiler”
dy
Previous y 
(at previous (x,y)) x
dx
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…

dy 2 dx 2
( )  ( ) dt
dt
dt
Formula for Speed in
Parametric equation?
Speedy
the
lightning
bolt
dy 2 dx 2
( ) ( )
dt
dt
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 )i  y (t ) j  z (t )k
Where x, y, and z are
treated the same as
parametric equations
Another notation for a
vector function is:
x(t ), y(t ), z(t )
What is the formula for the
velocity and acceleration vectors?
Velocity vector:

v (t )  x(t ), y(t ), z(t )
Acceleration vector:

a(t )  x(t ), y(t ), z(t )
(or use the i, j, k notation)
also...most AP vector problems will be
2-dimensional…so the third (z) component
will be omitted.
Work = ?

Force dx
Work in stretching
and/or contracting
springs?
Where:

b
a
kxdx
a = length of the spring when the work
begins
minus
the spring’s natural length
b = length of the spring when the work ends minus
the spring’s natural length
k = a constant peculiar to the spring in question
kx = force needed to maintain the spring at a length x units longer
(or shorter) than it’s natural length
Work in
pumping liquids
W 
Density · g · area of cross section · distance ·
dy
Density · g = weight
Average Value of J
b
J
dx

a
ba
You’re
done!
Created by:
Robert Jiang
Jake Ober
Class of ’07 rocks all.
Stay in school, kids.
Be sure to study the power points for :
1) Integrals
2) Derivatives
3) Pre-Calc Topics (on a separate page)
4) Sequences and Series
5) Miscellaneous Topics