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CALCULUS BCEXTRA 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?
)1(K
yky
dx
dy
Explain:
Calculus rocks!
This is the Logistic Equation, where
k= Growth rate
K= Carrying Capacity
Logistic Solution
P(t) = ?
ktAe
KtP
1
)(
Where:o
o
P
PKA
Next term using Euler’s Method = ?
“Oiler”
dx
dyy xPrevious (at previous (x,y))
Estimated “Change”
?LLength of a curve defined by f(x)…i.e. arc length…
Curve
leng
th
dxxf 2))((1How long is it?
?LLength of a parametric curve…
dtdt
dx
dt
dy 22 )()(
Formula for Speed in Parametric equation?
Speedy the lightning bolt
22 )()(dt
dx
dt
dy
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?
|)(| tv
?L(Polar)
Length of a polar curve…
d
d
drr 22 )(
?Aarea of a
region “inside” a polar graph...
dr 2
2
1
Master polar of equations
?dx
dy(Parametric)
More change
)(
)(
dtdxdt
dy
?2
2
dx
yd(Parametric)
The change of the change
dtdxdtdxdy
d )(
?dx
dy(Polar)
Polar Bear
cossin
sincos
ddr
r
ddr
r
♪ 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
ddx
ddy
dx
dy
?.. AS(Parametric)
Area
Surface
About Y-axis
About X-axis
dtdt
dy
dt
dxx 22 )()(2
dtdt
dy
dt
dxy 22 )()(2
?.. AS(Reg. Function)
dxdx
dyy 2)(12
dxdx
dyx 2)(12
About X-axis
About Y-axis
?)( tr
ktzjtyitx)()()(
Where x, y, and z are treated the same as parametric equations
Another notation for a vector function is:
)(),(),( tztytx
What is the formula for the velocity and acceleration vectors?
Velocity vector:
Acceleration vector:
)(),(),()( tztytxtv
)(),(),()( tztytxta
(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?
b
akxdx
Where:
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
ab
dxJb
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) Integrals2) Derivatives3) Pre-Calc Topics (on a separate page)4) Sequences and Series5) Miscellaneous Topics
