CALCULUS BC EXTRA TOPICS Developed by Susan Cantey and her students at Walnut Hills High School 2006

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