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

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

CALCULUS BC EXTRA TOPICS

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CALCULUS BC EXTRA TOPICS

CALCULUS BC EXTRA TOPICS

Presentation transcript:

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: Calculus rocks!

This is the Logistic Equation, where k= Growth rate K= Carrying Capacity

Logistic Solution P(t) = ?

Where:

Next term using Euler’s Method = ? “Oiler”

Previous(at previous (x,y)) Estimated “Change”

Length of a curve defined by f(x)…i.e. arc length… Curve length

How long is it?

Length of a parametric curve…

Formula for Speed in Parametric equation? Speedy the lightning bolt

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?

(Polar) Length of a polar curve…

area of a region “inside” a polar graph...

Master polar of equations

(Parametric) More change

(Parametric) The change of the change

(Polar) Polar Bear

♪ 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

(Parametric) Area Surface

About Y-axis About X-axis

(Reg. Function)

About X-axis About Y-axis

Where x, y, and z are treated the same as parametric equations

Another notation for a vector function is: What is the formula for the velocity and acceleration vectors?

Velocity vector: Acceleration vector: (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: 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

Density · g · area of cross section · distance · dy Density · g = weight

Average Value of J

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