2012 Parametric Functions AP Calculus : BC BONUS.

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Presentation transcript:

2012 Parametric Functions AP Calculus : BC BONUS

Parametric vs. Cartesian Graphs (x, y ) a position graph x = f (t) adds time, y = g (t) motion, and change ( f (t), g (t) ) is the ordered pair t and are called parameters. Adds- initial position and orientation

Parametric vs. Cartesian graphs (by hand) t x y

Parametric vs. Cartesian graphs (calculator) t x y 0  /2  3  /2 2  MODE: Parametric ZOOM: Square Try this. Parametric graphs are never unique!

Eliminate the Parameter Algebraic: Solve for t and substitute.

Eliminate the Parameter Trig: Use the Pythagorean Identities. Get the Trig function alone and square both sides.

Insert a Parameter Easiest: Let t equal some degree of x or y and plug in.

Calculus! The Derivative finds the RATE OF CHANGE. Words!

Example 1: Eliminate the parameter. and

Calculus! The Derivative finds the RATE OF CHANGE. x = f (t) then finds the rate of horizontal change with respect to time. y = g (t) then finds the rate of vertical change with respect to time. (( Think of a Pitcher and a Slider.)) still finds the slope of the tangent at any time.

Example 2: a) Find and interpret and at t = 2 b) Find and interpret at t = 2.

Example 3: Find the equation of the tangent at t = ( in terms of x and y ) Find the POINT. Find the SLOPE. Graph the curve and its tangent

Example 4: Find the points on the curve (in terms of x and y), if any, where the graph has horizontal and/or vertical tangents Vertical Tangent Slope is Undefined therefore, denominator = 0 Horizontal Tangents Slope = 0 therefore, numerator = 0

The Second Derivative Find the SECOND DERIVATIVE of the Parametric Function. 1). Find the derivative of the derivative w/ respect to t. 2). Divide by the original.

Example 1: Find the SECOND DERIVATIVE of the Parametric Function. =

Last Update: 10/19/07