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2012 Parametric Functions

2012 Parametric Functions. AP Calculus : BC BONUS. Parametric vs. Cartesian Graphs. Adds- initial position and orientation. (x , y ) a position graph . x = f (t) adds time , y = g (t) motion, and change

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2012 Parametric Functions

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  1. 2012 Parametric Functions AP Calculus : BC BONUS

  2. Parametric vs. Cartesian Graphs Adds- initial position and orientation (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.

  3. Parametric vs. Cartesian graphs (by hand) t x y -2 -1 0 1 2 3

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

  5. Eliminate the Parameter Algebraic: Solve for tand substitute.

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

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

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

  9. Example 1: Eliminate the parameter. and

  10. 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.)) Calculus! still finds the slope of the tangent at any time.

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

  12. 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

  13. 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 Horizontal Tangents Slope = 0 therefore, numerator = 0 Vertical Tangent Slope is Undefined therefore , denominator = 0

  14. 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 .

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

  16. Last Update: • 10/19/07

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