Lesson Plan #6



Lesson Plan #41

Class: AP Calculus Date: Wednesday December 15th, 2010

Topic: Derivatives of Parametric Equations Aim: How do we find the derivative of parametric equations?

Objectives:

1) Students will be able to graph parametric equations.

HW# 41:

1) Find [pic]for

A) [pic]and [pic] B) [pic]and [pic] C) [pic]and [pic]

2) The position of a particle moving along a straight line is given by [pic]

A) Find the interval(s) for which [pic]is increasing

B) Find the minimum value of the speed

C) Find the interval(s) for which acceleration is positive

D) Find the interval(s) for which the speed of the particle is decreasing

Do Now:

1) On your graphing calculator,

A) Set the mode to parametric

B) Set mode to radian

C) In Window

i. Set T min to 0

ii. Set T max to 2(

iii. Set T step to (/36

iv. Set X min to -5

v. Set X max to 5

vi. Set Y min to -5

vii. Set Y max to 5

viii. Set X scl to 1

ix. Set Y scl to 1

D) In X1= Type 3Cos(T)

E) In Y1= Type 4Sin(T)

2) We have graphed the equations [pic] and [pic]

Let’s try to eliminate the parameter, namely[pic]. Do this first by solving for [pic]and [pic]in each equation respectively.

What do we get? ______________________

Yes, we do get [pic]and [pic].

Now make use of the identity that [pic]

[pic]

We have now eliminated the parameter,(.

3) A) Use a graphing utility to sketch the graph of the curve represented by the parametric equations

[pic]and[pic].

B) Eliminate the parameter and write the corresponding rectangular equation.

Procedure:

Write the Aim and Do Now

Get students working!

Take attendance

Give back work

Go over the HW

Collect HW

Go over the Do Now

Assignment #1:

Simplify completely: [pic]

Yes, it simplifies to [pic]. So if you have a curve represented by parametric equations [pic]and [pic], then the slope of the curve at [pic], is represented by [pic].

Assignment # 2:

Find [pic]for the curve given by [pic]and [pic]

Assignment #3:

A) Find [pic]for the curve given by [pic]and [pic]

B) Find the slope of the curve at the [pic]

Assignment #4:

A) Find [pic]for the curve given by [pic]and [pic]

B) Find the slope of the curve at the [pic]

Particle Motion Along A Line:

If a particle moves along a line according to the law [pic], where [pic]represents the position of the particle [pic]on the line at time [pic], then the velocity [pic]of [pic]at time [pic]is given by [pic]and its acceleration [pic]by [pic]or by [pic]. The speed of the particle [pic], the magnitude of [pic].

NOTES:

1) If [pic], then [pic]is moving to the right and its distance [pic]is increasing. If [pic], then [pic]is moving to the left and its distance [pic]is decreasing.

2) If [pic], then [pic]is increasing; if [pic], then [pic]is decreasing;

3) If [pic]and [pic]are both positive or both negative, then the speed of [pic]is increasing or [pic]is accelerating.

If [pic]and [pic]are have opposite signs, then the speed of [pic]is decreasing or [pic]is decelerating.

4) If [pic]is a continuous function of [pic], then [pic]reverses direction whenever [pic]is zero and [pic]is different from zero. Note that zero velocity does not necessarily imply a change in direction.

Example #1:

A particle moves along a line according to the law [pic], where [pic]

A) Find all [pic]for which [pic]is increasing

B) Find all [pic]for which the velocity is increasing

C) Find all t for which the speed of the particle is increasing.

D) Find the speed when [pic]

E) Find the total distance traveled between [pic]and [pic]

Sample Test Questions:

A particle moves along a horizontal line and its position at time [pic]is [pic].

1) The particle is at rest when [pic]is equal to

A) 1 or 2 B) 0 C) [pic] D) 0,2, or 3 E) None of these

2) The velocity, v, is increasing when

A) [pic] B) [pic] C) [pic] D) [pic]or [pic] E) [pic]

3) The speed of the particle is increasing for

A) [pic] or [pic] B) [pic] C) [pic] D) [pic]or [pic] E) [pic]

4) The displacement from the origin of a particle moving on a line is given by [pic]. The maximum displacement during the time interval [pic]is

A) 27 B) 3 C) [pic] D) 48 E) None of these

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Solution: Substitute [pic]into the velocity function and then take the absolute value to get speed. Answer: 1.5

Solution: [pic]moves to the right when [pic], reverses its direction at [pic], moves to the left when [pic], reverses again at [pic], and continues to the right for all [pic]. The position of P at certain times [pic] are shown in the following table:

|[pic] |0 |1 |2 |4 |

|[pic] |-4 |1 |0 |28 |

From 0 to 1 it traveled 5 units, then from 1 to 2 it traveled 1 unit, then from 2 to 4 it traveled 28 units

Solution: The speed [pic]is increasing when both [pic]and [pic]are both positive and when [pic]and [pic]both negative. Those intervals are [pic]and [pic]

[pic]

Solution: [pic]. Set it equal to zero to get critical numbers; [pic], [pic]

|Interval |[pic] |[pic] |[pic] |

|Test Value |[pic] |[pic] |[pic] |

|Sign [pic]or [pic] |[pic] |- |+ |

|Conclusion |Increasing |Decreasing |Increasing |

Solution: [pic]. Set it equal to zero to get critical numbers; [pic]

|Interval |[pic] |[pic] |

|Test Value |[pic] |[pic] |

|Sign [pic]or [pic] |- |+ |

|Conclusion |Decreasing |Increasing |

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