Lesson Plan #6



Lesson Plan #35

Class: AP Calculus Date: Friday December 2nd, 2011

Topic: L’Hopital’s Rule Aim: How do we use L’Hoptital’s Rule to evaluate a limit?

Objectives:

1) Students will be able to use L’Hopital’s Rule to evaluate a limit

HW# 35:

Do Now:

1) Evaluate [pic]

2)

Let be the function defined above. Which of the following statements about are true?

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

In the above equation, we can find out where an object has been (basically its path), but we don’t know when it was at a certain location.

To determine the when an object has been at a location, what do we need?

Yes, you are right! We need a third variable to represent time. This variable could be [pic]. For example, we could have these two equations:

[pic] and

[pic].

These equations represent the location of the object at time[pic]. We pick a value for t, and then get the corresponding [pic]and [pic]values. [pic]is called the parameter. When we write [pic]and [pic]in terms of a third variable, we create parametric equations for [pic]and [pic].

To sketch a curve given by parametric equations, we can rely on the “old-school” method of picking a value for [pic], then finding the corresponding [pic]and [pic]values, then plot the points represented by those coordinates.

Example #1: Sketch the curve given by the parametric equations [pic]and[pic], where [pic]is in the range [pic].

|[pic] |-2 |-1 |0 |1 |2 |3 |

|[pic] | | | | | | |

|[pic] | | | | | | |

Of course, we could use our graphing utility to eliminate the need to pick points and evaluate, etc. On your calculator go to Mode and choose parametric.

Now we are going to choose our window. Go to Window and enter

Tmin=-2

Tmax =3

TStep = 1

You can do Xmax, Xmin, Ymax, Ymin as you see in the graph above.

Then go to the Y= button to enter in the equations.

First enter[pic] then enter [pic]

Example #2:

Using the graphing calculator, graph [pic], [pic], where [pic]

Go to Window and enter

Tmin=-2

Tmax =3

TStep = .5

What do you notice about the graphs in Example #1 and Example #2?

We see that two different sets of parametric equations can have the same graph. The difference between the Example #1 and Example #2, is that the first graph took longer (with respect to t) to graph the same thing.

Example #3:

Let’s try to find a rectangular equation that represents the same curve given by parametric equations [pic]and[pic].

We could do this by solving for [pic] in one equation and substituting in the other equation.

What do we get when we solve for [pic] in the 2nd equation?

Yes! We get [pic].

Now, substitute [pic]for [pic] in the 1st equation. What do we get?

Yes! We get [pic]. This is the rectangular equation for the parametric equations given above.

If you would like to graph this equation to check that it represents the same graph from the parametric equations, solve for y.

We get [pic]then [pic]then[pic]. Graph both equations in function mode in your calculator.

Do you get the same graph?

Exercise #1:

Sketch the curve represented by the parametric equations [pic]and [pic] (Use a graphing utility to check). Then eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.

Exercise #1:

Sketch the curve represented by the parametric equations [pic]and [pic] (Use a graphing utility to check). Then eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.

Up to this point we have been given parametric equations representing a curve and have found a rectangular equation that represents the same curve. Let’s see how we can do the reverse, in other words. Let’s find a set of parametric equations to represent the graph of[pic]. To do this we will need a parameter. Let’s use [pic]. Well there you have your first parametric equation, namely [pic]. To get the other one, substitute T for x, giving us [pic]

Sketch the graph of the equation (hopefully you get the same thing parametrically as you do rectangularly)

Try a different parameter, such as [pic]. Do we get the same thing?

Exercise #2:

Find a pair of parametric equations that represent the rectangular equation [pic]

L’Hopital’s Rule:

If the limit of [pic]produces the indeterminate form [pic]or [pic], then [pic]

Sample Test Questions:

1) Find the indicated limits using L’Hopital’s Rule. (There might be other ways to evaluate these limits)

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

2) Evaluate [pic]

A) -1 B) 0 C) 1 D) [pic] E) None of these

3) Evaluate [pic]

A) 1 B) 2 C) [pic] D) 0 E) [pic]

4) Evaluate [pic]

A) non-existent B) 1 C) 2 D) [pic] E) none of the other choices

If Enough Time:

1) Evaluate [pic]

A) [pic] B) 0 C) [pic] D) [pic] E) [pic]

2) [pic]is

A) [pic] B) 0 C) 1 D) [pic] E) non existent

3) The number of students N, taking AP Calculus at Brooklyn Technical High School is given in the table below:

|Year |1997 |1998 |1999 |2000 |

|[pic] |8 |10 |14 |22 |

A) 8 B) 12 C) 18 D) 40 E) 80

5) At how many points in the interval [pic]is a tangent to [pic]parallel to the secant line?

A) none B) 1 C) 2 D) 3 E) more than 3

6) Suppose [pic] If [pic]is the inverse function of [pic], then [pic]=

A) -1 B) [pic] C) [pic] D) 1 E) 3

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[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Alternate HW #35: {L’Hopital’s Rule Only} SHOW ALL WORK

1) [pic] 2) [pic] 3)[pic]

[pic]

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