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AP Calculus BC Tuesday , 27 August 2013PowerPoint Presentation

AP Calculus BC Tuesday , 27 August 2013

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AP Calculus BC Tuesday , 27 August 2013

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The

Student

Will

- OBJECTIVETSW (1) estimate a limit using a numerical and graphical approach; (2) learn different ways that a limit can fail to exist; and (3) study and use a formal definition of a limit.
- FORMS DUE (only if they are completed & signed)
- Information Sheet (wire basket)
- Acknowledgement Sheet (black tray)

- I will take T-Shirt orders at the beginning of class.

- Due any time between now and 20 September 2013.

- Angle measures are always in radians, not degrees.
- Unless directions tell otherwise, long decimals are rounded to three places (using conventional rounding or truncation).
- Always show work – Calculus is about communicating what you know, not just whether or not you can derive a correct answer.

You need to have these memorized for Friday’s quiz and for the rest of the year:

- Definition of the Six Trig Functions (including the pictures)
- Right Triangle Definitions
- Circular Function Definitions

- Reciprocal Identities
- Tangent and Cotangent Identities
- Pythagorean Identities

You need to have these memorized for Friday’s quiz and for the rest of the year:

- Unit Circle
- Special angles (in radians)
- Sines, cosines, and tangents of each special angle

- Double-Angle Formulas
- sin 2u
- cos 2u

You need to have these memorized for Friday’s quiz and for the rest of the year:

- Power-Reducing Formulas
- sin2u
- cos2u

Sec. 1.2: Finding Limits Graphically and Numerically

- An Introduction to Limits
Ex:

What is the value of

as x gets close to 2?

Undefined

???

- An Introduction to Limits
Ex:

2

2

12.06

11.41

11.94

11.994

12.61

undefined

12.006

12

12

- An Introduction to Limits
Ex:

“The limit as x approaches two of the quantity x cubed minus 8 divided by the quantity x minus 2 is 12”

“The limit as x approaches two of f(x) is 12”

- An Introduction to Limits
(informal) Definition: Limit

If f (x) becomes arbitrarily close to a single number L as x approaches c from both the left and the right, the limit as x approaches c is L.

- An Introduction to Limits
Ex:

- An Introduction to Limits
Ex:

1

- An Introduction to Limits
Ex:

In order for a limit to exist, it must approach a single number Lfrom both sides.

DNE

- An Introduction to Limits
Ex:

It would appear that the answer is –but this limit DNE because – is not a unique number.

DNE

- An Introduction to Limits
Ex:

DNE

ZOOM IN

ZOOM IN

ZOOM IN

- A Formal Definition of Limit
- Definition

Let f be a function defined on an open interval containing c (except possibly at c) and let L be a . The statement

means that for each > 0, a > 0 if

“There exists”

Epsilon

Delta

A real number

“Such That”

- A Formal Definition of Limit
- Definition

- A Formal Definition of Limit
Ex: Given that

Find such that

whenever

- A Formal Definition of Limit
Ex: Given that

Find such that

whenever