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# AP Calculus BC Tuesday , 27 August 2013 - PowerPoint PPT Presentation

T he S tudent W ill. AP Calculus BC Tuesday , 27 August 2013. OBJECTIVE TSW (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.

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The

Student

Will

APCalculus BCTuesday, 27 August 2013

• 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)

• 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

• 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