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AP Calculus BC Tuesday, 26 August 2014 OBJECTIVE TSW (1) estimate a limit using a numerical and graphical approach; (2) learn different ways that a limit.

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Presentation on theme: "AP Calculus BC Tuesday, 26 August 2014 OBJECTIVE TSW (1) estimate a limit using a numerical and graphical approach; (2) learn different ways that a limit."— Presentation transcript:

1 AP Calculus BC Tuesday, 26 August 2014 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. FORMS DUE (only if they are completed & signed) –Information Sheet & Acknowledgement Sheet (wire basket) I will take batteries and battery money at the beginning of the period. The Student Will

2 Things to Remember in Calculus 1.Angle measures are always in radians, not degrees. 2.Unless directions tell otherwise, long decimals are rounded to three places (using conventional rounding or truncation). 3.Always show work – Calculus is about communicating what you know, not just whether or not you can derive a correct answer.

3 Trigonometric Notes Sheet You need to have these memorized for Friday’s quiz and for the rest of the year: 1.Definition of the Six Trig Functions (including the pictures) a.Right Triangle Definitions b.Circular Function Definitions 2.Reciprocal Identities 3.Tangent and Cotangent Identities 4.Pythagorean Identities

4 Trigonometric Notes Sheet You need to have these memorized for Friday’s quiz and for the rest of the year: 5.Unit Circle a.Special angles (in radians) b.Sines, cosines, tangents, secants, cosecants, and cotangents of each special angle 6.Double-Angle Formulas a.sin 2u b.cos 2u

5 Trigonometric Notes Sheet You need to have these memorized for Friday’s quiz and for the rest of the year: 7.Power-Reducing Formulas a.sin 2 u b.cos 2 u

6 Sec. 1.2: Finding Limits Graphically and Numerically

7 An Introduction to Limits Ex: What is the value of as x gets close to 2? Undefined ???

8 Sec. 1.2: Finding Limits Graphically and Numerically An Introduction to Limits Ex: x f (x) undefined

9 Sec. 1.2: Finding Limits Graphically and Numerically 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”

10 Sec. 1.2: Finding Limits Graphically and Numerically 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.

11 Sec. 1.2: Finding Limits Graphically and Numerically An Introduction to Limits Ex:

12 Sec. 1.2: Finding Limits Graphically and Numerically An Introduction to Limits Ex: 1

13 Sec. 1.2: Finding Limits Graphically and Numerically An Introduction to Limits Ex: In order for a limit to exist, it must approach a single number L from both sides. DNE

14 An Introduction to Limits Ex: It would appear that the answer is –  but this limit DNE because –  is not a unique number. Sec. 1.2: Finding Limits Graphically and Numerically DNE

15 An Introduction to Limits Ex: Sec. 1.2: Finding Limits Graphically and Numerically DNE ZOOM IN

16 Sec. 1.2: Finding Limits Graphically and Numerically 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 A real number Epsilon “There exists” Delta “Such That”

17 A Formal Definition of Limit  -  Definition Sec. 1.2: Finding Limits Graphically and Numerically

18 A Formal Definition of Limit Ex: Given that Find  such that whenever

19 Sec. 1.2: Finding Limits Graphically and Numerically A Formal Definition of Limit Ex: Given that Find  such that whenever


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