Limits+and+Continuity

=Limits and Continuity= = =

1. Short lesson on the definition of the derivative

link to lesson

2. Lesson on the definition of the derivative (it is a bit long; you may wish to see only 13:15 to 20:40)

media type="youtube" key="MNk7RnyU0mk?feature=player_detailpage" height="360" width="640"

3. Review lesson on slopes of tangents and instantaneous rate of change (from MHF4U)

media type="youtube" key="yuEKC93wd_E?feature=player_detailpage" height="360" width="640"

An example requiring 'Rationalizing the Numerator

media type="custom" key="24070798"

4. Calculating Areas under a Curve (calculating an 'integral')

(i) Detailed calculations of "Upper" rectangles, "Lower" rectangles and rectangles formed from midpoints of function

media type="youtube" key="YPAljsPOu0w?feature=player_detailpage" height="360" width="640"

(ii) Another Example

media type="youtube" key="GNK8JonVcW4?feature=player_detailpage" height="360" width="640"

(iii) Still Another Example (ignore the fancy integral notation he introduces at the beginning; it isn't necessary)

media type="youtube" key="Gi8_a7NlKAQ?feature=player_detailpage" height="360" width="640"

The Limit of a Function

1. Introduction and definition

media type="youtube" key="jfEmtHyUmpY?feature=player_detailpage" height="360" width="640"

Another Overview

media type="youtube" key="HYSI-AHUqRM?feature=player_detailpage" height="360" width="640"

2. Evaluate limits by simplifying the expression or by factoring

media type="youtube" key="p8hQB11Ew1A?feature=player_detailpage" height="360" width="640"

3. Evaluate a limit by rationalizing the numerator / denominator

media type="youtube" key="gSWvhY5pE1w?feature=player_detailpage" height="360" width="640"

4. Evaluate a limit using left/right hand limits

(i) Limit of functions involving absolute value

media type="youtube" key="TUPkwGlrT8A?feature=player_detailpage" height="360" width="640"

(ii) Limits of piecewise defined functions

media type="youtube" key="duGU_Imgiug?feature=player_detailpage" height="360" width="640"

(iii) Determining limits using a (or if given a) graph

media type="youtube" key="UkjgJQaGx98?feature=player_detailpage" height="360" width="640"

5. Limits at Infinity ( determining end behaviour of functions, including sequences)

media type="youtube" key="FVJNuukADeQ?feature=player_detailpage" height="360" width="640"

6. More Examples (only view 2:53 to 14:02)

media type="youtube" key="IEfoQeWm8Fc?feature=player_detailpage" height="360" width="640"

7. Continuity

(i) Definition of Continuity

media type="youtube" key="ve3CEqnK8FQ?feature=player_detailpage" height="360" width="640"

(ii) Some sample calculations involving continuity

media type="youtube" key="VUEM6vWJvE4?feature=player_detailpage" height="360" width="640"

(iii) Types of Discontinuity (lots of examples) (Note: Her terminology is a little different from ours you can ignore 'essential' discontinuity)

1. Removable discontinuity = point discontinuity (or 'gap') 2. Infinite discontinuity = vertical asymptote

media type="youtube" key="KstJkCIWWho?feature=player_detailpage" height="360" width="640"

(iv) Another good example of a question involving continuity of a function

media type="youtube" key="YOuiXpLqDr0?feature=player_detailpage" height="360" width="640"

(v) A nice summary of functions we can ASSUME to be continuous (at each point in its domain) (note: add polynomial, rational, absolute value and trigonometric functions to his list)

link to the video

7. Review of Limits and Continuity

media type="youtube" key="YmuqoXnCSNI?feature=player_detailpage" height="360" width="640"