DefinitionsDefinitions 50 40 30 20 10LimitsLimitsOtherOther Misc.Misc. Chapter 2.6 to 2.9 50 40 30...

DefinitionsDefinitionsDefinitionsDefinitions

50

40

30

20

10

LimitsLimitsLimitsLimits OtherOtherOtherOther Misc.Misc.Misc.Misc.

Chapter 2.6 to 2.9Chapter 2.6 to 2.9

50

40

30

20

10

50

40

30

20

10

50

40

30

20

10

50

40

30

20

10

Review for the TestReview for the TestChapter 2.6 to 2.9Chapter 2.6 to 2.9

Final Round

GraphsGraphsGraphsGraphs

Definitions

CheckCheckWorkWork

Go HomeGo Home

10 Point Question10 Point Question

What is the limit definition for the What is the limit definition for the derivative of a function?derivative of a function?

ax

afxfxf

ax

)()(lim)('

True or False: If a function is True or False: If a function is differentiable, then it is differentiable, then it is

continuous.continuous.


CheckCheckWorkWork

Go HomeGo Home

TrueTrue

DD


Which of the following is not a valid Which of the following is not a valid interpretation of f’(t)?interpretation of f’(t)?

A)A) The instantaneous rate of change of f(t)The instantaneous rate of change of f(t)B)B) The slope of the tangent to f(t)The slope of the tangent to f(t)

C)C) The velocity, if f(t) gives the position as a The velocity, if f(t) gives the position as a function of timefunction of time

D)D) The area under f(t)The area under f(t)

CheckCheckWorkWork

Go HomeGo Home

Next category please!Next category please!


Daily Double: the team picking this Daily Double: the team picking this question gets 80 points!question gets 80 points!

CheckCheckWorkWork

Go HomeGo Home

What is the difference between the What is the difference between the two limits?two limits?


CheckCheckWorkWork

Go HomeGo Home

f’(a) = slope of a function at a point.f’(a) = slope of a function at a point.

f’(x) = a function where the output is the f’(x) = a function where the output is the slope of f(x) at any particular point.slope of f(x) at any particular point.

h

xfhxfxf

h

)()()(' lim

0

h

afhafaf

h

)()(lim)('

0

Limits

CheckCheckWorkWork

Go HomeGo Home

Find the Find the limit:limit:


525

5

2

xx

xLim


Find f’(5), if f(x) = 3x+1Find f’(5), if f(x) = 3x+1

CheckCheckWorkWork

Go HomeGo Home

33


CheckCheckWorkWork

Go HomeGo Home

Find f’(-1) if f(x) = 2x2.

-4

Find the derivative of Find the derivative of f(x) = -2xf(x) = -2x22 + 3x + 3x


f’(x) = -4x +3f’(x) = -4x +3

CheckCheckWorkWork

Go HomeGo Home

Sketch a graph with the following Sketch a graph with the following properties:properties:


CheckCheckWorkWork

Go HomeGo Home

Answers Answers may vary.may vary.

1)(3

xfLimx

2)(3

xfLimx

2)3( f

Graphing

If f’(a) is positive, what do we know If f’(a) is positive, what do we know about f(x) at a?about f(x) at a?


CheckCheckWorkWork

Go HomeGo Home

f(x) is f(x) is increasingincreasing

If f”(x) is negative, what do we know If f”(x) is negative, what do we know about the graph of f(x)?about the graph of f(x)?


CheckCheckWorkWork

Go HomeGo Home

It is concave It is concave down.down.

Sketch the graph of f’(x), for graph #1Sketch the graph of f’(x), for graph #1


CheckCheckWorkWork

Go HomeGo Home


Based on the graph, sketch f”(x), for Based on the graph, sketch f”(x), for graph #2.graph #2.

CheckCheckWorkWork

Go HomeGo Home

The graph in #3 is the derivative of f(x). The graph in #3 is the derivative of f(x). Sketch f(x), assuming f(0) = 0Sketch f(x), assuming f(0) = 0


CheckCheckWorkWork

Go HomeGo Home

On the test

There will be no derivative if the There will be no derivative if the function is discontinuous, has a function is discontinuous, has a sharp corner or has a vertical sharp corner or has a vertical

tangent line at the point.tangent line at the point.


In general, at what points does a In general, at what points does a function fail to have a derivative?function fail to have a derivative?

CheckCheckWorkWork

Go HomeGo Home

Daily Double: The team picking this Daily Double: The team picking this question gets 40 points!question gets 40 points!


CheckCheckWorkWork

Go HomeGo Home

No. Limit as x -> 5 is 3/8 (not No. Limit as x -> 5 is 3/8 (not 0.5)0.5)

Is this function continuous at x = 5?Is this function continuous at x = 5?30 Point Question30 Point Question

CheckCheckWorkWork

Go HomeGo Home

5

5

,5.0

,25

153)( 2

x

xx

xxf


What value must f(D) be to make What value must f(D) be to make the graph continuous?the graph continuous?

CheckWork

Go HomeGo Home

33


What value for c makes the function What value for c makes the function continuous at x = 3?continuous at x = 3?

CheckCheckWorkWork

Go HomeGo Home

c = 5c = 5

3

3

,

,1)( 2

x

x

cx

xxf

Miscelaneous

May 13th May 13th

When is Mother’s Day?When is Mother’s Day?


CheckCheckWorkWork

Go HomeGo Home

When does registration for Summer and When does registration for Summer and Fall start?Fall start?


Monday, May 7thMonday, May 7th

CheckCheckWorkWork

Go HomeGo Home


Who founded the Boeing Company?Who founded the Boeing Company?

William BoeingWilliam Boeing

CheckCheckWorkWork

Go HomeGo Home


Who is getting leaving “The Who is getting leaving “The View”?View”?

CheckCheckWorkWork

Go HomeGo Home

Rosie O’DonnelRosie O’Donnel

What is the penalty if you What is the penalty if you don’t make it to the don’t make it to the

Testing Center before 3 Testing Center before 3 PM on Friday?PM on Friday?


CheckCheckWorkWork

Go HomeGo Home

10% for taking the test late10% for taking the test late

Just kidding. It Just kidding. It really is really is infinity.infinity.

If If

Final QuestionFinal Question

CheckCheckWorkWork

Go HomeGo Home

28 )8(

1

xLimx

25 5

1

xLimx

ThenThen 5

DefinitionsDefinitions 50 40 30 20 10LimitsLimitsOtherOther Misc.Misc. Chapter 2.6 to 2.9 50 40 30...

Documents

Transcript of DefinitionsDefinitions 50 40 30 20 10LimitsLimitsOtherOther Misc.Misc. Chapter 2.6 to 2.9 50 40 30...