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

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Chapter 2.6 to 2.9Chapter 2.6 to 2.9

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

Final Round

GraphsGraphsGraphsGraphs

Definitions

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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.

20 Point Question20 Point Question

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TrueTrue

DD

30 Point Question30 Point Question

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)

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Next category please!Next category please!

40 Point Question40 Point Question

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

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

50 Point Question50 Point Question

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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

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Find the Find the limit:limit:

10 Point Question10 Point Question

525

5

2

xx

xLim

20 Point Question20 Point Question

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

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33

30 Point Question30 Point Question

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Find f’(-1) if f(x) = 2x2.

-4

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

40 Point Question40 Point Question

f’(x) = -4x +3f’(x) = -4x +3

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

50 Point Question50 Point Question

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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?

10 Point Question10 Point Question

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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)?

20 Point Question20 Point Question

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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

30 Point Question30 Point Question

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40 Point Question40 Point Question

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

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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

50 Point Question50 Point Question

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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.

10 Point Question10 Point Question

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?

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

20 Point Question20 Point Question

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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

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5

5

,5.0

,25

153)( 2

x

xx

xxf

40 Point Question40 Point Question

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

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33

50 Point Question50 Point Question

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

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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?

10 Point Question10 Point Question

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

20 Point Question20 Point Question

Monday, May 7thMonday, May 7th

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30 Point Question30 Point Question

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

William BoeingWilliam Boeing

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40 Point Question40 Point Question

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

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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?

50 Point Question50 Point Question

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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

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28 )8(

1

xLimx

25 5

1

xLimx

ThenThen 5