the mean value theorem and rolles theorem lesson 3.2 i wonder how mean this theorem really is?
TRANSCRIPT
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The Mean Value Theoremand Rolle’s Theorem
Lesson 3.2
I wonder how mean this
theorem really is?
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3.2 Bellwork
• Locate the COORDINATES of the absolute extrema of the function on the closed interval given.
• Verify the absolute extrema you found by graphing the function in an appropriate viewing window.
• Find the equation of the tangent line to the curve when x = 2.
• Graph the curve and the tangent line in an appropriate viewing window on your calculator.
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Finding the equation of the tangent line.
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The absolute maximum and absolute minimum are clearly shown in this viewing window.
You can also see that each critical point represents a local extrema for the graph.
(0,0) is the location of a relative maximum, also known as a local maximum of f(x).
(1,-1/2) is the location of a relative minimum, also known as a local minimum of f(x).
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Notice that the slope of the tangent line clearly matches the slope of the curve at the point (2,2).
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The Mean Value Theoremand Rolle’s Theorem
Lesson 3.2
I wonder how mean this
theorem really is?
![Page 9: The Mean Value Theorem and Rolles Theorem Lesson 3.2 I wonder how mean this theorem really is?](https://reader035.vdocument.in/reader035/viewer/2022081421/55155de455034685568b59e2/html5/thumbnails/9.jpg)
This is Really Mean
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Think About It
• Consider a trip of two hours that is 120 miles in distance … You have averaged 60 miles per hour
• What reading on your speedometer would you have expected to see at least once?
60
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Rolle’s Theorem
• Given f(x) on closed interval [a, b] Differentiable on open interval (a, b)
• If f(a) = f(b) … then There exists at least one number
a < c < b such that f ’(c) = 0
f(a) = f(b)
a bc
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Mean Value Theorem
• We can “tilt” the picture of Rolle’s Theorem Stipulating that f(a) ≠ f(b)
• Then there exists a c such that
a bc
( ) ( )'( )
f b f af c
b a
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Mean Value Theorem
• Applied to a cubic equation
Note Geogebera Example
Note Geogebera Example
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Finding c
• Given a function f(x) = 2x3 – x2 Find all points on the interval [0, 2] where
• Strategy Find slope of line from f(0) to f(2) Find f ‘(x) Set equal to slope … solve for x
( ) ( )'( )
f b f af c
b a
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Modeling Problem• Two police cars are located at fixed points 6
miles apart on a long straight road. The speed limit is 55 mph A car passes the first point at 53 mph Five minutes later he passes the second at 48
mph Yuk! Yuk! I think he was
speeding, EnosWe need to
prove it, Rosco