![Page 1: Lecture 18: Rolle’s Theorem and Rectilinear Motionparnachev/Lecture18_14.pdfRolle’s Theorem and Rectilinear Motion. Example Find the radius and height of the right circular cylinder](https://reader035.vdocuments.mx/reader035/viewer/2022062610/61154a5cd9cd2d25b7577494/html5/thumbnails/1.jpg)
Lecture 18: Rolle’s Theorem and Rectilinear Motion
![Page 2: Lecture 18: Rolle’s Theorem and Rectilinear Motionparnachev/Lecture18_14.pdfRolle’s Theorem and Rectilinear Motion. Example Find the radius and height of the right circular cylinder](https://reader035.vdocuments.mx/reader035/viewer/2022062610/61154a5cd9cd2d25b7577494/html5/thumbnails/2.jpg)
Example Find the radius and height of the right circular cylinder of largest volume that can be inscribed in a right circular cone with radius 6 inches and height 10 inches
Solution r = radius (in inches) of the cylinder h = height (in inches) of the cylinder V = volume (in cubic inches) of the cylinder
occurs when the inscribed cylinder has radius 4 in
the maximum volume
![Page 3: Lecture 18: Rolle’s Theorem and Rectilinear Motionparnachev/Lecture18_14.pdfRolle’s Theorem and Rectilinear Motion. Example Find the radius and height of the right circular cylinder](https://reader035.vdocuments.mx/reader035/viewer/2022062610/61154a5cd9cd2d25b7577494/html5/thumbnails/3.jpg)
Example A closed cylindrical can is to hold 1 liter of liquid. How should we choose the height and radius to minimize the amount of material needed to manufacture the can?
Solution h = height (in cm) of the can r = radius (in cm) of the can S = surface area (in cm2) of the can
S does have a minimum on the interval (0, +∞)
![Page 4: Lecture 18: Rolle’s Theorem and Rectilinear Motionparnachev/Lecture18_14.pdfRolle’s Theorem and Rectilinear Motion. Example Find the radius and height of the right circular cylinder](https://reader035.vdocuments.mx/reader035/viewer/2022062610/61154a5cd9cd2d25b7577494/html5/thumbnails/4.jpg)
Rolle's Theorem
ROLLE'S THEOREM
![Page 5: Lecture 18: Rolle’s Theorem and Rectilinear Motionparnachev/Lecture18_14.pdfRolle’s Theorem and Rectilinear Motion. Example Find the radius and height of the right circular cylinder](https://reader035.vdocuments.mx/reader035/viewer/2022062610/61154a5cd9cd2d25b7577494/html5/thumbnails/5.jpg)
Example The differentiability requirement in Rolle's Theorem is critical.
roots at
yet there is no horizontal tangent to the graph of f over the interval
![Page 6: Lecture 18: Rolle’s Theorem and Rectilinear Motionparnachev/Lecture18_14.pdfRolle’s Theorem and Rectilinear Motion. Example Find the radius and height of the right circular cylinder](https://reader035.vdocuments.mx/reader035/viewer/2022062610/61154a5cd9cd2d25b7577494/html5/thumbnails/6.jpg)
!THE MEAN-VALUE THEOREM
![Page 7: Lecture 18: Rolle’s Theorem and Rectilinear Motionparnachev/Lecture18_14.pdfRolle’s Theorem and Rectilinear Motion. Example Find the radius and height of the right circular cylinder](https://reader035.vdocuments.mx/reader035/viewer/2022062610/61154a5cd9cd2d25b7577494/html5/thumbnails/7.jpg)
MOTIVATION FOR THE PROOF
![Page 8: Lecture 18: Rolle’s Theorem and Rectilinear Motionparnachev/Lecture18_14.pdfRolle’s Theorem and Rectilinear Motion. Example Find the radius and height of the right circular cylinder](https://reader035.vdocuments.mx/reader035/viewer/2022062610/61154a5cd9cd2d25b7577494/html5/thumbnails/8.jpg)
VELOCITY INTERPRETATION OF THE MEAN-VALUE THEOREM
is the position versus time curve for a car moving along a straight road.
the right side is the average velocity of the car over the time interval
the left side is the instantaneous velocity at time
Thus, the Mean-Value Theorem implies that at least once during the time interval the instantaneous velocity must equal the average velocity. This agrees with our real-world experience—if the average velocity for a trip is 40 mi/h, then sometime during the trip the speedometer has to read 40 mi/h.
![Page 9: Lecture 18: Rolle’s Theorem and Rectilinear Motionparnachev/Lecture18_14.pdfRolle’s Theorem and Rectilinear Motion. Example Find the radius and height of the right circular cylinder](https://reader035.vdocuments.mx/reader035/viewer/2022062610/61154a5cd9cd2d25b7577494/html5/thumbnails/9.jpg)
THE CONSTANT DIFFERENCE THEOREM
![Page 10: Lecture 18: Rolle’s Theorem and Rectilinear Motionparnachev/Lecture18_14.pdfRolle’s Theorem and Rectilinear Motion. Example Find the radius and height of the right circular cylinder](https://reader035.vdocuments.mx/reader035/viewer/2022062610/61154a5cd9cd2d25b7577494/html5/thumbnails/10.jpg)
RECTILINEAR MOTION
velocity function
speed function
acceleration function
![Page 11: Lecture 18: Rolle’s Theorem and Rectilinear Motionparnachev/Lecture18_14.pdfRolle’s Theorem and Rectilinear Motion. Example Find the radius and height of the right circular cylinder](https://reader035.vdocuments.mx/reader035/viewer/2022062610/61154a5cd9cd2d25b7577494/html5/thumbnails/11.jpg)
![Page 12: Lecture 18: Rolle’s Theorem and Rectilinear Motionparnachev/Lecture18_14.pdfRolle’s Theorem and Rectilinear Motion. Example Find the radius and height of the right circular cylinder](https://reader035.vdocuments.mx/reader035/viewer/2022062610/61154a5cd9cd2d25b7577494/html5/thumbnails/12.jpg)