Lecture 18: Rolle’s Theorem and Rectilinear Motion

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

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, +∞)

Rolle's Theorem

ROLLE'S THEOREM

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

!THE MEAN-VALUE THEOREM

MOTIVATION FOR THE PROOF

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.

THE CONSTANT DIFFERENCE THEOREM

RECTILINEAR MOTION

velocity function

speed function

acceleration function