ENTRY TASK
•SKETCH A RECTANGULAR COORDINATE PLANE ON A PIECE OF PAPER.
LABEL THE POINTS (1, 3) AND
(5, 3). USE YOUR PENCIL TO DRAW THE GRAPH OF A DIFFERENTIABLE
FUNCTION F THAT STARTS AT (1, 3) AND ENDS AT (5, 3).
• IS THERE AT LEAST ONE POINT WHERE 𝑓 ′ = 0?
• IS IT POSSIBLE TO DRAW THE GRAPH SO THAT THERE ISN’T A POINT FOR
WHICH 𝑓 ′ = 0? EXPLAIN YOUR REASONING.
LEARNING TARGETS

I UNDERSTAND ROLLE’S THEOREM AND THE MEAN VALUE THEOREM AND HOW THEY RELATE TO
DERIVATIVES OF FUNCTIONS.
 I SOLVED A VARIETY OF PROBLEMS INVOLVING APPLICATIONS OF DERIVATIVES WITH ROLLE’S
THEOREM AND THE MVT.
SECTION 3.1 – EXTREMA ON AND INTERVAL
• WHAT ARE EXTREMA?
• THE EXTREME VALUE THEOREM
• IF F IS CONTINUOUS ON A CLOSED INTERVAL [A, B], THEN F HAS BOTH A MAXIMUM AND A MINIMUM.
ROLLE’S THEOREM
• REVISIT EXIT TICKET
ROLLE’S THEOREM
• ROLLE’S THEOREM
• LET 𝑓 BE CONTINUOUS ON THE CLOSED INTERVAL [𝑎, 𝑏] AND DIFFERENTIABLE ON
THE OPEN INTERVAL (𝑎, 𝑏). IF 𝑓 𝑎 = 𝑓(𝑏), THEN THERE IS AT LEAST ONE NUMBER
𝑐 IN (𝑎, 𝑏) SUCH THAT : 𝑓 ′ 𝑐 = 0.
MVT
•THE MEAN VALUE THEOREM
• IF 𝑓 IS CONTINUOUS ON THE CLOSED INTERVAL
𝑎, 𝑏 AND
DIFFERENTIABLE ON THE OPEN INTERVAL (𝑎, 𝑏), THEN THERE EXISTS A
NUMBER 𝑐 IN (𝑎, 𝑏) SUCH THAT:
𝑓′
𝑓 𝑏 −𝑓 𝑎
𝑐 =
𝑏−𝑎
• TWO STATIONARY PATROL CARS
EQUIPPED WITH RADAR ARE 5 MILES
APART ON A HIGHWAY. AS A TRUCK
PASSES THE FIRST PATROL CAR, ITS SPEED
IS CLOCKED AT 55 MPH. FOUR MINUTES
LATER, WHEN THE TRUCK PASSES THE
SECOND PATROL CAR, ITS SPEED IS CLOCKED
AT 50 MPH. PROVE THAT THE TRUCK MUST HAVE EXCEEDED THE SPEED LIMIT (55 MPH) AT
SOME TIME DURING THE FOUR MINUTES.
ASSIGNMENT #10
•PAGE 176-178: 15, 17, 21, 41, 44, 59, 64, 77-80