Videos about optimization, motion!

http://www.epsilen.com/MyPortal/Public/CustomTab.aspx?
tabID=28919&prefix=ashleyhubbl&systemName=Calculus%20BC%20Keys

videos solving problems about optimization, motion, limits!

http://www.epsilen.com/MyPortal/Public/CustomTab.aspx?tabID=28919&prefix=ashleyhubbl&systemName=Calculus%20BC%20Keys

MEAN value theorem and Rolle's Theorem

MEANie VALUE THEOREM

if f is continuous on interval[a,b] and differentiable on (a,b) then there is a VALUE of f(x) where the slope is equal to the average slope.

f '(c) = f(b)-f(a)

        b-a

in other words, there is some point of f(c) where the SECANT LINE SLOPE is equal to the slope at a certain point 

*IF the function is continuous and differentiable*

EXAMPLE: If a car accelerating from zero takes 8 sec to go 352 ft, its average

velocity for the 8-sec interval is 352/8=44 ft/sec.

What does the MVT(mean value theorem) say about this reading on the speedometer? 

there must be some time b/w 0 and 8 seconds where the car is traveling 44 ft/sec

ROLLE'S THEOREM :

if the y-values of two points on some closed interval are the same then there must be a point where the slope is ZERO ( or the derivative is equal to zero)

b/c

f '(c) = f(b)- f(a)

             b-a

if f(b)=f(a) then f(b)-f(a) = 0

SO SLOPE (or derivative at c) is ZERO

Find extrema on CLOSED interval!

Extreme Value Theorem

If f is continuous on a [closed] interval, then f has an ABSOLUTE MAXIMUM & MINIMUM

wait, there is a max and min even when it is a straight line?

like: y=1

YES!!! IF it is defined on a closed interval

the max and min of y=1 is 1!

f(c) is the max if f(c) >= f(x) for all x values

f(c) is the min if f(c) <= f(x) for all x values

1. all x values means it is continuous