Onesided Limits, Continuity, and IVT Notes - rangercalculusab

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10 Οκτ 2013 (πριν από 3 χρόνια και 10 μήνες)

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One sided Limits,

Continuity of Closed Intervals,
Continuity Theorems
, and IVT










Ex: If
and

then













Is the function

contin
uous over the interval [0,1]


Is the function

continuous over the interval [0,0.5]


Is the function f(x) =

continuous over the interval [2, 12]




























f(x) = x
2

+ 2x


g(x) =


h(x) =

j(x) = sin(x)


Is f(x) + j(x) continuous?


Is

continuous?


Is f(g(x)) continuous at x =
2
?

Is f(h(x)) continuous at x = 0
?


Is j(h(x)) continuous at x =
?


Intermediate Value Theorem










Another way to look at the Intermediate Value Theorem: Let suppose the length of your
hair is currently 2 inches long. In one year your hair ends up being 6 inches long. The
IVT would state that there
must be a time were your hair was exactly 3 inches long.


We can use the IVT to approximate zeros of an equation. Lets say the function f(x) is
continuous and has the following x/y chart.


X:
-
3
-
2
-
1 0 1 2 3 4

Y: 2 3 1
-
1 2

3 4
-
2

Approximate where the x
-
intercepts for this continuous function would fall?



Textbook Example: Show that the IVT applies to the indicated interval and find the
value of c guaranteed by the theorem.

Ex.
,


,