1.4 calculus bc.notebook
TRANSCRIPT
1.4 calculus bc.notebook
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August 26, 2009
Do now as a warmup:
Is there some number a, such that this limit exists?
If so, find the value of a and find the limit. If not, explain why not.
1.4 calculus bc.notebook
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existsf(c) exists
1.4 Continuity
not continuous at c
continuous at c
not continuous at c
not continuous at c
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f defined at c not continuous at c
f undefined at c
not continuous at c
f defined at c butnot continuous at c
f defined at c and they're =continuous at c
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Defn. A function f is continuous at c if all these hold:
f(c) is defined
Defn. A function f is continuous on an open interval (a,b) if it is continuous at every point in (a,b).
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Classifying Discontinuities
Removable or point (Holes) 2 sided limit exists
Essential or Nonremovable
Jump1 sided limits exist
Infinite (vertical asymptotes) at least one of the 1 sided limits don't exist
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ex. Evaluate:
f(1)
Is f continuous at x=1?
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ex. Evaluate these limits:
f(0)
Is f continuous at x=0?
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f is a function
and
f is continuous on (a,b)
Thm. c∈R and L∈R
given:
Thm.
f is continuous on [a,b]
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ex. Find a and b so that f(x) is everywhere continuous.
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Discuss the continuity of:
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Thm.
f(x) and g(x) are continuous at x=cb∈R
These are continuous at c(b﴿﴾f(x)) f+g fg
fg f/g (as long as g(c) ≠ 0)
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Thm. Continuity of a composite
g is continuous at cf is continuous at g(c) f(g(x)) is continuous at c
gf Is f(g(x))
cont's at...0?7/3?2?5?3?
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ex. Discuss the continuity of the composite function f(g(x)) if...
and
and
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ex. Find the domain for each function. Where is each function not cont's? What kind of discontinuity is this?
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ex. Find the domain. Where is the function not cont's? What kind of discontinuity is this?
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Thm. The Intermediate Value Theorem (IVT)
[a,b]where f(c)=m
ba cf(b)
f(a)m
f is continuous on [a,b]m∈Rf(a)<m<f(b)
Try it! Draw a cont's function between(a,f(a)) and (b,f(b)).Pick any y value between f(a) and f(b).A horizontal line through that y value has to cross your function.
http://www.calculusapplets.com/ivt.html
This is an existence theorem.It says something exists, but not where.
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The IVT holds for these:
The IVT even holds for this:
It says there's at least one c value that works. There could be many!
But, if f is not continuous on [a,b], the theorem doesn't tell us anything.
f is continuous on [a,b]m∈Rf(a)<m<f(b)
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Apply the IVT, if possible, on [0,5] so that f(c)= 11 for the function
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Apply the IVT, if possible, on [0,3π/2] for y=sin x so that f(c)=0.