10/13/2015mrs. liu's precalc day191 cp calculus block 19 hw-review p88, 84;90;96;108;114;...
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04/19/23 Mrs. Liu's PreCalc Day19 1
CP Calculus Block 19 HW-Review p88,
84;90;96;108;114; 120;126;
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CP Calculus Block 19
HW p98,x3’s
3-48
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Ch 2.4 Continuity and One-sided Limit
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Do Now:
2
x 2, x 0
f (1
, x
xx0
) 1
x
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2
1x
x 2 11
x 1 x 1
2
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2
xlim f ( x )
0
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2
xlim f ( x )
1
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2x 0lim f ( x )
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2x 0lim f ( x )
2
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Objective 1: Determine the
continuity of the functions
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Definition A: Continuity at an interior point
x clim f(x) = f(c)
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Definition B: Continuity on closed interval [a,b], left endpoint a or right endpoint b
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b
+
-
x a
x
lim f(x) = f( )
lim f(x)
a
= f(b)
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Definition C: A function is continuous if it is continuous at each point of its domain
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The Continuity Test The function y=f(x) is continuous at x=c if and only if all of the following are true:
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x c
x c
ii ) limf(x) exists
iii) li
i
m
) f(c) ex
f(x
itst
) = c)
s
f(
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Existence of Limit
cx x clim f ( x ) lim f
L
( x )
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Properties of Continuity I Scalar Multiple: bf(x)
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Properties of Continuity ii Sum or difference f + g, or f - g
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Properties of Continuity iii Product f(x)g(x)
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Properties of Continuity iv Quotient f(x)/g(x)g( x ) 0
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Continuity of Composite Functions f(g(x)) or g(f(x))
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Theorem: If f is continuous at c and g is continuous at f©, then the composite gof is continuous at c.
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Example 0 Given f(x)
21 x , 0 x
x
1,
f ( x
,
)
x
1 x 1
0
1
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21 x , 0 x
x
1,
f ( x
,
)
x
1 x 1
0
1
a) Determine a complete graph of f(x)
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21 x , 0 x
x
1,
f ( x
,
)
x
1 x 1
0
1
b) Is f(x) continuous? Explain.
Yes. At all points
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Example 1 Find the points at which the function is not continuous
2
1f ( x )
( x 2 )
x 2
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Example 2 Find the points at which the function is not continuous
2
x 3f ( x )
x 3x 10
x 2,5
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Example 3 Find the points at which the function is not continuous 2
1f ( x )
x 1
No discontinuities
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Example 4 Find the points at which the function is not continuous
No discontinuities
f ( x ) 2x 3
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Example 5 Find the points at which the function is not continuous
x=0
xf ( x )
x
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Example 6 Find the points at which the function is not continuous
4f ( x ) 3x 1 x<1/3
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Example 7 Find the points at which the function is not continuousNo discontinuities
5f ( x ) 2 x
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Example 8 Define g(x) so that g(x)=(x2-9)/(x-3) is continuous at x=3
g(3)=6
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Example 9 Define f(1) so that f(x)=(x3-1)/(x2-1) is continuous at x=1
3f (1)
2
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Example 10 Given g(x)
3
2
g( x )1
bx
1x
,
, x
2
2
x
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What value should be assigned to b to make g(x) continuous at x=1/2?
1b
2
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Example 11: Find the limit
x 0limtan x
0
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Example 12: Find the limit
x 0lim sin( cos(tan x ))
2
1
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Example 13:Example 13: Find the Find the vertical asymptote(s) ofvertical asymptote(s) of
3f(x) =
(x + 2)
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Example 14:Example 14: Find Find the vertical the vertical asymptote(s) ofasymptote(s) of
2
1f(x) =
x - x - 2
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2
1f(x) =
x - x - 21
=(x + 1)(x - 2)
x = 2
x = -1
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Example 15:Example 15: Find the Find the end behavior end behavior asymptote(s) ofasymptote(s) of
2
x + 1f(x) =
x - x - 6
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vertical asymptotes
end behavior asymptote (horizontal)
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Example 16
3
2
x
3
Find limf(x) if
2x 7f
(x) =x 7
x x
a )2
b )2
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Example 17
2
xFind limf(x) if
3x 7f(x)
=x 2
a )0
b )0
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Example 18
4
3
xFind limf(x) i
xf(x) =
f
x 1
a )
b )
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Example 19
5
6
x
4
Find limf(x) if
10x
x 31x
f(x) =
a )0
b )0
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Example 20
4
4 3 2
xFind limf(x) if
xf(x) =
x 7 x 7 x 9
a )
1
1
b )
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Lesson quiz 1: Given
2x 1, 1 x 0
2x , 0 x 1
f ( x ) 1, x 1
2x 4, 1 x 2
0, 2 x 3
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2x 1, 1 x 0
2x , 0 x 1
f ( x ) 1, x 1
2x 4, 1 x 2
0, 2 x 3
a) Does f(1) exist?yes,f (1) 1
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2x 1, 1 x 0
2x , 0 x 1
f ( x ) 1, x 1
2x 4, 1 x 2
0, 2 x 3
x 1b) lim f(x) exist?
x 1yes,limf ( x ) 2
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2x 1, 1 x 0
2x , 0 x 1
f ( x ) 1, x 1
2x 4, 1 x 2
0, 2 x 3
x 1c) Does lim f(x) = f(1)?
No
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2x 1, 1 x 0
2x , 0 x 1
f ( x ) 1, x 1
2x 4, 1 x 2
0, 2 x 3
d) Is f(x) continuous at x = 1?
No
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2x 1, 1 x 0
2x , 0 x 1
f ( x ) 1, x 1
2x 4, 1 x 2
0, 2 x 3e) At what values of x
is f(x) continuous? all points in [-1, 3]
except x = 0, 1, 2
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2x 1, 1 x 0
2x , 0 x 1
f ( x ) 1, x 1
2x 4, 1 x 2
0, 2 x 3f) How should h be
defined to make h a
continuous extension
of f to the point x = 1?
h( x ) f ( x )
x 1,h(1) 2