Calculus: Continuity by: javier
Calculus: Continuity
by: javier
Review Limits: Methods to determine
Review: Limits: Methods to determine
□ Plug-in-Method [PIM] □ Engeneer’s Method [EM] □ Graphs □ Re-Write [RW] □ Recognize Famous limits
Review: Limits: Methods to determine
□ Plug-in-Method [PIM]
( ) lim x3 + 2
x→3
□ Engeneer’s Method [EM] □ Graphs □ Re-Write [RW] □ Recognize Famous limits
Review: Limits: Methods to determine
□ Plug-in-Method [PIM] □ Engeneer’s Method [EM]
lim ln(x) − ln(5)x − 5
x→5
□ Graphs □ Re-Write [RW] □ Recognize Famous limits
Review: Limits: Methods to determine □ Plug-in-Method [PIM] □ Engeneer’s Method [EM] □ Graphs 1
0.5
0
−0.5
∞ 0
10
□ Re-Write [RW] □ Recognize Famous limits
20
30
∞
Review: Limits: Methods to determine
□ Plug-in-Method [PIM] □ Engeneer’s Method [EM] □ Graphs □ Re-Write [RW] □ Recognize Famous limits
lim xx
x→0+
Review Limits: Definitions
Review: Limits: Definitions
□ (lim at c) □ (lim at infinity)
Review: Limits: Definitions
□ (lim at c) lim f(x) = L ⇐⇒ ∀ϵ > 0 ∃ δ > 0 ∋ 0|x − c| < δ =⇒ |f(x) − L| < ϵ
x→c
□ (lim at infinity)
Review: Limits: Definitions
□ (lim at c) lim f(x) = L ⇐⇒ ∀ϵ > 0 ∃ δ > 0 ∋ 0|x − c| < δ =⇒ |f(x) − L| < ϵ
x→c
□ (lim at infinity) lim f(x) = L ⇐⇒ ∀ϵ > 0 ∃ M ∋ x > M =⇒ |f(x) − L| < ϵ
x→∞
Review Limits: properties
Review: Limits: properties
□ (lim of k)
limx→c k = k
□ (lim of x)
limx→c x = c
□ (pull k)
limx→c kf(x) = k limx→c f(x)
□
(LS=SL)∗
limx→c (g(x) + f(x)) = limx→c g(x) + limx→c f(x)
□
(LP=PL)∗
limx→c (g(x) · f(x)) = limx→c g(x) · limx→c f(x)
□ (LQ=QL)∗
limx→c
g(x) f(x)
=
limx→c g(x) limx→c f(x)
Review: Limits: properties
□ (lim of k)
limx→c k = k
□ (lim of x)
limx→c x = c
□ (pull k)
limx→c kf(x) = k limx→c f(x)
□
(LS=SL)∗
limx→c (g(x) + f(x)) = limx→c g(x) + limx→c f(x)
□
(LP=PL)∗
limx→c (g(x) · f(x)) = limx→c g(x) · limx→c f(x)
□ (LQ=QL)∗
limx→c
g(x) f(x)
∗ each must exists and be finite
=
limx→c g(x) limx→c f(x)
Review Limits: Reason for being
Review: Limits: Reason for being
□ We need limits to evaluate and make sense of indeterminate forms. □ We need to evaluate indeterminate forms to compute areas, infinite many , each infitely small... and ... □ (also) We need to evaluate indeterminate forms to compute slopes, infinitely small rise over infinitely small
runs. □ We need to compute areas and slopes because when a meaninful label is put on the x and the y coordinates the area/slope represents something meaningful, many real life science quesions are solved gracefully this way.
Review: Limits: Reason for being
□ We need limits to evaluate and make sense of indeterminate forms. ∞ · 0, 00 etc □ We need to evaluate indeterminate forms to compute areas, infinite many , each infitely small... and ... □ (also) We need to evaluate indeterminate forms to compute slopes, infinitely small rise over infinitely small
runs. □ We need to compute areas and slopes because when a meaninful label is put on the x and the y coordinates the area/slope represents something meaningful, many real life science quesions are solved gracefully this way.
Continuity layman definition
Continuity: layman definition
Is this function continous at x = 2 ? □ Can we ’trace’ the graph around x = 2 □ without lifting our pencil □ without running out of ink
6
4
2
0 −4
−2
0
2
4
Continuity: layman definition
Is this function continous at x = 0 ? □ Can we ’trace’ the graph around x = 0 □ without lifting our pencil □ without running out of ink
6
4
2
0 −4
−2
0
2
4
Continuity: layman definition
Is this function continous at x = 0 ? □ Can we ’trace’ the graph around x = 0 □ without lifting our pencil □ without running out of ink
4 3 2 1 0 −4
−2
0
2
4
Continuity: layman definition
Is this function continous at x = 0 ? □ Can we ’trace’ the graph around x = 0 □ without lifting our pencil □ without running out of ink
4 3 2 1 0 −1 −4
−2
0
2
4
Continuity: layman definition
Y
Is this function continous at x = 0 ? □ Can we ’trace’ the graph around x = 0 □ without lifting our pencil □ without running out of ink
y = tan(x)
X
Continuity: layman definition
Is this function continous at x = π2 ? □ Can we ’trace’ the graph around x = □ without lifting our pencil □ without running out of ink
Y
y = tan(x)
X
π 2
Continuity: layman definition
Is this function continous at x = 0 ? □ Can we ’trace’ the graph around x = 0 □ without lifting our pencil □ without running out of ink
1 0.5 0 −0.5 f(x) = sin
−1 −0.5
0
0.5
(1) x
1
Continuity rigorous definition
Continuity: rigorous definition
DEFINITION: f(x) is said to be ”continous at c” IFF? □ limx→c f(x) exists and is finite □ f(c) exists and is finite □ the above two are equal
10
5 f(x) 0
−1
0
1
2
Continuity what can you do with it
Continuity: what can you do with it
DEFINITION: f(x) is said to be ”continous at c” IFF? □ many nice things happen when for continous functions □ continous functions make continous functions when added, multiplied, etc □ continous functions allow limits through
Continuity: what can you do with it
DEFINITION: f(x) is said to be ”continous at c” IFF? □ many nice things happen when for continous functions □ continous functions make continous functions when added, multiplied, etc □ continous functions allow limits through √ √ limx→3 x + 1 = limx→3 x + 1
Continuity: what can you do with it
DEFINITION: f(x) is said to be ”continous at c” IFF? □ many nice things happen when for continous functions □ continous functions make continous functions when added, multiplied, etc □ continous functions allow limits through √ √ limx→3 x + 1 = limx→3 x + 1 √ √ limx→−5 x + 1 = limx→−5 x + 1
Continuity famous types of discontinuities
Continuity: famous types of discontinuities
DEFINITION: f(x) is said to be ”continous at c” IFF? □ JUMP Discontinuities □ ASYMPTOTE Discontinuities □ REMOVABLE Discontinuities □ OTHER Discontinuities
Continuity famously continous everywhere functions
Continuity: famously continous everywhere functions
some functions that are fmaously continous at every real number: □ polynomials □ sin(x) and cos(x) □ exponential functions, ex □ combinations of these
by: javier
Review Limits: Methods to determine
Review: Limits: Methods to determine
□ Plug-in-Method [PIM] □ Engeneer’s Method [EM] □ Graphs □ Re-Write [RW] □ Recognize Famous limits
Review: Limits: Methods to determine
□ Plug-in-Method [PIM]
( ) lim x3 + 2
x→3
□ Engeneer’s Method [EM] □ Graphs □ Re-Write [RW] □ Recognize Famous limits
Review: Limits: Methods to determine
□ Plug-in-Method [PIM] □ Engeneer’s Method [EM]
lim ln(x) − ln(5)x − 5
x→5
□ Graphs □ Re-Write [RW] □ Recognize Famous limits
Review: Limits: Methods to determine □ Plug-in-Method [PIM] □ Engeneer’s Method [EM] □ Graphs 1
0.5
0
−0.5
∞ 0
10
□ Re-Write [RW] □ Recognize Famous limits
20
30
∞
Review: Limits: Methods to determine
□ Plug-in-Method [PIM] □ Engeneer’s Method [EM] □ Graphs □ Re-Write [RW] □ Recognize Famous limits
lim xx
x→0+
Review Limits: Definitions
Review: Limits: Definitions
□ (lim at c) □ (lim at infinity)
Review: Limits: Definitions
□ (lim at c) lim f(x) = L ⇐⇒ ∀ϵ > 0 ∃ δ > 0 ∋ 0|x − c| < δ =⇒ |f(x) − L| < ϵ
x→c
□ (lim at infinity)
Review: Limits: Definitions
□ (lim at c) lim f(x) = L ⇐⇒ ∀ϵ > 0 ∃ δ > 0 ∋ 0|x − c| < δ =⇒ |f(x) − L| < ϵ
x→c
□ (lim at infinity) lim f(x) = L ⇐⇒ ∀ϵ > 0 ∃ M ∋ x > M =⇒ |f(x) − L| < ϵ
x→∞
Review Limits: properties
Review: Limits: properties
□ (lim of k)
limx→c k = k
□ (lim of x)
limx→c x = c
□ (pull k)
limx→c kf(x) = k limx→c f(x)
□
(LS=SL)∗
limx→c (g(x) + f(x)) = limx→c g(x) + limx→c f(x)
□
(LP=PL)∗
limx→c (g(x) · f(x)) = limx→c g(x) · limx→c f(x)
□ (LQ=QL)∗
limx→c
g(x) f(x)
=
limx→c g(x) limx→c f(x)
Review: Limits: properties
□ (lim of k)
limx→c k = k
□ (lim of x)
limx→c x = c
□ (pull k)
limx→c kf(x) = k limx→c f(x)
□
(LS=SL)∗
limx→c (g(x) + f(x)) = limx→c g(x) + limx→c f(x)
□
(LP=PL)∗
limx→c (g(x) · f(x)) = limx→c g(x) · limx→c f(x)
□ (LQ=QL)∗
limx→c
g(x) f(x)
∗ each must exists and be finite
=
limx→c g(x) limx→c f(x)
Review Limits: Reason for being
Review: Limits: Reason for being
□ We need limits to evaluate and make sense of indeterminate forms. □ We need to evaluate indeterminate forms to compute areas, infinite many , each infitely small... and ... □ (also) We need to evaluate indeterminate forms to compute slopes, infinitely small rise over infinitely small
runs. □ We need to compute areas and slopes because when a meaninful label is put on the x and the y coordinates the area/slope represents something meaningful, many real life science quesions are solved gracefully this way.
Review: Limits: Reason for being
□ We need limits to evaluate and make sense of indeterminate forms. ∞ · 0, 00 etc □ We need to evaluate indeterminate forms to compute areas, infinite many , each infitely small... and ... □ (also) We need to evaluate indeterminate forms to compute slopes, infinitely small rise over infinitely small
runs. □ We need to compute areas and slopes because when a meaninful label is put on the x and the y coordinates the area/slope represents something meaningful, many real life science quesions are solved gracefully this way.
Continuity layman definition
Continuity: layman definition
Is this function continous at x = 2 ? □ Can we ’trace’ the graph around x = 2 □ without lifting our pencil □ without running out of ink
6
4
2
0 −4
−2
0
2
4
Continuity: layman definition
Is this function continous at x = 0 ? □ Can we ’trace’ the graph around x = 0 □ without lifting our pencil □ without running out of ink
6
4
2
0 −4
−2
0
2
4
Continuity: layman definition
Is this function continous at x = 0 ? □ Can we ’trace’ the graph around x = 0 □ without lifting our pencil □ without running out of ink
4 3 2 1 0 −4
−2
0
2
4
Continuity: layman definition
Is this function continous at x = 0 ? □ Can we ’trace’ the graph around x = 0 □ without lifting our pencil □ without running out of ink
4 3 2 1 0 −1 −4
−2
0
2
4
Continuity: layman definition
Y
Is this function continous at x = 0 ? □ Can we ’trace’ the graph around x = 0 □ without lifting our pencil □ without running out of ink
y = tan(x)
X
Continuity: layman definition
Is this function continous at x = π2 ? □ Can we ’trace’ the graph around x = □ without lifting our pencil □ without running out of ink
Y
y = tan(x)
X
π 2
Continuity: layman definition
Is this function continous at x = 0 ? □ Can we ’trace’ the graph around x = 0 □ without lifting our pencil □ without running out of ink
1 0.5 0 −0.5 f(x) = sin
−1 −0.5
0
0.5
(1) x
1
Continuity rigorous definition
Continuity: rigorous definition
DEFINITION: f(x) is said to be ”continous at c” IFF? □ limx→c f(x) exists and is finite □ f(c) exists and is finite □ the above two are equal
10
5 f(x) 0
−1
0
1
2
Continuity what can you do with it
Continuity: what can you do with it
DEFINITION: f(x) is said to be ”continous at c” IFF? □ many nice things happen when for continous functions □ continous functions make continous functions when added, multiplied, etc □ continous functions allow limits through
Continuity: what can you do with it
DEFINITION: f(x) is said to be ”continous at c” IFF? □ many nice things happen when for continous functions □ continous functions make continous functions when added, multiplied, etc □ continous functions allow limits through √ √ limx→3 x + 1 = limx→3 x + 1
Continuity: what can you do with it
DEFINITION: f(x) is said to be ”continous at c” IFF? □ many nice things happen when for continous functions □ continous functions make continous functions when added, multiplied, etc □ continous functions allow limits through √ √ limx→3 x + 1 = limx→3 x + 1 √ √ limx→−5 x + 1 = limx→−5 x + 1
Continuity famous types of discontinuities
Continuity: famous types of discontinuities
DEFINITION: f(x) is said to be ”continous at c” IFF? □ JUMP Discontinuities □ ASYMPTOTE Discontinuities □ REMOVABLE Discontinuities □ OTHER Discontinuities
Continuity famously continous everywhere functions
Continuity: famously continous everywhere functions
some functions that are fmaously continous at every real number: □ polynomials □ sin(x) and cos(x) □ exponential functions, ex □ combinations of these
