SlideShare uma empresa Scribd logo
1 de 99
Baixar para ler offline
. . . . . .
Section	2.1
The	Derivative	and	Rates	of	Change
V63.0121.034, Calculus	I
September	23, 2009
Announcements
WebAssignments	due	Monday. Email	me	if	you	need	an
extension	for	Yom	Kippur.
. . . . . .
Regarding	WebAssign
We	feel	your	pain
. . . . . .
Explanations
From	the	syllabus:
Graders	will	be	expecting	you	to	express	your	ideas
clearly, legibly, and	completely, often	requiring
complete	English	sentences	rather	than	merely	just	a
long	string	of	equations	or	unconnected	mathematical
expressions. This	means	you	could	lose	points	for
unexplained	answers.
. . . . . .
Rubric
Points Description	of	Work
3 Work	 is	 completely	 accurate	 and	 essentially	 perfect.
Work	is	thoroughly	developed, neat, and	easy	to	read.
Complete	sentences	are	used.
2 Work	 is	 good, but	 incompletely	 developed, hard	 to
read, unexplained, or	 jumbled. Answers	 which	 are
not	explained, even	if	correct, will	generally	receive	2
points. Work	contains	“right	idea”	but	is	flawed.
1 Work	is	sketchy. There	is	some	correct	work, but	most
of	work	is	incorrect.
0 Work	minimal	or	non-existent. Solution	is	completely
incorrect.
. . . . . .
Outline
Rates	of	Change
Tangent	Lines
Velocity
Population	growth
Marginal	costs
The	derivative, defined
Derivatives	of	(some)	power	functions
What	does f tell	you	about f′
?
How	can	a	function	fail	to	be	differentiable?
Other	notations
The	second	derivative
. . . . . .
The	tangent	problem
Problem
Given	a	curve	and	a	point	on	the	curve, find	the	slope	of	the	line
tangent	to	the	curve	at	that	point.
. . . . . .
The	tangent	problem
Problem
Given	a	curve	and	a	point	on	the	curve, find	the	slope	of	the	line
tangent	to	the	curve	at	that	point.
Example
Find	the	slope	of	the	line	tangent	to	the	curve y = x2
at	the	point
(2, 4).
. . . . . .
Graphically	and	numerically
. .x
.y
.
.2
..4 .
x m
. . . . . .
Graphically	and	numerically
. .x
.y
.
.2
..4 .
.
.
.3
..9
x m
3 5
. . . . . .
Graphically	and	numerically
. .x
.y
.
.2
..4 .
.
.
.2.5
..6.25
x m
3 5
2.5 4.25
. . . . . .
Graphically	and	numerically
. .x
.y
.
.2
..4 .
.
.
.2.1
..4.41
x m
3 5
2.5 4.25
2.1 4.1
. . . . . .
Graphically	and	numerically
. .x
.y
.
.2
..4 ..
.
.2.01
..4.0401
x m
3 5
2.5 4.25
2.1 4.1
2.01 4.01
. . . . . .
Graphically	and	numerically
. .x
.y
.
.2
..4 .
.
.
.1
..1
x m
3 5
2.5 4.25
2.1 4.1
2.01 4.01
1 3
. . . . . .
Graphically	and	numerically
. .x
.y
.
.2
..4 .
.
.
.1.5
..2.25
x m
3 5
2.5 4.25
2.1 4.1
2.01 4.01
1.5 3.5
1 3
. . . . . .
Graphically	and	numerically
. .x
.y
.
.2
..4 .
.
.
.1.9
..3.61
x m
3 5
2.5 4.25
2.1 4.1
2.01 4.01
1.9 3.9
1.5 3.5
1 3
. . . . . .
Graphically	and	numerically
. .x
.y
.
.2
..4 ..
.
.1.99
..3.9601
x m
3 5
2.5 4.25
2.1 4.1
2.01 4.01
1.99 3.99
1.9 3.9
1.5 3.5
1 3
. . . . . .
Graphically	and	numerically
. .x
.y
.
.2
..4 .
.
.
.3
..9
.
.
.2.5
..6.25
.
.
.2.1
..4.41 .
.
.2.01
..4.0401
.
.
.1
..1
.
.
.1.5
..2.25
.
.
.1.9
..3.61
.
.
.1.99
..3.9601
x m
3 5
2.5 4.25
2.1 4.1
2.01 4.01
limit 4
1.99 3.99
1.9 3.9
1.5 3.5
1 3
. . . . . .
The	tangent	problem
Problem
Given	a	curve	and	a	point	on	the	curve, find	the	slope	of	the	line
tangent	to	the	curve	at	that	point.
Example
Find	the	slope	of	the	line	tangent	to	the	curve y = x2
at	the	point
(2, 4).
Upshot
If	the	curve	is	given	by y = f(x), and	the	point	on	the	curve	is
(a, f(a)), then	the	slope	of	the	tangent	line	is	given	by
mtangent = lim
x→a
f(x) − f(a)
x − a
. . . . . .
Velocity
Problem
Given	the	position	function	of	a	moving	object, find	the	velocity
of	the	object	at	a	certain	instant	in	time.
Example
Drop	a	ball	off	the	roof	of	the	Silver	Center	so	that	its	height	can
be	described	by
h(t) = 50 − 5t2
where t is	seconds	after	dropping	it	and h is	meters	above	the
ground. How	fast	is	it	falling	one	second	after	we	drop	it?
. . . . . .
Numerical	evidence
t vave =
h(t) − h(1)
t − 1
2 − 15
. . . . . .
Numerical	evidence
t vave =
h(t) − h(1)
t − 1
2 − 15
1.5
. . . . . .
Numerical	evidence
t vave =
h(t) − h(1)
t − 1
2 − 15
1.5 − 12.5
. . . . . .
Numerical	evidence
t vave =
h(t) − h(1)
t − 1
2 − 15
1.5 − 12.5
1.1
. . . . . .
Numerical	evidence
t vave =
h(t) − h(1)
t − 1
2 − 15
1.5 − 12.5
1.1 − 10.5
. . . . . .
Numerical	evidence
t vave =
h(t) − h(1)
t − 1
2 − 15
1.5 − 12.5
1.1 − 10.5
1.01
. . . . . .
Numerical	evidence
t vave =
h(t) − h(1)
t − 1
2 − 15
1.5 − 12.5
1.1 − 10.5
1.01 − 10.05
. . . . . .
Numerical	evidence
t vave =
h(t) − h(1)
t − 1
2 − 15
1.5 − 12.5
1.1 − 10.5
1.01 − 10.05
1.001
. . . . . .
Numerical	evidence
t vave =
h(t) − h(1)
t − 1
2 − 15
1.5 − 12.5
1.1 − 10.5
1.01 − 10.05
1.001 − 10.005
. . . . . .
Velocity
Problem
Given	the	position	function	of	a	moving	object, find	the	velocity
of	the	object	at	a	certain	instant	in	time.
Example
Drop	a	ball	off	the	roof	of	the	Silver	Center	so	that	its	height	can
be	described	by
h(t) = 50 − 5t2
where t is	seconds	after	dropping	it	and h is	meters	above	the
ground. How	fast	is	it	falling	one	second	after	we	drop	it?
Solution
The	answer	is
v = lim
t→1
(50 − 5t2) − 45
t − 1
= lim
t→1
5 − 5t2
t − 1
= lim
t→1
5(1 − t)(1 + t)
t − 1
= (−5) lim
t→1
(1 + t) = −5 · 2 = −10
. . . . . .
Upshot
If	the	height	function	is	given
by h(t), the	instantaneous
velocity	at	time t0 is	given	by
v = lim
t→t0
h(t) − h(t0)
t − t0
= lim
∆t→0
h(t0 + ∆t) − h(t0)
∆t
. .t
.y = h(t)
.
.
.
.t0
.
.t
.∆t
. . . . . .
Population	growth
Problem
Given	the	population	function	of	a	group	of	organisms, find	the
rate	of	growth	of	the	population	at	a	particular	instant.
. . . . . .
Population	growth
Problem
Given	the	population	function	of	a	group	of	organisms, find	the
rate	of	growth	of	the	population	at	a	particular	instant.
Example
Suppose	the	population	of	fish	in	the	East	River	is	given	by	the
function
P(t) =
3et
1 + et
where t is	in	years	since	2000	and P is	in	millions	of	fish. Is	the
fish	population	growing	fastest	in	1990, 2000, or	2010? (Estimate
numerically)?
. . . . . .
Derivation
Let ∆t be	an	increment	in	time	and ∆P the	corresponding	change
in	population:
∆P = P(t + ∆t) − P(t)
This	depends	on ∆t, so	we	want
lim
∆t→0
∆P
∆t
= lim
∆t→0
1
∆t
(
3et+∆t
1 + et+∆t
−
3et
1 + et
)
. . . . . .
Derivation
Let ∆t be	an	increment	in	time	and ∆P the	corresponding	change
in	population:
∆P = P(t + ∆t) − P(t)
This	depends	on ∆t, so	we	want
lim
∆t→0
∆P
∆t
= lim
∆t→0
1
∆t
(
3et+∆t
1 + et+∆t
−
3et
1 + et
)
Too	hard! Try	a	small ∆t to	approximate.
. . . . . .
Numerical	evidence
r1990 ≈
P(−10 + 0.1) − P(−10)
0.1
≈
. . . . . .
Numerical	evidence
r1990 ≈
P(−10 + 0.1) − P(−10)
0.1
≈ 0.000136
. . . . . .
Numerical	evidence
r1990 ≈
P(−10 + 0.1) − P(−10)
0.1
≈ 0.000136
r2000 ≈
P(0.1) − P(0)
0.1
≈
. . . . . .
Numerical	evidence
r1990 ≈
P(−10 + 0.1) − P(−10)
0.1
≈ 0.000136
r2000 ≈
P(0.1) − P(0)
0.1
≈ 0.75
. . . . . .
Numerical	evidence
r1990 ≈
P(−10 + 0.1) − P(−10)
0.1
≈ 0.000136
r2000 ≈
P(0.1) − P(0)
0.1
≈ 0.75
r2010 ≈
P(10 + 0.1) − P(10)
0.1
≈
. . . . . .
Numerical	evidence
r1990 ≈
P(−10 + 0.1) − P(−10)
0.1
≈ 0.000136
r2000 ≈
P(0.1) − P(0)
0.1
≈ 0.75
r2010 ≈
P(10 + 0.1) − P(10)
0.1
≈ 0.000136
. . . . . .
Population	growth
Problem
Given	the	population	function	of	a	group	of	organisms, find	the
rate	of	growth	of	the	population	at	a	particular	instant.
Example
Suppose	the	population	of	fish	in	the	East	River	is	given	by	the
function
P(t) =
3et
1 + et
where t is	in	years	since	2000	and P is	in	millions	of	fish. Is	the
fish	population	growing	fastest	in	1990, 2000, or	2010? (Estimate
numerically)?
Solution
The	estimated	rates	of	growth	are 0.000136, 0.75, and 0.000136.
. . . . . .
Upshot
The	instantaneous	population	growth	is	given	by
lim
∆t→0
P(t + ∆t) − P(t)
∆t
. . . . . .
Marginal	costs
Problem
Given	the	production	cost	of	a	good, find	the	marginal	cost	of
production	after	having	produced	a	certain	quantity.
. . . . . .
Marginal	costs
Problem
Given	the	production	cost	of	a	good, find	the	marginal	cost	of
production	after	having	produced	a	certain	quantity.
Example
Suppose	the	cost	of	producing q tons	of	rice	on	our	paddy	in	a
year	is
C(q) = q3
− 12q2
+ 60q
We	are	currently	producing 5 tons	a	year. Should	we	change	that?
. . . . . .
Comparisons
q C(q)
4
5
6
. . . . . .
Comparisons
q C(q)
4 112
5
6
. . . . . .
Comparisons
q C(q)
4 112
5 125
6
. . . . . .
Comparisons
q C(q)
4 112
5 125
6 144
. . . . . .
Comparisons
q C(q) AC(q) = C(q)/q
4 112
5 125
6 144
. . . . . .
Comparisons
q C(q) AC(q) = C(q)/q
4 112 28
5 125
6 144
. . . . . .
Comparisons
q C(q) AC(q) = C(q)/q
4 112 28
5 125 25
6 144
. . . . . .
Comparisons
q C(q) AC(q) = C(q)/q
4 112 28
5 125 25
6 144 24
. . . . . .
Comparisons
q C(q) AC(q) = C(q)/q ∆C = C(q + 1) − C(q)
4 112 28
5 125 25
6 144 24
. . . . . .
Comparisons
q C(q) AC(q) = C(q)/q ∆C = C(q + 1) − C(q)
4 112 28 13
5 125 25
6 144 24
. . . . . .
Comparisons
q C(q) AC(q) = C(q)/q ∆C = C(q + 1) − C(q)
4 112 28 13
5 125 25 19
6 144 24
. . . . . .
Comparisons
q C(q) AC(q) = C(q)/q ∆C = C(q + 1) − C(q)
4 112 28 13
5 125 25 19
6 144 24 31
. . . . . .
Marginal	costs
Problem
Given	the	production	cost	of	a	good, find	the	marginal	cost	of
production	after	having	produced	a	certain	quantity.
Example
Suppose	the	cost	of	producing q tons	of	rice	on	our	paddy	in	a
year	is
C(q) = q3
− 12q2
+ 60q
We	are	currently	producing 5 tons	a	year. Should	we	change	that?
Example
If q = 5, then C = 125, ∆C = 19, while AC = 25. So	we	should
produce	more	to	lower	average	costs.
. . . . . .
Upshot
The	incremental	cost
∆C = C(q + 1) − C(q)
is	useful, but	depends	on	units.
. . . . . .
Upshot
The	incremental	cost
∆C = C(q + 1) − C(q)
is	useful, but	depends	on	units.
The	marginal	cost	after	producing q given	by
MC = lim
∆q→0
C(q + ∆q) − C(q)
∆q
is	more	useful	since	it’s	unit-independent.
. . . . . .
Outline
Rates	of	Change
Tangent	Lines
Velocity
Population	growth
Marginal	costs
The	derivative, defined
Derivatives	of	(some)	power	functions
What	does f tell	you	about f′
?
How	can	a	function	fail	to	be	differentiable?
Other	notations
The	second	derivative
. . . . . .
The	definition
All	of	these	rates	of	change	are	found	the	same	way!
. . . . . .
The	definition
All	of	these	rates	of	change	are	found	the	same	way!
Definition
Let f be	a	function	and a a	point	in	the	domain	of f. If	the	limit
f′
(a) = lim
h→0
f(a + h) − f(a)
h
exists, the	function	is	said	to	be differentiable	at a and f′
(a) is	the
derivative	of f at a.
. . . . . .
Derivative	of	the	squaring	function
Example
Suppose f(x) = x2
. Use	the	definition	of	derivative	to	find f′
(a).
. . . . . .
Derivative	of	the	squaring	function
Example
Suppose f(x) = x2
. Use	the	definition	of	derivative	to	find f′
(a).
Solution
f′
(a) = lim
h→0
f(a + h) − f(a)
h
= lim
h→0
(a + h)2 − a2
h
= lim
h→0
(a2 + 2ah + h2
) − a2
h
= lim
h→0
2ah + h2
h
= lim
h→0
(2a + h) = 2a.
. . . . . .
Derivative	of	the	reciprocal	function
Example
Suppose f(x) =
1
x
. Use	the
definition	of	the	derivative	to
find f′
(2).
. . . . . .
Derivative	of	the	reciprocal	function
Example
Suppose f(x) =
1
x
. Use	the
definition	of	the	derivative	to
find f′
(2).
Solution
f′
(2) = lim
x→2
1/x − 1/2
x − 2
= lim
x→2
2 − x
2x(x − 2)
= lim
x→2
−1
2x
= −
1
4
. .x
.x
.
. . . . . .
The	Sure-Fire	Sally	Rule	(SFSR) for	adding	Fractions
In	anticipation	of	the	question, “How	did	you	get	that?”
a
b
±
c
d
=
ad ± bc
bd
So
1
x
−
1
2
x − 2
=
2 − x
2x
x − 2
=
2 − x
2x(x − 2)
. . . . . .
The	Sure-Fire	Sally	Rule	(SFSR) for	adding	Fractions
In	anticipation	of	the	question, “How	did	you	get	that?”
a
b
±
c
d
=
ad ± bc
bd
So
1
x
−
1
2
x − 2
=
2 − x
2x
x − 2
=
2 − x
2x(x − 2)
. . . . . .
What	does f tell	you	about f′
?
If f is	a	function, we	can	compute	the	derivative f′
(x) at	each
point x where f is	differentiable, and	come	up	with	another
function, the	derivative	function.
What	can	we	say	about	this	function f′
?
. . . . . .
What	does f tell	you	about f′
?
If f is	a	function, we	can	compute	the	derivative f′
(x) at	each
point x where f is	differentiable, and	come	up	with	another
function, the	derivative	function.
What	can	we	say	about	this	function f′
?
If f is	decreasing	on	an	interval, f′
is	negative	(well,
nonpositive)	on	that	interval
. . . . . .
Derivative	of	the	reciprocal	function
Example
Suppose f(x) =
1
x
. Use	the
definition	of	the	derivative	to
find f′
(2).
Solution
f′
(2) = lim
x→2
1/x − 1/2
x − 2
= lim
x→2
2 − x
2x(x − 2)
= lim
x→2
−1
2x
= −
1
4
. .x
.x
.
. . . . . .
What	does f tell	you	about f′
?
If f is	a	function, we	can	compute	the	derivative f′
(x) at	each
point x where f is	differentiable, and	come	up	with	another
function, the	derivative	function.
What	can	we	say	about	this	function f′
?
If f is	decreasing	on	an	interval, f′
is	negative	(well,
nonpositive)	on	that	interval
If f is	increasing	on	an	interval, f′
is	positive	(well,
nonnegative)	on	that	interval
. . . . . .
Graphically	and	numerically
. .x
.y
.
.2
..4 .
.
.
.3
..9
.
.
.2.5
..6.25
.
.
.2.1
..4.41 .
.
.2.01
..4.0401
.
.
.1
..1
.
.
.1.5
..2.25
.
.
.1.9
..3.61
.
.
.1.99
..3.9601
x m
3 5
2.5 4.25
2.1 4.1
2.01 4.01
limit 4
1.99 3.99
1.9 3.9
1.5 3.5
1 3
. . . . . .
What	does f tell	you	about f′
?
Fact
If f is	decreasing	on (a, b), then f′
≤ 0 on (a, b).
Proof.
If f is	decreasing	on (a, b), and ∆x > 0, then
f(x + ∆x) < f(x) =⇒
f(x + ∆x) − f(x)
∆x
< 0
. . . . . .
What	does f tell	you	about f′
?
Fact
If f is	decreasing	on (a, b), then f′
≤ 0 on (a, b).
Proof.
If f is	decreasing	on (a, b), and ∆x > 0, then
f(x + ∆x) < f(x) =⇒
f(x + ∆x) − f(x)
∆x
< 0
But	if ∆x < 0, then x + ∆x < x, and
f(x + ∆x) > f(x) =⇒
f(x + ∆x) − f(x)
∆x
< 0
still!
. . . . . .
What	does f tell	you	about f′
?
Fact
If f is	decreasing	on (a, b), then f′
≤ 0 on (a, b).
Proof.
If f is	decreasing	on (a, b), and ∆x > 0, then
f(x + ∆x) < f(x) =⇒
f(x + ∆x) − f(x)
∆x
< 0
But	if ∆x < 0, then x + ∆x < x, and
f(x + ∆x) > f(x) =⇒
f(x + ∆x) − f(x)
∆x
< 0
still! Either	way,
f(x + ∆x) − f(x)
∆x
< 0, so
f′
(x) = lim
∆x→0
f(x + ∆x) − f(x)
∆x
≤ 0
. . . . . .
Outline
Rates	of	Change
Tangent	Lines
Velocity
Population	growth
Marginal	costs
The	derivative, defined
Derivatives	of	(some)	power	functions
What	does f tell	you	about f′
?
How	can	a	function	fail	to	be	differentiable?
Other	notations
The	second	derivative
. . . . . .
Differentiability	is	super-continuity
Theorem
If f is	differentiable	at a, then f is	continuous	at a.
. . . . . .
Differentiability	is	super-continuity
Theorem
If f is	differentiable	at a, then f is	continuous	at a.
Proof.
We	have
lim
x→a
(f(x) − f(a)) = lim
x→a
f(x) − f(a)
x − a
· (x − a)
= lim
x→a
f(x) − f(a)
x − a
· lim
x→a
(x − a)
= f′
(a) · 0 = 0
. . . . . .
Differentiability	is	super-continuity
Theorem
If f is	differentiable	at a, then f is	continuous	at a.
Proof.
We	have
lim
x→a
(f(x) − f(a)) = lim
x→a
f(x) − f(a)
x − a
· (x − a)
= lim
x→a
f(x) − f(a)
x − a
· lim
x→a
(x − a)
= f′
(a) · 0 = 0
Note	the	proper	use	of	the	limit	law: if the	factors	each	have	a
limit	at a, the	limit	of	the	product	is	the	product	of	the	limits.
. . . . . .
How	can	a	function	fail	to	be	differentiable?
Kinks
. .x
.f(x)
. . . . . .
How	can	a	function	fail	to	be	differentiable?
Kinks
. .x
.f(x)
. .x
.f′
(x)
.
. . . . . .
How	can	a	function	fail	to	be	differentiable?
Kinks
. .x
.f(x)
. .x
.f′
(x)
.
.
. . . . . .
How	can	a	function	fail	to	be	differentiable?
Cusps
. .x
.f(x)
. . . . . .
How	can	a	function	fail	to	be	differentiable?
Cusps
. .x
.f(x)
. .x
.f′
(x)
. . . . . .
How	can	a	function	fail	to	be	differentiable?
Cusps
. .x
.f(x)
. .x
.f′
(x)
. . . . . .
How	can	a	function	fail	to	be	differentiable?
Vertical	Tangents
. .x
.f(x)
. . . . . .
How	can	a	function	fail	to	be	differentiable?
Vertical	Tangents
. .x
.f(x)
. .x
.f′
(x)
. . . . . .
How	can	a	function	fail	to	be	differentiable?
Vertical	Tangents
. .x
.f(x)
. .x
.f′
(x)
. . . . . .
How	can	a	function	fail	to	be	differentiable?
Weird, Wild, Stuff
. .x
.f(x)
This	function	is	differentiable
at 0.
. . . . . .
How	can	a	function	fail	to	be	differentiable?
Weird, Wild, Stuff
. .x
.f(x)
This	function	is	differentiable
at 0.
. .x
.f′
(x)
But	the	derivative	is	not
continuous	at 0!
. . . . . .
Outline
Rates	of	Change
Tangent	Lines
Velocity
Population	growth
Marginal	costs
The	derivative, defined
Derivatives	of	(some)	power	functions
What	does f tell	you	about f′
?
How	can	a	function	fail	to	be	differentiable?
Other	notations
The	second	derivative
. . . . . .
Notation
Newtonian	notation
f′
(x) y′
(x) y′
Leibnizian	notation
dy
dx
d
dx
f(x)
df
dx
These	all	mean	the	same	thing.
. . . . . .
Meet	the	Mathematician: Isaac	Newton
English, 1643–1727
Professor	at	Cambridge
(England)
Philosophiae	Naturalis
Principia	Mathematica
published	1687
. . . . . .
Meet	the	Mathematician: Gottfried	Leibniz
German, 1646–1716
Eminent	philosopher	as
well	as	mathematician
Contemporarily
disgraced	by	the
calculus	priority	dispute
. . . . . .
Outline
Rates	of	Change
Tangent	Lines
Velocity
Population	growth
Marginal	costs
The	derivative, defined
Derivatives	of	(some)	power	functions
What	does f tell	you	about f′
?
How	can	a	function	fail	to	be	differentiable?
Other	notations
The	second	derivative
. . . . . .
The	second	derivative
If f is	a	function, so	is f′
, and	we	can	seek	its	derivative.
f′′
= (f′
)′
It	measures	the	rate	of	change	of	the	rate	of	change!
. . . . . .
The	second	derivative
If f is	a	function, so	is f′
, and	we	can	seek	its	derivative.
f′′
= (f′
)′
It	measures	the	rate	of	change	of	the	rate	of	change! Leibnizian
notation:
d2
y
dx2
d2
dx2
f(x)
d2
f
dx2
. . . . . .
function, derivative, second	derivative
. .x
.y
.f(x) = x2
.f′
(x) = 2x
.f′′
(x) = 2

Mais conteúdo relacionado

Mais procurados

MinFill_Presentation
MinFill_PresentationMinFill_Presentation
MinFill_Presentation
Anna Lasota
 
14 unit tangent and normal vectors
14 unit tangent and normal vectors14 unit tangent and normal vectors
14 unit tangent and normal vectors
math267
 

Mais procurados (20)

26 Machine Learning Unsupervised Fuzzy C-Means
26 Machine Learning Unsupervised Fuzzy C-Means26 Machine Learning Unsupervised Fuzzy C-Means
26 Machine Learning Unsupervised Fuzzy C-Means
 
Dynamic Programming - Matrix Chain Multiplication
Dynamic Programming - Matrix Chain MultiplicationDynamic Programming - Matrix Chain Multiplication
Dynamic Programming - Matrix Chain Multiplication
 
Calculus 10th edition anton solutions manual
Calculus 10th edition anton solutions manualCalculus 10th edition anton solutions manual
Calculus 10th edition anton solutions manual
 
MinFill_Presentation
MinFill_PresentationMinFill_Presentation
MinFill_Presentation
 
DISCRETE LOGARITHM PROBLEM
DISCRETE LOGARITHM PROBLEMDISCRETE LOGARITHM PROBLEM
DISCRETE LOGARITHM PROBLEM
 
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...
 
Sect2 1
Sect2 1Sect2 1
Sect2 1
 
3 grechnikov
3 grechnikov3 grechnikov
3 grechnikov
 
Daa unit 4
Daa unit 4Daa unit 4
Daa unit 4
 
Backtracking
BacktrackingBacktracking
Backtracking
 
013_20160328_Topological_Measurement_Of_Protein_Compressibility
013_20160328_Topological_Measurement_Of_Protein_Compressibility013_20160328_Topological_Measurement_Of_Protein_Compressibility
013_20160328_Topological_Measurement_Of_Protein_Compressibility
 
10 arc length parameter and curvature
10 arc length parameter and curvature10 arc length parameter and curvature
10 arc length parameter and curvature
 
Euclid's Algorithm for Greatest Common Divisor - Time Complexity Analysis
Euclid's Algorithm for Greatest Common Divisor - Time Complexity AnalysisEuclid's Algorithm for Greatest Common Divisor - Time Complexity Analysis
Euclid's Algorithm for Greatest Common Divisor - Time Complexity Analysis
 
Divergence center-based clustering and their applications
Divergence center-based clustering and their applicationsDivergence center-based clustering and their applications
Divergence center-based clustering and their applications
 
14 unit tangent and normal vectors
14 unit tangent and normal vectors14 unit tangent and normal vectors
14 unit tangent and normal vectors
 
Giáo trình Phân tích và thiết kế giải thuật - CHAP 8
Giáo trình Phân tích và thiết kế giải thuật - CHAP 8Giáo trình Phân tích và thiết kế giải thuật - CHAP 8
Giáo trình Phân tích và thiết kế giải thuật - CHAP 8
 
Tutorial of topological_data_analysis_part_1(basic)
Tutorial of topological_data_analysis_part_1(basic)Tutorial of topological_data_analysis_part_1(basic)
Tutorial of topological_data_analysis_part_1(basic)
 
Divergence clustering
Divergence clusteringDivergence clustering
Divergence clustering
 
Classification with mixtures of curved Mahalanobis metrics
Classification with mixtures of curved Mahalanobis metricsClassification with mixtures of curved Mahalanobis metrics
Classification with mixtures of curved Mahalanobis metrics
 
cyclic_code.pdf
cyclic_code.pdfcyclic_code.pdf
cyclic_code.pdf
 

Destaque

Inverse matrix pptx
Inverse matrix pptxInverse matrix pptx
Inverse matrix pptx
Kimguan Tan
 

Destaque (7)

Pc 8.5 notes_appsof_det
Pc 8.5 notes_appsof_detPc 8.5 notes_appsof_det
Pc 8.5 notes_appsof_det
 
Консалтинговое агентство "Дмира" / Consulting Agency "Dmira"
Консалтинговое агентство "Дмира" / Consulting Agency "Dmira"Консалтинговое агентство "Дмира" / Consulting Agency "Dmira"
Консалтинговое агентство "Дмира" / Consulting Agency "Dmira"
 
Pc 8.5 Notes Application of Determinants
Pc 8.5 Notes Application of DeterminantsPc 8.5 Notes Application of Determinants
Pc 8.5 Notes Application of Determinants
 
Matrices - Cramer's Rule
Matrices - Cramer's RuleMatrices - Cramer's Rule
Matrices - Cramer's Rule
 
Lecture 5 inverse of matrices - section 2-2 and 2-3
Lecture 5   inverse of matrices - section 2-2 and 2-3Lecture 5   inverse of matrices - section 2-2 and 2-3
Lecture 5 inverse of matrices - section 2-2 and 2-3
 
Cramer's Rule
Cramer's RuleCramer's Rule
Cramer's Rule
 
Inverse matrix pptx
Inverse matrix pptxInverse matrix pptx
Inverse matrix pptx
 

Semelhante a Lesson 7: The Derivative

Introdution to differential forms
Introdution to differential formsIntrodution to differential forms
Introdution to differential forms
Dunga Pessoa
 
T2311 - Ch 4_Part1.pptx
T2311 - Ch 4_Part1.pptxT2311 - Ch 4_Part1.pptx
T2311 - Ch 4_Part1.pptx
GadaFarhan
 
Algorithms - Rocksolid Tour 2013
Algorithms  - Rocksolid Tour 2013Algorithms  - Rocksolid Tour 2013
Algorithms - Rocksolid Tour 2013
Gary Short
 
Mock cat solutions paper no 1
Mock cat solutions paper no 1Mock cat solutions paper no 1
Mock cat solutions paper no 1
Vandan Kashyap
 
Divide-and-Conquer & Dynamic ProgrammingDivide-and-Conqu.docx
Divide-and-Conquer & Dynamic ProgrammingDivide-and-Conqu.docxDivide-and-Conquer & Dynamic ProgrammingDivide-and-Conqu.docx
Divide-and-Conquer & Dynamic ProgrammingDivide-and-Conqu.docx
jacksnathalie
 

Semelhante a Lesson 7: The Derivative (20)

Introdution to differential forms
Introdution to differential formsIntrodution to differential forms
Introdution to differential forms
 
T2311 - Ch 4_Part1.pptx
T2311 - Ch 4_Part1.pptxT2311 - Ch 4_Part1.pptx
T2311 - Ch 4_Part1.pptx
 
AppsDiff3c.pdf
AppsDiff3c.pdfAppsDiff3c.pdf
AppsDiff3c.pdf
 
Unit-1 DAA_Notes.pdf
Unit-1 DAA_Notes.pdfUnit-1 DAA_Notes.pdf
Unit-1 DAA_Notes.pdf
 
Algorithms - Rocksolid Tour 2013
Algorithms  - Rocksolid Tour 2013Algorithms  - Rocksolid Tour 2013
Algorithms - Rocksolid Tour 2013
 
CMSC 56 | Lecture 8: Growth of Functions
CMSC 56 | Lecture 8: Growth of FunctionsCMSC 56 | Lecture 8: Growth of Functions
CMSC 56 | Lecture 8: Growth of Functions
 
M112rev
M112revM112rev
M112rev
 
Big oh Representation Used in Time complexities
Big oh Representation Used in Time complexitiesBig oh Representation Used in Time complexities
Big oh Representation Used in Time complexities
 
Mock cat solutions paper no 1
Mock cat solutions paper no 1Mock cat solutions paper no 1
Mock cat solutions paper no 1
 
Theoryofcomp science
Theoryofcomp scienceTheoryofcomp science
Theoryofcomp science
 
time_complexity_list_02_04_2024_22_pages.pdf
time_complexity_list_02_04_2024_22_pages.pdftime_complexity_list_02_04_2024_22_pages.pdf
time_complexity_list_02_04_2024_22_pages.pdf
 
Newton's Laws of motion Lec2
Newton's Laws of motion Lec2Newton's Laws of motion Lec2
Newton's Laws of motion Lec2
 
Mechanics Class Notes
Mechanics Class NotesMechanics Class Notes
Mechanics Class Notes
 
Lesson 5: Tangents, Velocity, the Derivative
Lesson 5: Tangents, Velocity, the DerivativeLesson 5: Tangents, Velocity, the Derivative
Lesson 5: Tangents, Velocity, the Derivative
 
pradeepbishtLecture13 div conq
pradeepbishtLecture13 div conqpradeepbishtLecture13 div conq
pradeepbishtLecture13 div conq
 
3.-SEQUENCES-AND-SERIES-THEORY.hhsssspdf
3.-SEQUENCES-AND-SERIES-THEORY.hhsssspdf3.-SEQUENCES-AND-SERIES-THEORY.hhsssspdf
3.-SEQUENCES-AND-SERIES-THEORY.hhsssspdf
 
Divide-and-Conquer & Dynamic ProgrammingDivide-and-Conqu.docx
Divide-and-Conquer & Dynamic ProgrammingDivide-and-Conqu.docxDivide-and-Conquer & Dynamic ProgrammingDivide-and-Conqu.docx
Divide-and-Conquer & Dynamic ProgrammingDivide-and-Conqu.docx
 
Daa chapter 3
Daa chapter 3Daa chapter 3
Daa chapter 3
 
ECES302 Exam 1 Solutions
ECES302 Exam 1 SolutionsECES302 Exam 1 Solutions
ECES302 Exam 1 Solutions
 
differential-calculus-1-23.pdf
differential-calculus-1-23.pdfdifferential-calculus-1-23.pdf
differential-calculus-1-23.pdf
 

Mais de Matthew Leingang

Mais de Matthew Leingang (20)

Making Lesson Plans
Making Lesson PlansMaking Lesson Plans
Making Lesson Plans
 
Streamlining assessment, feedback, and archival with auto-multiple-choice
Streamlining assessment, feedback, and archival with auto-multiple-choiceStreamlining assessment, feedback, and archival with auto-multiple-choice
Streamlining assessment, feedback, and archival with auto-multiple-choice
 
Electronic Grading of Paper Assessments
Electronic Grading of Paper AssessmentsElectronic Grading of Paper Assessments
Electronic Grading of Paper Assessments
 
Lesson 27: Integration by Substitution (slides)
Lesson 27: Integration by Substitution (slides)Lesson 27: Integration by Substitution (slides)
Lesson 27: Integration by Substitution (slides)
 
Lesson 26: The Fundamental Theorem of Calculus (slides)
Lesson 26: The Fundamental Theorem of Calculus (slides)Lesson 26: The Fundamental Theorem of Calculus (slides)
Lesson 26: The Fundamental Theorem of Calculus (slides)
 
Lesson 26: The Fundamental Theorem of Calculus (slides)
Lesson 26: The Fundamental Theorem of Calculus (slides)Lesson 26: The Fundamental Theorem of Calculus (slides)
Lesson 26: The Fundamental Theorem of Calculus (slides)
 
Lesson 27: Integration by Substitution (handout)
Lesson 27: Integration by Substitution (handout)Lesson 27: Integration by Substitution (handout)
Lesson 27: Integration by Substitution (handout)
 
Lesson 26: The Fundamental Theorem of Calculus (handout)
Lesson 26: The Fundamental Theorem of Calculus (handout)Lesson 26: The Fundamental Theorem of Calculus (handout)
Lesson 26: The Fundamental Theorem of Calculus (handout)
 
Lesson 25: Evaluating Definite Integrals (slides)
Lesson 25: Evaluating Definite Integrals (slides)Lesson 25: Evaluating Definite Integrals (slides)
Lesson 25: Evaluating Definite Integrals (slides)
 
Lesson 25: Evaluating Definite Integrals (handout)
Lesson 25: Evaluating Definite Integrals (handout)Lesson 25: Evaluating Definite Integrals (handout)
Lesson 25: Evaluating Definite Integrals (handout)
 
Lesson 24: Areas and Distances, The Definite Integral (handout)
Lesson 24: Areas and Distances, The Definite Integral (handout)Lesson 24: Areas and Distances, The Definite Integral (handout)
Lesson 24: Areas and Distances, The Definite Integral (handout)
 
Lesson 24: Areas and Distances, The Definite Integral (slides)
Lesson 24: Areas and Distances, The Definite Integral (slides)Lesson 24: Areas and Distances, The Definite Integral (slides)
Lesson 24: Areas and Distances, The Definite Integral (slides)
 
Lesson 23: Antiderivatives (slides)
Lesson 23: Antiderivatives (slides)Lesson 23: Antiderivatives (slides)
Lesson 23: Antiderivatives (slides)
 
Lesson 23: Antiderivatives (slides)
Lesson 23: Antiderivatives (slides)Lesson 23: Antiderivatives (slides)
Lesson 23: Antiderivatives (slides)
 
Lesson 22: Optimization Problems (slides)
Lesson 22: Optimization Problems (slides)Lesson 22: Optimization Problems (slides)
Lesson 22: Optimization Problems (slides)
 
Lesson 22: Optimization Problems (handout)
Lesson 22: Optimization Problems (handout)Lesson 22: Optimization Problems (handout)
Lesson 22: Optimization Problems (handout)
 
Lesson 21: Curve Sketching (slides)
Lesson 21: Curve Sketching (slides)Lesson 21: Curve Sketching (slides)
Lesson 21: Curve Sketching (slides)
 
Lesson 21: Curve Sketching (handout)
Lesson 21: Curve Sketching (handout)Lesson 21: Curve Sketching (handout)
Lesson 21: Curve Sketching (handout)
 
Lesson 20: Derivatives and the Shapes of Curves (slides)
Lesson 20: Derivatives and the Shapes of Curves (slides)Lesson 20: Derivatives and the Shapes of Curves (slides)
Lesson 20: Derivatives and the Shapes of Curves (slides)
 
Lesson 20: Derivatives and the Shapes of Curves (handout)
Lesson 20: Derivatives and the Shapes of Curves (handout)Lesson 20: Derivatives and the Shapes of Curves (handout)
Lesson 20: Derivatives and the Shapes of Curves (handout)
 

Último

Jual Obat Aborsi Hongkong ( Asli No.1 ) 085657271886 Obat Penggugur Kandungan...
Jual Obat Aborsi Hongkong ( Asli No.1 ) 085657271886 Obat Penggugur Kandungan...Jual Obat Aborsi Hongkong ( Asli No.1 ) 085657271886 Obat Penggugur Kandungan...
Jual Obat Aborsi Hongkong ( Asli No.1 ) 085657271886 Obat Penggugur Kandungan...
ZurliaSoop
 
Russian Escort Service in Delhi 11k Hotel Foreigner Russian Call Girls in Delhi
Russian Escort Service in Delhi 11k Hotel Foreigner Russian Call Girls in DelhiRussian Escort Service in Delhi 11k Hotel Foreigner Russian Call Girls in Delhi
Russian Escort Service in Delhi 11k Hotel Foreigner Russian Call Girls in Delhi
kauryashika82
 
Seal of Good Local Governance (SGLG) 2024Final.pptx
Seal of Good Local Governance (SGLG) 2024Final.pptxSeal of Good Local Governance (SGLG) 2024Final.pptx
Seal of Good Local Governance (SGLG) 2024Final.pptx
negromaestrong
 
Spellings Wk 3 English CAPS CARES Please Practise
Spellings Wk 3 English CAPS CARES Please PractiseSpellings Wk 3 English CAPS CARES Please Practise
Spellings Wk 3 English CAPS CARES Please Practise
AnaAcapella
 
Activity 01 - Artificial Culture (1).pdf
Activity 01 - Artificial Culture (1).pdfActivity 01 - Artificial Culture (1).pdf
Activity 01 - Artificial Culture (1).pdf
ciinovamais
 

Último (20)

2024-NATIONAL-LEARNING-CAMP-AND-OTHER.pptx
2024-NATIONAL-LEARNING-CAMP-AND-OTHER.pptx2024-NATIONAL-LEARNING-CAMP-AND-OTHER.pptx
2024-NATIONAL-LEARNING-CAMP-AND-OTHER.pptx
 
Dyslexia AI Workshop for Slideshare.pptx
Dyslexia AI Workshop for Slideshare.pptxDyslexia AI Workshop for Slideshare.pptx
Dyslexia AI Workshop for Slideshare.pptx
 
microwave assisted reaction. General introduction
microwave assisted reaction. General introductionmicrowave assisted reaction. General introduction
microwave assisted reaction. General introduction
 
Unit-IV; Professional Sales Representative (PSR).pptx
Unit-IV; Professional Sales Representative (PSR).pptxUnit-IV; Professional Sales Representative (PSR).pptx
Unit-IV; Professional Sales Representative (PSR).pptx
 
Jual Obat Aborsi Hongkong ( Asli No.1 ) 085657271886 Obat Penggugur Kandungan...
Jual Obat Aborsi Hongkong ( Asli No.1 ) 085657271886 Obat Penggugur Kandungan...Jual Obat Aborsi Hongkong ( Asli No.1 ) 085657271886 Obat Penggugur Kandungan...
Jual Obat Aborsi Hongkong ( Asli No.1 ) 085657271886 Obat Penggugur Kandungan...
 
ICT Role in 21st Century Education & its Challenges.pptx
ICT Role in 21st Century Education & its Challenges.pptxICT Role in 21st Century Education & its Challenges.pptx
ICT Role in 21st Century Education & its Challenges.pptx
 
Key note speaker Neum_Admir Softic_ENG.pdf
Key note speaker Neum_Admir Softic_ENG.pdfKey note speaker Neum_Admir Softic_ENG.pdf
Key note speaker Neum_Admir Softic_ENG.pdf
 
Russian Escort Service in Delhi 11k Hotel Foreigner Russian Call Girls in Delhi
Russian Escort Service in Delhi 11k Hotel Foreigner Russian Call Girls in DelhiRussian Escort Service in Delhi 11k Hotel Foreigner Russian Call Girls in Delhi
Russian Escort Service in Delhi 11k Hotel Foreigner Russian Call Girls in Delhi
 
Third Battle of Panipat detailed notes.pptx
Third Battle of Panipat detailed notes.pptxThird Battle of Panipat detailed notes.pptx
Third Battle of Panipat detailed notes.pptx
 
Mixin Classes in Odoo 17 How to Extend Models Using Mixin Classes
Mixin Classes in Odoo 17  How to Extend Models Using Mixin ClassesMixin Classes in Odoo 17  How to Extend Models Using Mixin Classes
Mixin Classes in Odoo 17 How to Extend Models Using Mixin Classes
 
How to Create and Manage Wizard in Odoo 17
How to Create and Manage Wizard in Odoo 17How to Create and Manage Wizard in Odoo 17
How to Create and Manage Wizard in Odoo 17
 
Seal of Good Local Governance (SGLG) 2024Final.pptx
Seal of Good Local Governance (SGLG) 2024Final.pptxSeal of Good Local Governance (SGLG) 2024Final.pptx
Seal of Good Local Governance (SGLG) 2024Final.pptx
 
Spellings Wk 3 English CAPS CARES Please Practise
Spellings Wk 3 English CAPS CARES Please PractiseSpellings Wk 3 English CAPS CARES Please Practise
Spellings Wk 3 English CAPS CARES Please Practise
 
Food safety_Challenges food safety laboratories_.pdf
Food safety_Challenges food safety laboratories_.pdfFood safety_Challenges food safety laboratories_.pdf
Food safety_Challenges food safety laboratories_.pdf
 
PROCESS RECORDING FORMAT.docx
PROCESS      RECORDING        FORMAT.docxPROCESS      RECORDING        FORMAT.docx
PROCESS RECORDING FORMAT.docx
 
Explore beautiful and ugly buildings. Mathematics helps us create beautiful d...
Explore beautiful and ugly buildings. Mathematics helps us create beautiful d...Explore beautiful and ugly buildings. Mathematics helps us create beautiful d...
Explore beautiful and ugly buildings. Mathematics helps us create beautiful d...
 
Making communications land - Are they received and understood as intended? we...
Making communications land - Are they received and understood as intended? we...Making communications land - Are they received and understood as intended? we...
Making communications land - Are they received and understood as intended? we...
 
Introduction to Nonprofit Accounting: The Basics
Introduction to Nonprofit Accounting: The BasicsIntroduction to Nonprofit Accounting: The Basics
Introduction to Nonprofit Accounting: The Basics
 
This PowerPoint helps students to consider the concept of infinity.
This PowerPoint helps students to consider the concept of infinity.This PowerPoint helps students to consider the concept of infinity.
This PowerPoint helps students to consider the concept of infinity.
 
Activity 01 - Artificial Culture (1).pdf
Activity 01 - Artificial Culture (1).pdfActivity 01 - Artificial Culture (1).pdf
Activity 01 - Artificial Culture (1).pdf
 

Lesson 7: The Derivative

  • 1. . . . . . . Section 2.1 The Derivative and Rates of Change V63.0121.034, Calculus I September 23, 2009 Announcements WebAssignments due Monday. Email me if you need an extension for Yom Kippur.
  • 2. . . . . . . Regarding WebAssign We feel your pain
  • 3. . . . . . . Explanations From the syllabus: Graders will be expecting you to express your ideas clearly, legibly, and completely, often requiring complete English sentences rather than merely just a long string of equations or unconnected mathematical expressions. This means you could lose points for unexplained answers.
  • 4. . . . . . . Rubric Points Description of Work 3 Work is completely accurate and essentially perfect. Work is thoroughly developed, neat, and easy to read. Complete sentences are used. 2 Work is good, but incompletely developed, hard to read, unexplained, or jumbled. Answers which are not explained, even if correct, will generally receive 2 points. Work contains “right idea” but is flawed. 1 Work is sketchy. There is some correct work, but most of work is incorrect. 0 Work minimal or non-existent. Solution is completely incorrect.
  • 5. . . . . . . Outline Rates of Change Tangent Lines Velocity Population growth Marginal costs The derivative, defined Derivatives of (some) power functions What does f tell you about f′ ? How can a function fail to be differentiable? Other notations The second derivative
  • 6. . . . . . . The tangent problem Problem Given a curve and a point on the curve, find the slope of the line tangent to the curve at that point.
  • 7. . . . . . . The tangent problem Problem Given a curve and a point on the curve, find the slope of the line tangent to the curve at that point. Example Find the slope of the line tangent to the curve y = x2 at the point (2, 4).
  • 8. . . . . . . Graphically and numerically . .x .y . .2 ..4 . x m
  • 9. . . . . . . Graphically and numerically . .x .y . .2 ..4 . . . .3 ..9 x m 3 5
  • 10. . . . . . . Graphically and numerically . .x .y . .2 ..4 . . . .2.5 ..6.25 x m 3 5 2.5 4.25
  • 11. . . . . . . Graphically and numerically . .x .y . .2 ..4 . . . .2.1 ..4.41 x m 3 5 2.5 4.25 2.1 4.1
  • 12. . . . . . . Graphically and numerically . .x .y . .2 ..4 .. . .2.01 ..4.0401 x m 3 5 2.5 4.25 2.1 4.1 2.01 4.01
  • 13. . . . . . . Graphically and numerically . .x .y . .2 ..4 . . . .1 ..1 x m 3 5 2.5 4.25 2.1 4.1 2.01 4.01 1 3
  • 14. . . . . . . Graphically and numerically . .x .y . .2 ..4 . . . .1.5 ..2.25 x m 3 5 2.5 4.25 2.1 4.1 2.01 4.01 1.5 3.5 1 3
  • 15. . . . . . . Graphically and numerically . .x .y . .2 ..4 . . . .1.9 ..3.61 x m 3 5 2.5 4.25 2.1 4.1 2.01 4.01 1.9 3.9 1.5 3.5 1 3
  • 16. . . . . . . Graphically and numerically . .x .y . .2 ..4 .. . .1.99 ..3.9601 x m 3 5 2.5 4.25 2.1 4.1 2.01 4.01 1.99 3.99 1.9 3.9 1.5 3.5 1 3
  • 17. . . . . . . Graphically and numerically . .x .y . .2 ..4 . . . .3 ..9 . . .2.5 ..6.25 . . .2.1 ..4.41 . . .2.01 ..4.0401 . . .1 ..1 . . .1.5 ..2.25 . . .1.9 ..3.61 . . .1.99 ..3.9601 x m 3 5 2.5 4.25 2.1 4.1 2.01 4.01 limit 4 1.99 3.99 1.9 3.9 1.5 3.5 1 3
  • 18. . . . . . . The tangent problem Problem Given a curve and a point on the curve, find the slope of the line tangent to the curve at that point. Example Find the slope of the line tangent to the curve y = x2 at the point (2, 4). Upshot If the curve is given by y = f(x), and the point on the curve is (a, f(a)), then the slope of the tangent line is given by mtangent = lim x→a f(x) − f(a) x − a
  • 19. . . . . . . Velocity Problem Given the position function of a moving object, find the velocity of the object at a certain instant in time. Example Drop a ball off the roof of the Silver Center so that its height can be described by h(t) = 50 − 5t2 where t is seconds after dropping it and h is meters above the ground. How fast is it falling one second after we drop it?
  • 20. . . . . . . Numerical evidence t vave = h(t) − h(1) t − 1 2 − 15
  • 21. . . . . . . Numerical evidence t vave = h(t) − h(1) t − 1 2 − 15 1.5
  • 22. . . . . . . Numerical evidence t vave = h(t) − h(1) t − 1 2 − 15 1.5 − 12.5
  • 23. . . . . . . Numerical evidence t vave = h(t) − h(1) t − 1 2 − 15 1.5 − 12.5 1.1
  • 24. . . . . . . Numerical evidence t vave = h(t) − h(1) t − 1 2 − 15 1.5 − 12.5 1.1 − 10.5
  • 25. . . . . . . Numerical evidence t vave = h(t) − h(1) t − 1 2 − 15 1.5 − 12.5 1.1 − 10.5 1.01
  • 26. . . . . . . Numerical evidence t vave = h(t) − h(1) t − 1 2 − 15 1.5 − 12.5 1.1 − 10.5 1.01 − 10.05
  • 27. . . . . . . Numerical evidence t vave = h(t) − h(1) t − 1 2 − 15 1.5 − 12.5 1.1 − 10.5 1.01 − 10.05 1.001
  • 28. . . . . . . Numerical evidence t vave = h(t) − h(1) t − 1 2 − 15 1.5 − 12.5 1.1 − 10.5 1.01 − 10.05 1.001 − 10.005
  • 29. . . . . . . Velocity Problem Given the position function of a moving object, find the velocity of the object at a certain instant in time. Example Drop a ball off the roof of the Silver Center so that its height can be described by h(t) = 50 − 5t2 where t is seconds after dropping it and h is meters above the ground. How fast is it falling one second after we drop it? Solution The answer is v = lim t→1 (50 − 5t2) − 45 t − 1 = lim t→1 5 − 5t2 t − 1 = lim t→1 5(1 − t)(1 + t) t − 1 = (−5) lim t→1 (1 + t) = −5 · 2 = −10
  • 30. . . . . . . Upshot If the height function is given by h(t), the instantaneous velocity at time t0 is given by v = lim t→t0 h(t) − h(t0) t − t0 = lim ∆t→0 h(t0 + ∆t) − h(t0) ∆t . .t .y = h(t) . . . .t0 . .t .∆t
  • 31. . . . . . . Population growth Problem Given the population function of a group of organisms, find the rate of growth of the population at a particular instant.
  • 32. . . . . . . Population growth Problem Given the population function of a group of organisms, find the rate of growth of the population at a particular instant. Example Suppose the population of fish in the East River is given by the function P(t) = 3et 1 + et where t is in years since 2000 and P is in millions of fish. Is the fish population growing fastest in 1990, 2000, or 2010? (Estimate numerically)?
  • 33. . . . . . . Derivation Let ∆t be an increment in time and ∆P the corresponding change in population: ∆P = P(t + ∆t) − P(t) This depends on ∆t, so we want lim ∆t→0 ∆P ∆t = lim ∆t→0 1 ∆t ( 3et+∆t 1 + et+∆t − 3et 1 + et )
  • 34. . . . . . . Derivation Let ∆t be an increment in time and ∆P the corresponding change in population: ∆P = P(t + ∆t) − P(t) This depends on ∆t, so we want lim ∆t→0 ∆P ∆t = lim ∆t→0 1 ∆t ( 3et+∆t 1 + et+∆t − 3et 1 + et ) Too hard! Try a small ∆t to approximate.
  • 35. . . . . . . Numerical evidence r1990 ≈ P(−10 + 0.1) − P(−10) 0.1 ≈
  • 36. . . . . . . Numerical evidence r1990 ≈ P(−10 + 0.1) − P(−10) 0.1 ≈ 0.000136
  • 37. . . . . . . Numerical evidence r1990 ≈ P(−10 + 0.1) − P(−10) 0.1 ≈ 0.000136 r2000 ≈ P(0.1) − P(0) 0.1 ≈
  • 38. . . . . . . Numerical evidence r1990 ≈ P(−10 + 0.1) − P(−10) 0.1 ≈ 0.000136 r2000 ≈ P(0.1) − P(0) 0.1 ≈ 0.75
  • 39. . . . . . . Numerical evidence r1990 ≈ P(−10 + 0.1) − P(−10) 0.1 ≈ 0.000136 r2000 ≈ P(0.1) − P(0) 0.1 ≈ 0.75 r2010 ≈ P(10 + 0.1) − P(10) 0.1 ≈
  • 40. . . . . . . Numerical evidence r1990 ≈ P(−10 + 0.1) − P(−10) 0.1 ≈ 0.000136 r2000 ≈ P(0.1) − P(0) 0.1 ≈ 0.75 r2010 ≈ P(10 + 0.1) − P(10) 0.1 ≈ 0.000136
  • 41. . . . . . . Population growth Problem Given the population function of a group of organisms, find the rate of growth of the population at a particular instant. Example Suppose the population of fish in the East River is given by the function P(t) = 3et 1 + et where t is in years since 2000 and P is in millions of fish. Is the fish population growing fastest in 1990, 2000, or 2010? (Estimate numerically)? Solution The estimated rates of growth are 0.000136, 0.75, and 0.000136.
  • 42. . . . . . . Upshot The instantaneous population growth is given by lim ∆t→0 P(t + ∆t) − P(t) ∆t
  • 43. . . . . . . Marginal costs Problem Given the production cost of a good, find the marginal cost of production after having produced a certain quantity.
  • 44. . . . . . . Marginal costs Problem Given the production cost of a good, find the marginal cost of production after having produced a certain quantity. Example Suppose the cost of producing q tons of rice on our paddy in a year is C(q) = q3 − 12q2 + 60q We are currently producing 5 tons a year. Should we change that?
  • 45. . . . . . . Comparisons q C(q) 4 5 6
  • 46. . . . . . . Comparisons q C(q) 4 112 5 6
  • 47. . . . . . . Comparisons q C(q) 4 112 5 125 6
  • 48. . . . . . . Comparisons q C(q) 4 112 5 125 6 144
  • 49. . . . . . . Comparisons q C(q) AC(q) = C(q)/q 4 112 5 125 6 144
  • 50. . . . . . . Comparisons q C(q) AC(q) = C(q)/q 4 112 28 5 125 6 144
  • 51. . . . . . . Comparisons q C(q) AC(q) = C(q)/q 4 112 28 5 125 25 6 144
  • 52. . . . . . . Comparisons q C(q) AC(q) = C(q)/q 4 112 28 5 125 25 6 144 24
  • 53. . . . . . . Comparisons q C(q) AC(q) = C(q)/q ∆C = C(q + 1) − C(q) 4 112 28 5 125 25 6 144 24
  • 54. . . . . . . Comparisons q C(q) AC(q) = C(q)/q ∆C = C(q + 1) − C(q) 4 112 28 13 5 125 25 6 144 24
  • 55. . . . . . . Comparisons q C(q) AC(q) = C(q)/q ∆C = C(q + 1) − C(q) 4 112 28 13 5 125 25 19 6 144 24
  • 56. . . . . . . Comparisons q C(q) AC(q) = C(q)/q ∆C = C(q + 1) − C(q) 4 112 28 13 5 125 25 19 6 144 24 31
  • 57. . . . . . . Marginal costs Problem Given the production cost of a good, find the marginal cost of production after having produced a certain quantity. Example Suppose the cost of producing q tons of rice on our paddy in a year is C(q) = q3 − 12q2 + 60q We are currently producing 5 tons a year. Should we change that? Example If q = 5, then C = 125, ∆C = 19, while AC = 25. So we should produce more to lower average costs.
  • 58. . . . . . . Upshot The incremental cost ∆C = C(q + 1) − C(q) is useful, but depends on units.
  • 59. . . . . . . Upshot The incremental cost ∆C = C(q + 1) − C(q) is useful, but depends on units. The marginal cost after producing q given by MC = lim ∆q→0 C(q + ∆q) − C(q) ∆q is more useful since it’s unit-independent.
  • 60. . . . . . . Outline Rates of Change Tangent Lines Velocity Population growth Marginal costs The derivative, defined Derivatives of (some) power functions What does f tell you about f′ ? How can a function fail to be differentiable? Other notations The second derivative
  • 61. . . . . . . The definition All of these rates of change are found the same way!
  • 62. . . . . . . The definition All of these rates of change are found the same way! Definition Let f be a function and a a point in the domain of f. If the limit f′ (a) = lim h→0 f(a + h) − f(a) h exists, the function is said to be differentiable at a and f′ (a) is the derivative of f at a.
  • 63. . . . . . . Derivative of the squaring function Example Suppose f(x) = x2 . Use the definition of derivative to find f′ (a).
  • 64. . . . . . . Derivative of the squaring function Example Suppose f(x) = x2 . Use the definition of derivative to find f′ (a). Solution f′ (a) = lim h→0 f(a + h) − f(a) h = lim h→0 (a + h)2 − a2 h = lim h→0 (a2 + 2ah + h2 ) − a2 h = lim h→0 2ah + h2 h = lim h→0 (2a + h) = 2a.
  • 65. . . . . . . Derivative of the reciprocal function Example Suppose f(x) = 1 x . Use the definition of the derivative to find f′ (2).
  • 66. . . . . . . Derivative of the reciprocal function Example Suppose f(x) = 1 x . Use the definition of the derivative to find f′ (2). Solution f′ (2) = lim x→2 1/x − 1/2 x − 2 = lim x→2 2 − x 2x(x − 2) = lim x→2 −1 2x = − 1 4 . .x .x .
  • 67. . . . . . . The Sure-Fire Sally Rule (SFSR) for adding Fractions In anticipation of the question, “How did you get that?” a b ± c d = ad ± bc bd So 1 x − 1 2 x − 2 = 2 − x 2x x − 2 = 2 − x 2x(x − 2)
  • 68. . . . . . . The Sure-Fire Sally Rule (SFSR) for adding Fractions In anticipation of the question, “How did you get that?” a b ± c d = ad ± bc bd So 1 x − 1 2 x − 2 = 2 − x 2x x − 2 = 2 − x 2x(x − 2)
  • 69. . . . . . . What does f tell you about f′ ? If f is a function, we can compute the derivative f′ (x) at each point x where f is differentiable, and come up with another function, the derivative function. What can we say about this function f′ ?
  • 70. . . . . . . What does f tell you about f′ ? If f is a function, we can compute the derivative f′ (x) at each point x where f is differentiable, and come up with another function, the derivative function. What can we say about this function f′ ? If f is decreasing on an interval, f′ is negative (well, nonpositive) on that interval
  • 71. . . . . . . Derivative of the reciprocal function Example Suppose f(x) = 1 x . Use the definition of the derivative to find f′ (2). Solution f′ (2) = lim x→2 1/x − 1/2 x − 2 = lim x→2 2 − x 2x(x − 2) = lim x→2 −1 2x = − 1 4 . .x .x .
  • 72. . . . . . . What does f tell you about f′ ? If f is a function, we can compute the derivative f′ (x) at each point x where f is differentiable, and come up with another function, the derivative function. What can we say about this function f′ ? If f is decreasing on an interval, f′ is negative (well, nonpositive) on that interval If f is increasing on an interval, f′ is positive (well, nonnegative) on that interval
  • 73. . . . . . . Graphically and numerically . .x .y . .2 ..4 . . . .3 ..9 . . .2.5 ..6.25 . . .2.1 ..4.41 . . .2.01 ..4.0401 . . .1 ..1 . . .1.5 ..2.25 . . .1.9 ..3.61 . . .1.99 ..3.9601 x m 3 5 2.5 4.25 2.1 4.1 2.01 4.01 limit 4 1.99 3.99 1.9 3.9 1.5 3.5 1 3
  • 74. . . . . . . What does f tell you about f′ ? Fact If f is decreasing on (a, b), then f′ ≤ 0 on (a, b). Proof. If f is decreasing on (a, b), and ∆x > 0, then f(x + ∆x) < f(x) =⇒ f(x + ∆x) − f(x) ∆x < 0
  • 75. . . . . . . What does f tell you about f′ ? Fact If f is decreasing on (a, b), then f′ ≤ 0 on (a, b). Proof. If f is decreasing on (a, b), and ∆x > 0, then f(x + ∆x) < f(x) =⇒ f(x + ∆x) − f(x) ∆x < 0 But if ∆x < 0, then x + ∆x < x, and f(x + ∆x) > f(x) =⇒ f(x + ∆x) − f(x) ∆x < 0 still!
  • 76. . . . . . . What does f tell you about f′ ? Fact If f is decreasing on (a, b), then f′ ≤ 0 on (a, b). Proof. If f is decreasing on (a, b), and ∆x > 0, then f(x + ∆x) < f(x) =⇒ f(x + ∆x) − f(x) ∆x < 0 But if ∆x < 0, then x + ∆x < x, and f(x + ∆x) > f(x) =⇒ f(x + ∆x) − f(x) ∆x < 0 still! Either way, f(x + ∆x) − f(x) ∆x < 0, so f′ (x) = lim ∆x→0 f(x + ∆x) − f(x) ∆x ≤ 0
  • 77. . . . . . . Outline Rates of Change Tangent Lines Velocity Population growth Marginal costs The derivative, defined Derivatives of (some) power functions What does f tell you about f′ ? How can a function fail to be differentiable? Other notations The second derivative
  • 78. . . . . . . Differentiability is super-continuity Theorem If f is differentiable at a, then f is continuous at a.
  • 79. . . . . . . Differentiability is super-continuity Theorem If f is differentiable at a, then f is continuous at a. Proof. We have lim x→a (f(x) − f(a)) = lim x→a f(x) − f(a) x − a · (x − a) = lim x→a f(x) − f(a) x − a · lim x→a (x − a) = f′ (a) · 0 = 0
  • 80. . . . . . . Differentiability is super-continuity Theorem If f is differentiable at a, then f is continuous at a. Proof. We have lim x→a (f(x) − f(a)) = lim x→a f(x) − f(a) x − a · (x − a) = lim x→a f(x) − f(a) x − a · lim x→a (x − a) = f′ (a) · 0 = 0 Note the proper use of the limit law: if the factors each have a limit at a, the limit of the product is the product of the limits.
  • 81. . . . . . . How can a function fail to be differentiable? Kinks . .x .f(x)
  • 82. . . . . . . How can a function fail to be differentiable? Kinks . .x .f(x) . .x .f′ (x) .
  • 83. . . . . . . How can a function fail to be differentiable? Kinks . .x .f(x) . .x .f′ (x) . .
  • 84. . . . . . . How can a function fail to be differentiable? Cusps . .x .f(x)
  • 85. . . . . . . How can a function fail to be differentiable? Cusps . .x .f(x) . .x .f′ (x)
  • 86. . . . . . . How can a function fail to be differentiable? Cusps . .x .f(x) . .x .f′ (x)
  • 87. . . . . . . How can a function fail to be differentiable? Vertical Tangents . .x .f(x)
  • 88. . . . . . . How can a function fail to be differentiable? Vertical Tangents . .x .f(x) . .x .f′ (x)
  • 89. . . . . . . How can a function fail to be differentiable? Vertical Tangents . .x .f(x) . .x .f′ (x)
  • 90. . . . . . . How can a function fail to be differentiable? Weird, Wild, Stuff . .x .f(x) This function is differentiable at 0.
  • 91. . . . . . . How can a function fail to be differentiable? Weird, Wild, Stuff . .x .f(x) This function is differentiable at 0. . .x .f′ (x) But the derivative is not continuous at 0!
  • 92. . . . . . . Outline Rates of Change Tangent Lines Velocity Population growth Marginal costs The derivative, defined Derivatives of (some) power functions What does f tell you about f′ ? How can a function fail to be differentiable? Other notations The second derivative
  • 93. . . . . . . Notation Newtonian notation f′ (x) y′ (x) y′ Leibnizian notation dy dx d dx f(x) df dx These all mean the same thing.
  • 94. . . . . . . Meet the Mathematician: Isaac Newton English, 1643–1727 Professor at Cambridge (England) Philosophiae Naturalis Principia Mathematica published 1687
  • 95. . . . . . . Meet the Mathematician: Gottfried Leibniz German, 1646–1716 Eminent philosopher as well as mathematician Contemporarily disgraced by the calculus priority dispute
  • 96. . . . . . . Outline Rates of Change Tangent Lines Velocity Population growth Marginal costs The derivative, defined Derivatives of (some) power functions What does f tell you about f′ ? How can a function fail to be differentiable? Other notations The second derivative
  • 97. . . . . . . The second derivative If f is a function, so is f′ , and we can seek its derivative. f′′ = (f′ )′ It measures the rate of change of the rate of change!
  • 98. . . . . . . The second derivative If f is a function, so is f′ , and we can seek its derivative. f′′ = (f′ )′ It measures the rate of change of the rate of change! Leibnizian notation: d2 y dx2 d2 dx2 f(x) d2 f dx2
  • 99. . . . . . . function, derivative, second derivative . .x .y .f(x) = x2 .f′ (x) = 2x .f′′ (x) = 2