text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
laws for degree distributions for
networks indicate the existence of a certain kind of structure
for the network?
• No, power laws are consistent with a wide variety of
networks having various structures and some without
central hubs (Li et al)
• Moreover, power laws are the equivalent of normal
distributions at h... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
existence of a power law for degree distributions for
networks indicate existence of a specific mechanism for
formation?
• No, power laws are consistent with a wide variety of
mechanisms for network formation (Newman, “Power
laws, Pareto distributions and Zipf’s law”2004/5)
• Does the existence of power laws for d... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
-degree and
an out-degree and the degree distribution becomes a
function of two variables (j and k for in and out
degrees). Since in and out degrees can be strongly
correlated, the joint distribution also contains
information about the network.
• Maximum Degree (Power Law)
k
max ~
/(1
)1
−αn
Professor C. Magee, 20... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
enlightenment
(the 2 transitivity coefficients example)
Professor C. Magee, 2006
Page 46
References for Lecture 6
• Overall key references
• Wasserman and Faust, Social Network Analysis: Methods and
Applications, Cambridge University Press (1994)
• M. E. J. Newman, “The structure and function of complex networks”
SI... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
18.034 ;
Feb 6, 2004
Lecture 2
1. Set-up model for a mixing problem
Rate of mass of chemical in
=(conc. in)× (rate of flow of liquid)= a.c(t)
Rate of mass out = (conc. out) × (rate of flow) =
q ⋅
So
y
='
( )
tca
⋅
−
a
v
y
'
y
+
a
v
y
=
( )tca
.
where a,
0>V
are constants
( )
ty
v
.
2. Discussed method of ... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/3f60608b2696971c5f2fea7198a1b0e5_lec2.pdf |
form.
of
( )
( )tqtp ,
( )
+'
ytpy
are defined and cts. on (
defined on all of (
( )tq
)ba, ⊂ IR , then there exists a
)ba,
, the solution is unique,
=
( )
ty
=
e
( )
tp
−
t
∫
0
e
sp
( ) ( )
dssq
+
ey
0
−
tp
)(
,
where
3. Used this method to solve the mixing problem:
( )tp
( )
'
tp
=
( ) 0
0 =
p
( )
ty
=
e
α
v... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/3f60608b2696971c5f2fea7198a1b0e5_lec2.pdf |
=λ , get
v
2
ωλ
+
analyze the solution.
2
aA
t
λ
e
sin
)
(
tan,
φω
−
t
( )
φ
=
ω
λ
. Didn’t have time to really
( )
ty
=
aA
2
2
ωλ
+
sin
(
)
φω
−
t
+
be
−
t
λ
for some b
4. Particular solution method. To find the general solution of
( )
(i) Find general solution of undriven/ homo system
ytp
(ii) Find a particul... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/3f60608b2696971c5f2fea7198a1b0e5_lec2.pdf |
Key Concepts for section IV (Electrokinetics and Forces)
1: Debye layer, Zeta potential, Electrokinetics
2: Electrophoresis, Electroosmosis
3: Dielectrophoresis
4: Inter-Debye layer force, Van-Der Waals forces
5: Coupled systems, Scaling, Dimensionless Number
Goals of Part IV:
(1) Understand electrokinetic phenomena... | https://ocw.mit.edu/courses/20-330j-fields-forces-and-flows-in-biological-systems-spring-2007/3f79682d8f684ab43e9dbd4208737945_electrokin_lec2.pdf |
Source: Prof. Han’s Ph.D thesis
DNA electrophoresis in a channel
)
c
e
s
/
m
μ
(
y
t
i
c
o
e
V
l
40
20
0
-20
-40
-60
Velocity of DNA in obstacle-free channel
10X TBE
0
0.25
0.5
0.75
1
0.5X TBE
T7
T2
Buffer concentration(M)
Electroosmosis
• The oxide or glass surface
become unprotonated (pK ~ 2)
when they are in c... | https://ocw.mit.edu/courses/20-330j-fields-forces-and-flows-in-biological-systems-spring-2007/3f79682d8f684ab43e9dbd4208737945_electrokin_lec2.pdf |
ates
E-field (ΔΦ)
+
v
E field (ΔΦ)
generates
particle
motion (v)
+
+
+
+ +
+
+
+
v
E
v
+
+
+
+ +
+
+
+
v
E
E
v
particle
motion (v)
generates
E-field (ΔΦ)
Electrophoresis
Sedimentation potential
Figure by MIT OCW.
Fick’s law of diffusion
Maxwell’s equation
Concentration(c)
(ρ)
Electrophoresis
ρ, J : source
E and B ... | https://ocw.mit.edu/courses/20-330j-fields-forces-and-flows-in-biological-systems-spring-2007/3f79682d8f684ab43e9dbd4208737945_electrokin_lec2.pdf |
p
r
v
0 (
incompressible
)
( )
v r
z
= −
2
ε
(
μ
ζ
− Φ
)
( )
r E
P
zo
L
144424443 1442443
⎞
⎟
⎠
Poiseuille flow
⎛
− ⎜
⎝
electroosmotic flow
R
−
r
μ
Δ
4
2
Δ ≠
P
0,
E
z
=
0 :
Poiseuille flow
parabolic flow profile
Δ =
P
0,
E
z
≠
0 :
Electroosmotic flow
flat (plug-like) profile
v
EEO
= −
εζ
μ
=
E
z
μ
EEO
E (outside of th... | https://ocw.mit.edu/courses/20-330j-fields-forces-and-flows-in-biological-systems-spring-2007/3f79682d8f684ab43e9dbd4208737945_electrokin_lec2.pdf |
18.465 March 29, 2005, revised May 2
M-estimators and their consistency
This handout is adapted from Section 3.3 of 18.466 lecture notes on mathematical
statistics, available on OCW.
A sequence of estimators Tn , one for each sample size n, possibly only defined for
n large enough, is called consistent if for X1, X... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
variables with values in X and distribution
P , specifically, coordinates on the countable product (X ∞ , A∞, P ∞) of copies of (X, A, P )
(RAP, Sec. 8.2). A statistic Tn = Tn(X1, ..., Xn) with values in Θ will be called an M-
estimator if
�n
1
n
h(Tn , Xi) = inf θ∈Θ
1
n
i=1
h(θ, Xi).
i=1
�n
Thus, in the log li... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
a list of as-
sumptions as follows.
(A-1) h(θ, x) is a separable stochastic process, meaning that there is a set A ⊂ X with
P (A) = 0 and a countable subset S ⊂ Θ such that for every open set U ⊂ Θ and every
closed set J ⊂ [−∞, ∞],
{x : h(θ, x) ∈ J for all θ ∈ S ∩ U } ⊂ A ∪ {x : h(θ, x) ∈ J for all θ ∈ U }.
1
Th... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
is unknown,
the separability is more clearly attainable in case h has at least a one-sided continuity
property as just mentioned.
Instead of continuity, here is a weaker assumption:
(A-2) For each x in X, the function h(·, x) is lower semicontinuous on Θ, meaning that
h(θ, x) ≤ lim inf φ→θ h(φ, x) for all θ.
Ofte... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
called adjusted for P if
(3.3.2)
(3.3.3)
Eh(θ, x)− < ∞ for all θ ∈ Θ, and
Eh(θ, x)+ < ∞ for some θ ∈ Θ.
To say that h is adjusted is equivalent to saying that Eh(θ, ·) is well-defined (possibly +∞)
and not −∞ for all θ, and for some θ, also Eh(θ, ·) < +∞, so it is some finite real number.
If a(·) is a measurable r... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
|θ|), so γ(θ) is defined and finite for all θ. Thus |x| is in fact an
adjustment function for any P .
:= γa(θ)
2
i=1
The example illustrates an idea of Huber (1967,1981) who seems to have invented the
notion of adjustment. An estimator is defined by minimizing or approximately minimizing
�n h(θ, Xi). If ∫ h(θ, x)dP ... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
“If” is clear. To prove “only if,” we have E((h(θ, x) − ai(x))−) < ∞ for all θ and
i = 1, 2, while E((h(θi, x) − ai(x))+) < ∞ for some θi and i = 1, 2. We can write for θ = θ1
or θ2,
(a1 − a2)(x) = h(θ, x) − a2(x) − [h(θ, x) − a1(x)]
for P -almost all x. To check this we need to take account that h can have values ... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
(A-4) There is a θ0 ∈ Θ such that γ(θ) > γ(θ0) for all θ (cid:12)= θ0.
Here θ0 is called the M-functional of P . In the log likelihood case it is sometimes
called the pseudo-true value of θ. Then h(θ, x) = − log f (θ, x) where for fixed θ, f is a
density or probability mass function for a probability measure Pθ. The ... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
> γa(θ0) and
E{lim inf θ→∞ (h(θ, x) − a(x))/b(θ)} ≥ 1.
This completes the list of assumptions. Here (3.3.5) and (3.3.7) may depend on the
choice of adjustment function. In the example where X = Θ = R, h(θ, x) = |x − θ| and
a(x) := |x|, all the assumptions hold if b(θ) := |θ| + 1 and P is any law on R with a
unique... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
of θ converges to {θ},
E(inf{h(φ, x) − a(x) : φ ∈ Uk }) → γ(θ) ≤ +∞.
Proof. Separability (A-1) applied to sets J = [q, +∞) for all rational q and joint measur-
ability of h(·, ·) imply that the infimum in (A-2(cid:6)) is equal almost surely to a measurable
function of x. By (A-2), the integrand on the left converges ... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
(3.3.1), the definition of approximate M-estimator, is not affected by sub-
tracting a(x) from h(θ, x).
By the alternate formulation given for separability (A-1), h(θ, x) − a(x) is separable
and since b(θ) is continuous and strictly positive, (h(θ, x) − a(x))/b(θ) is also separable.
For any adjustable h(·, ·) and adju... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
3.3.7), (A-1), and monotone convergence as in the
last proof, C can be chosen large enough so that
E(inf{ha (θ, x)/b(θ) : θ /∈ C}) ≥ 1 − ε/2.
�n
Then by the strong law of large numbers (RAP, Sec. 8.3), where a function with expectation
+∞ can be replaced by a smaller function with large positive expectation, a.s. f... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
Tn} be a sequence of approximate M-estimators. Assume either
(a) (A-1) through (A-5) hold, or (b) (A-1), (A-2(cid:6)), (A-3) and (A-4) hold, and for some
compact C, (3.3.11) holds. Then Tn → θ0 almost uniformly.
Proof. Assumptions (a) imply (A-2(cid:6)) by Lemma 3.3.9, and (3.3.11) by Lemma 3.3.10. So
assumptions (... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
ε := (γ(θ∞) − γ(θ0))/4, or if γ(θ∞) = +∞ let ε := 1. By (A-2(cid:6)), each θ ∈ C \ U
has an open neighborhood Uθ such that
E(inf{ha(φ, x) : φ ∈ Uθ }) ≥ γ(θ0) + 3ε.
Again, the infimum is measurable since by separability it can be restricted to a countable
dense set in Uθ . Take finitely many points θ(j), j = 1, . . . ... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
Let P and Q be two probability
measures on the same sample space S. Then there always exists some measure µ such that
both P and Q have densities with respect to µ, where µ is a σ-finite measure, in other
words there is a countable sequence of sets An whose union is all of S with µ(An) < ∞ for
each n. For example, i... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
for different µ’s, then R = S except possibly on some set A with
P (A) = Q(A) = 0. This is shown in Appendix A of the 18.466 Mathematical Statistics
notes on the MIT OCW site.
To apply Theorem 3.3.13 to the case of maximum likelihood estimation the following
will help. Let P and Q be two laws on a sample space (X, B... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
under suitable condi-
tions, does follow from Theorem 3.3.13, and assumption (A-3), and (A-4) for the true θ0,
will follow from Theorem 3.3.15 rather than having to be assumed:
3.3.16 Theorem. Assume (A-1) holds in the log likelihood case, for a measurable family
{Pθ , θ ∈ Θ} dominated by a σ-finite measure v, with (... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
< ∞ a.s. for P , and so −∞ < log f (θ0, x) <
∞. Thus h(θ, x) − a(x) is well-defined a.s. and equals
− log(f (θ, x)/f (θ0, x)) = − log RPθ /Pθ0
as shown in Appendix A of the 18.466 OCW notes. Thus for all θ,
γ(θ) := E[h(θ, x) − a(x)] = I(Pθ0, Pθ ) ≥ 0 > −∞
by Theorem 3.3.15 and γ(θ0) = 0, so (A-3) holds. Also by The... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
h(θ, x) := |x − θ|, show that conditions (A-1) through (A-5) hold for some a(·)
and b(·) (suggested in the text).
7
NOTES
An early result relating to consistency of maximum likelihood estimators was given
by Cram´er (1946), §33.3, namely, that under some hypotheses, there exist roots of the
likelihood equation(s) ... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
M. (1998). Consistency of M -estimators and one-sided bracketing. In High
Dimensional Probability, Progress in Probability 43, Birkh¨auser, Basel.
Haughton, D. M.-A. (1983). On the choice of a model to fit data from an exponential
family. Ph. D. dissertation, Mathematics, M.I.T.
Haughton, D. M.-A. (1988). On the cho... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
. 22, 79-86.
Wald, A. (1949). Note on the consistency of the maximum likelihood estimate. Ann.
Math. Statist. 20, 595-601.
8 | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
6.S096 Lecture 2 – Subtleties of C
Data structures and Floating-point arithmetic
Andre Kessler
Andre Kessler
6.S096 Lecture 2 – Subtleties of C
1 / 16
Outline
1
Memory Model
2
Data structures
3
Floating Point
4
Wrap-up
Andre Kessler
6.S096 Lecture 2 – Subtleties of C
2 / 16
... | https://ocw.mit.edu/courses/6-s096-effective-programming-in-c-and-c-january-iap-2014/3f8f2bc5d4ee804b3bce651a210a940f_MIT6_S096IAP14_Lecture2.pdf |
2 – Subtleties of C
4 / 16
Memory Model
malloc and the Heap
Statically allocated
int array[10];
int array2[] = { 1, 2, 3, 4, 5 };
char str[] = "Static string";
Dynamically allocated
#include <stdlib.h>
int *array = malloc( 10 * sizeof( int ) );
// do stuff
array... | https://ocw.mit.edu/courses/6-s096-effective-programming-in-c-and-c-january-iap-2014/3f8f2bc5d4ee804b3bce651a210a940f_MIT6_S096IAP14_Lecture2.pdf |
Memory Model
Let’s go over that again...
“I promise to always free
each chunk of memory that I
allocate.”
Don’t be the cause of memory leaks!
It’s a bad practice.
Andre Kessler
6.S096 Lecture 2 – Subtleties of C
7 / 16
... | https://ocw.mit.edu/courses/6-s096-effective-programming-in-c-and-c-january-iap-2014/3f8f2bc5d4ee804b3bce651a210a940f_MIT6_S096IAP14_Lecture2.pdf |
pair.second = 2;
struct IntPair_s *pairPtr = &pair;
// use pairPtr->first and pairPtr->second
// to access elements
Andre Kessler
6.S096 Lecture 2 – Subtleties of C
9 / 16
... | https://ocw.mit.edu/courses/6-s096-effective-programming-in-c-and-c-january-iap-2014/3f8f2bc5d4ee804b3bce651a210a940f_MIT6_S096IAP14_Lecture2.pdf |
2 – Subtleties of C
11 / 16
Floating Point
Let’s see some pictures...
float (32 bits)
Sign
Exponent
(8-bit)
Fraction (23-bit)
0 0
1 1 1 1 1
0 0 0
1
0
0
0 0 0 0 0 0 0 ... | https://ocw.mit.edu/courses/6-s096-effective-programming-in-c-and-c-january-iap-2014/3f8f2bc5d4ee804b3bce651a210a940f_MIT6_S096IAP14_Lecture2.pdf |
)
Andre Kessler
6.S096 Lecture 2 – Subtleties of C
15 // 16
Wrap-up & Monday
Wrap-up
Class on Monday
is back
Two shorter guest lectures:
Daniel Kang presenting x86 Assembly
Lef
presenting Secure C
Ioannidis
Questions?
Andre Kessler
6.S096 ... | https://ocw.mit.edu/courses/6-s096-effective-programming-in-c-and-c-january-iap-2014/3f8f2bc5d4ee804b3bce651a210a940f_MIT6_S096IAP14_Lecture2.pdf |
6.252 NONLINEAR PROGRAMMING
LECTURE 8
OPTIMIZATION OVER A CONVEX SET;
OPTIMALITY CONDITIONS
Problem: minx∈X(cid:160)f (x),(cid:160)where:
(a) X(cid:160)⊂ (cid:2)n(cid:160)is nonempty, convex, and closed.
(b) f(cid:160)is continuously differentiable over X.
• Local and global minima. If f(cid:160) is convex local
mi... | https://ocw.mit.edu/courses/6-252j-nonlinear-programming-spring-2003/3fab988e50bec05b31655d6cccd8af53_6252_slides08.pdf |
a local min but we have
∇f (x(cid:160)∗)(cid:2)(x(cid:160)− x(cid:160)∗) <(cid:160) 0 for
the feasible vector x(cid:160)shown.
PROOF
Proof: (a) Suppose that ∇f (x(cid:160)∗)(cid:3)(x(cid:160)− x(cid:160)∗) <(cid:160)0 for
some x(cid:160)∈ X. By the Mean Value Theorem, for
every (cid:1) > (cid:160)0 there exists a... | https://ocw.mit.edu/courses/6-252j-nonlinear-programming-spring-2003/3fab988e50bec05b31655d6cccd8af53_6252_slides08.pdf |
60)∗) is feasible for all (cid:1)(cid:160)∈ [0,(cid:160)1] because
X(cid:160)is convex, so the local optimality of x(cid:160)∗ is con-
tradicted.
(b) Using the convexity of f(cid:160)
f (x) ≥ f (x(cid:160)∗) + ∇f (x(cid:160)∗)(cid:3)(x(cid:160)− x(cid:160)∗)
for every x(cid:160)∈ X. If the condition ∇f (x(cid:160)∗)... | https://ocw.mit.edu/courses/6-252j-nonlinear-programming-spring-2003/3fab988e50bec05b31655d6cccd8af53_6252_slides08.pdf |
f(x*)
x*
∇f(x*)
x* = 0
OPTIMIZATION OVER A SIMPLEX
(cid:4)
X =
x
(cid:6)
(cid:5)
(cid:5)
(cid:5) x ≥ 0,
n(cid:3)
i=1
xi = r
where r > 0 is a given scalar.
• Necessary condition for x∗ = (x∗
a local min:
1, . . . , x∗
n) to be
n(cid:3)
i=1
∂f (x∗)
∂xi
∗
i ) ≥ 0,
(xi−x
∀ xi ≥ 0 with
n(cid:3)
i=1
xi = r.
i > 0 and let j ... | https://ocw.mit.edu/courses/6-252j-nonlinear-programming-spring-2003/3fab988e50bec05b31655d6cccd8af53_6252_slides08.pdf |
p > 0 =⇒ ∂D(x∗)
x∗
∂xp
≤ ∂D(x∗)
∂xp(cid:1)
,
∀ p(cid:3) ∈ Pw.
TRAFFIC ASSIGNMENT
(cid:15)(cid:16)
• Transportation network with OD pairs w. Each
w has paths p ∈ Pw and traffic rw. Let xp be the
(cid:17)
flow of path p and let Tij
p: crossing (i,j) xp
be the travel time of link (i, j).
• User-optimization principle: Traf... | https://ocw.mit.edu/courses/6-252j-nonlinear-programming-spring-2003/3fab988e50bec05b31655d6cccd8af53_6252_slides08.pdf |
x∗)(cid:2)(x − x∗) ≤ 0
• If X is a subspace, z − x∗ ⊥ X.
• The mapping f : (cid:2)n (cid:12)→ X defined by f (x) =
[x]+ is continuous and nonexpansive, that is,
(cid:10)[x]+ − [y]+(cid:10) ≤ (cid:10)x − y(cid:10),
∀ x, y ∈ (cid:2)n. | https://ocw.mit.edu/courses/6-252j-nonlinear-programming-spring-2003/3fab988e50bec05b31655d6cccd8af53_6252_slides08.pdf |
Introduction to C++
Massachusetts Institute of Technology
January 26, 2011
6.096
Lecture 10 Notes: Advanced Topics II
1 Stuff You May Want to Use in Your Project
1.1 File handling
File handling in C++ works almost identically to terminal input/output. To use files, you
write #include <fstream> at the top of your so... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/3fb8a922fc4283863e29eeed55d7e664_MIT6_096IAP11_lec10.pdf |
to open a file by calling the open method on it: source.open("other-file.txt");.
Close your files using the close() method when you’re done using them. This is automat
ically done for you in the object’s destructor, but you often want to close the file ASAP,
without waiting for the destructor.
You can specify a secon... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/3fb8a922fc4283863e29eeed55d7e664_MIT6_096IAP11_lec10.pdf |
ints, each of which is an ID for a particular suit. If
you wanted to print the suit name for a particular ID, you might write this:
1 const int CLUBS = 0 , DIAMONDS = 1 , HEARTS = 2 , SPADES = 3;
2 void print_suit ( const int suit ) {
3
4
5
6 }
const char * names [] = { " Clubs " , " Diamonds " ,
" Hearts " , ... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/3fb8a922fc4283863e29eeed55d7e664_MIT6_096IAP11_lec10.pdf |
can specify which integers you want them to be:
1 enum suit_t { CLUBS =18 , DIAMONDS =91 , HEARTS =241 , SPADES =13};
The following rules are used by default to determine the values of the enum constants:
•
The first item defaults to 0.
• Every other item defaults to the previous item plus 1.
Just like any other ty... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/3fb8a922fc4283863e29eeed55d7e664_MIT6_096IAP11_lec10.pdf |
pointers. For instance, just like int * is the
“pointer to an integer” type, int & is the “reference to an integer” type. References can be
passed as arguments to functions, returned from functions, and otherwise manipulated just
like any other type.
References are just pointers internally; when you declare a refer... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/3fb8a922fc4283863e29eeed55d7e664_MIT6_096IAP11_lec10.pdf |
}
6
7
8
9
// ... Somewhere in main
int & gRef = getG () ; // gRef is now an alias for g
gRef = 7; // Modifies g
// reference * to * doesn ’t get an & in front of it
If you’re writing a class method that needs to return some internal object, it’s often best to
return it by reference, since that avoids copying ... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/3fb8a922fc4283863e29eeed55d7e664_MIT6_096IAP11_lec10.pdf |
a non-const reference, yet we are trying to set it to a const
reference (the reference returned by getG).
In short, the compiler will not let you convert a const value into a non-const value unless
you’re just making a copy (which leaves the original const value safe).
2.2.2 const functions
For simple values like ... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/3fb8a922fc4283863e29eeed55d7e664_MIT6_096IAP11_lec10.pdf |
exits
immediately as well, the exception passes up to the next function, and so on up the call
stack (the chain of function calls that got us to the exception).
An example:
if ( y == 0)
return x / y ;
throw DIV_BY_0 ;
1 const int DIV_BY_0 = 0;
2 int divide ( const int x , const int y ) {
3
4
5
6 }
7
8 voi... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/3fb8a922fc4283863e29eeed55d7e664_MIT6_096IAP11_lec10.pdf |
number of catch blocks after a try block:
1
2
3
4
int divide ( const int x , const int y ) {
if ( y == 0)
throw std :: runtime_exception ( " Divide
by 0! " ) ;
return x / y ;
6
* arrPtr = new int [ divide (5 , x ) ];
}
catch ( bad_alloc & error ) { // new throws exceptions of this type
try {
5 }
6
7 v... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/3fb8a922fc4283863e29eeed55d7e664_MIT6_096IAP11_lec10.pdf |
throw exception
objects. Most exception classes inherit from class std::exception in header file
<stdexcept>.
• The standard exception classes all have a constructor taking a string that describes the
problem. That description can be accessed by calling the what method on an exception
object.
• You should always u... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/3fb8a922fc4283863e29eeed55d7e664_MIT6_096IAP11_lec10.pdf |
of class A should be fully
available to class B, we’d write:
1 class A {
2
3
4 };
friend class B ;
// More code ...
5 Preprocessor Macros
We’ve seen how to define constants using the preprocessor command #define. We can also
define macros, small snippets of code that depend on arguments. For instance, we can wr... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/3fb8a922fc4283863e29eeed55d7e664_MIT6_096IAP11_lec10.pdf |
type type. For
instance, we could cast a Vehicle * called v to a Car * by writing dynamic cast<Car
*>(v). If v is in fact a pointer to a Car, not a Vehicle of some other type such as
Truck, this returns a valid pointer of type Car *. If v does not point to a Car, it
returns null.
dynamic cast can also be used with... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/3fb8a922fc4283863e29eeed55d7e664_MIT6_096IAP11_lec10.pdf |
STL
containers and iterators
9
• void pointers – pointers to data of an unknown type
• virtual inheritance – the solution to the “dreaded diamond” problem described in
Lecture 8
• String streams – allow you to input from and output to string objects as though they
were streams like cin and cout
• Run-time type... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/3fb8a922fc4283863e29eeed55d7e664_MIT6_096IAP11_lec10.pdf |
18.175: Lecture 10
Zero-one laws and maximal inequalities
Scott Sheffield
MIT
18.175 Lecture 10
1 Outline
Recollections
Kolmogorov zero-one law and three-series theorem
18.175 Lecture 10
2 Outline
Recollections
Kolmogorov zero-one law and three-series theorem
18.175 Lecture 10
3 Recall Borel-Cantelli lemm... | https://ocw.mit.edu/courses/18-175-theory-of-probability-spring-2014/3fed030f143799a396c58ad269a4b19b_MIT18_175S14_Lecture10.pdf |
space. Write F � = σ(Xn, Xn1 , . . .) and T = ∩nF � .
n
T is called the tail σ-algebra. It contains the information you
can observe by looking only at stuff arbitrarily far into the
future. Intuitively, membership in tail event doesn’t change
when finitely many Xn are changed.
Event that Xn converge to a limit is exa... | https://ocw.mit.edu/courses/18-175-theory-of-probability-spring-2014/3fed030f143799a396c58ad269a4b19b_MIT18_175S14_Lecture10.pdf |
.
MIT OpenCourseWare
http://ocw.mit.edu
18.175 Theory of Probability
Spring 2014
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . | https://ocw.mit.edu/courses/18-175-theory-of-probability-spring-2014/3fed030f143799a396c58ad269a4b19b_MIT18_175S14_Lecture10.pdf |
18.01 Calculus
Jason Starr
Fall 2005
Math 18.01 Lecture Summaries
Homework. These are the problems from the assigned Problem Set which can be completed using
the material from that date’s lecture.
8 Velocity and derivatives
9 Limits
Practice Problems. Practice problems are not to be written up or turned in. These ... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
Lecture 22. Nov.
Lecture 23. Nov.
Lecture 24. Nov. 15
Lecture 25. Nov. 17
Lecture 26. Nov. 18 Partial fraction decomposition
Lecture 27. Nov. 22
4 Related rates problems
6 Newton’s method
13 Antidifferentiation
14 Riemann integrals
18 The Fundamental Theorem of Calculus
20 Properties of the Riemann integral
21 Sepa... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
). Increment from t0 to t0 + Δt is,
Δs = s(t0 + Δt) − s(t0).
Average velocity from t0 to t0 + Δt is,
vave =
Δs
Δt
=
s(t0 + Δt) − s(t0)
.
Δt
Velocity, or instantaneous velocity, at t0 is,
v(t0 ) = lim vave = lim
→
→
0
0
Δt
Δt
s(t0 + Δt) − s(t0)
.
Δt
This is a derivative, v(t) equals s�(t) = ds/dt. The deriv... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
005
The difference quotient or average rateofchange of y from x0 to x0 + Δx is,
Δy
Δx
=
f (x0 + Δx) − f (x0)
.
Δx
The derivative of y (or f (x)) with respect to x at x0 is,
Δy
lim
→
Δx
= lim
→
0 Δx Δx 0
f (x0 + Δx) − f (x0)
Δx
.
3. Examples in science and math.
(i) Economics. Marginal cost is the derivat... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
V = A × h.
3
where A is the base area of the cone and h is the height of the cone. The radius r of the base
is proportional to the height,
r(h) = ch,
3
18.01 Calculus
Jason Starr
Fall 2005
for some constant c. Since A = πr2, this gives,
The derivative is,
V (h) = c 2h3 .
π
3
dV
dh
= πc 2h2 = πr 2 =
A.... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
x2
0.
For instance, the equation of the tangent line through (2, 4) is,
y = 4x − 4.
2
Given a point (x, y), what are all points (x0, x0) on the parabola whose tangent line contains
(x, y)? To solve, consider x and y as constants and solve for x0. For instance, if (x, y) =
(1, −3), this gives,
or,
2
(−3) = 2x0(1... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
centered on x0 parallel to
the yaxis (but with the line x = x0 deleted) so that only the portion of the graph in the
fieldofview is illuminated. If for every magnification of the microscope, the illuminator can
succeed, then the limit is defined and equals L.
→x0
There is a beautiful Java applet on the webpage of ... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
. We need general
methods to simplify computations.
f �(x
) =
− x
3(3 + 1)
−3/2/2 .
5
√
3x + 1
18.01 Calculus
Jason Starr
Fall 2005
2. The binomial theorem. For a positive integer n, the factorial,
n! = n × (n − 1) × (n − 2) × · · · × 3 × 2 × 1,
is the number of ways of arranging n distinct objects in a lin... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
easy when n = 1; it just says
that (a + b)1 equals a1 + b1 . Next, make the induction hypothesis that the theorem is true for the
integer n. The goal is to deduce the theorem for n + 1,
�
n+1−k bk
a
= a + (n + 1)a nb + · · · +
+ · · · + (n + 1)abn + bn+1 .
(a + b)n+1
�
n+1
n + 1
k
By the definition of the (n... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
n
n
n
6
� �
n
k
)an−kbk+1 +
. . . + (1 + n)ab
n
+ bn+1.
18.01 Calculus
Jason Starr
Fall 2005
Using Pascal’s formula, this simplifies to,
�
an+1 + (n + 1)anb +
. . . +
�
n+1 an+1−k bk +
k
�
�
n+1 an−k bk+1
k+1
+
. . . + (n + 1)abn +
bn+1
.
This proves the theorem for n + 1, assuming the theorem for ... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
· · +
a n−k hk−1 + · · · + hn−1 .
Every summand except the first is divisible by h. The limit of such a term as h
→
0 is 0. Thus,
lim
→
h 0
f (a + h) − f (a)
h
= na n−1 + 0 + · · · + 0 = na
n−1
.
So f �(x) equals nxn−1 .
3. Linearity. For differentiable functions f (x) and g(x) and for constants b and c, bf (x... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
,
(f (x)/g(x))� = [f �(x)g(x) − f (x)g�(x)]/g(x)2 .
7
18.01 Calculus
Jason Starr
Fall 2005
One way to deduce this formula is to set q(x) = f (x)/g(x) so that f (x) = q(x)g(x), and the apply
the Leibniz formula to get,
f �(x) = q�(x)g(x) + q(x)g�(x) = q�(x)g(x) + f (x)g�(x)/g(x).
Solving for q�(x) gives,
q�(x... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
can be replaced,
d(xn+1)
dx
= x + x(nx n−1) = x + nx n = (n + 1)x .
n
n
n
Thus the formula for n implies the formula for n + 1. Therefore, by mathematical induction, the
formula holds for every positive integer n.
Lecture 4. September 15, 2005
Homework. No new problems.
Practice Problems. Course Reader: 1F1, 1F6... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
1). For that integer, suppose the result is known,
d(un)
dx
= nu
n−1 du
.
dx
The goal is to prove the result for n + 1, that is,
Let v = un . Then un+1 equals uv. So, by the product rule,
d(u
n+1)
dx
= (n + 1)u n
du
.
dx
d(u
n+1)
dx
=
d(uv)
dx
=
du
dx
v + u
dv
.
dx
Plugging in v = un, this is,
n+... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
1
n(xm/n)n−1
.
One of the basic rules of exponents is that (a
equals nxm/n(n−1), which equals nx
b)c
. Thus,
m−m/n
equals abc . Thus the denominator n(xm/n)n−1
du mxm−1
nxm−m/n
dx
=
m m−1 m/n−m
·
.
x
= x
n
9
18.01 Calculus
Jason Starr
Fall 2005
Another basic rule of exponents is that ab a equals ab+c . T... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
0 Δy
= lim
→
Δu Δy
·
Δx
,
Δu
where y0 equals f (x0), u0 equals g(y0) = g(f (x0)), Δy equals f (x0 + Δx) − f (x0) = f (x0 + Δx) − y0,
and Δu equals g(y0 +Δy)−g(y0) = g(f (x0 +Δx))−g(f (x0)). So long as Δy is nonzero, the fraction
in the limit is defined. And, as Δx approaches 0, also Δy approaches 0. Thus the limi... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
005
Lecture 5. September 16, 2005
Homework. Problem Set 2 Part I: (a)–(e); Part II: Problem 2.
Practice Problems. Course Reader: 1I1, 1I4, 1I5
1. Example of implicit differentiation. Let y = f (x) be the unique function satisfying the
equation,
1
x
1
y
+ = 2.
What is slope of the tangent line to the graph of y =... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
ials are as follows.
Rule 1. If ab equals B and a equals C, then ab+c equals B · C, i.e.,
c
b+c
c
a = a a .
·
b
Rule 2. If ab equals B and Bd equals D, then abd equals D, i.e.,
(a = a .
b)d
bd
If ab equals B, the logarithm with base a of B is defined to be b. This is written loga(B) = b. The
function B → loga(B... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
h 0
By Rule 1, ax0+h equals a
x0 ah
. Thus the limit factors as,
− ax0
ax0 ah
h
lim
→
h 0
= a x0 lim a h − 1h.
h 0
→
Therefore, for every x, the derivative of ax is,
d(ax)
dx
= L(a)a x .
What is L(a)? To figure this out, consider how L(a) changes as a changes. First of all,
By Rule 2, (ab)h equals abh . So the ... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
positive (though we have not
proved that L(a0) is even defined). Thus the graph of L(a) looks qualitatively like the graph of
loga0 (a). In particular, for a less than 1, L(a) is negative. The value L(1) equals 0. And L(a)
approaches +∞ and a increases. Therefore, there must be a number where L takes the value 1.
By... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
1.
du
ln(a)a u = 1.
dx
13
18.01 Calculus
Solving gives,
Jason Starr
Fall 2005
d loga(x)
dx
=
1
1
ln(a) au
=
1/(ln(a)x) .
In particular, for a = e, this gives,
d ln(x)
dx
=
1/x .
What is the derivative of ln(x) at x = 1? On the one hand, since the derivative of ln(x) equals 1/x,
the derivative at x... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
entiation. There is a method of computing derivatives of products of
functions that is often useful. If y is a product of n factors, say f1(x)·
fn(x), the derivative
of y can be computed by the product rule. However, it seems to be a fact that multiplication is
more errorprone than addition. Thus introduce,
f2(x)·... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
3x2
(1 + x3)
+
1
x(1 +
√
√
2
x)
−
3
.
7x
x) −
√
3
7
x)�/(1+
ln(x).
√
√
x)� = (
x) = (1/2x−1/2)/(1+
√
x).
So, finally,
y� = yu� =
(1 + x3)(1 +
√
x3/7
x) �
3x2
(1 + x3)
+
1
x(1 +
√
√
2
x)
−
�
.
3
7x
Lecture 6. September 20, 2005
Homework. Problem Set 2 Part I: (f)–(j); Part II: Problems 1, 3 and 4.
Pr... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
of the form mπ/2n, with m and n integers. Every angle can
15
18.01 Calculus
Jason Starr
Fall 2005
be approximated arbitrarily well by such angles. Thus, for every continuous function of an angle,
every value of the function can be computed.
The basic functions are sin(θ), cos(θ), tan(θ), sec(θ), csc(θ) and cot(θ... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
length
by the chord length tends to 1. This was not proved in lecture, nor is it proved in your textbook
in §2.1 (despite the author’s claim). However, it is geometrically reasonable. And, of course, it can
be proved.
This limit implies another limit,
To see this, rewrite the term as,
lim
→
0
θ
cos(θ) − 1
θ
=... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
)
sin(h)
h
.
Taking the limit gives,
lim
h→0
sin(a + h) − sin(a)
h
a
= sin( ) lim
h→0
cos(h) − 1
h
a
+ cos( ) lim
h→0
sin(h)
.
h
Using the limits from above, this gives,
sin�(a) = sin(a) × 0 + cos(a) × 1 = cos(a).
Thus the derivative of sin(x) equals,
d sin(x)
dx
=
cos(x).
An entirely similar com... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
)
dx
= − csc(x) cot(x).
Review for Exam 1. No new material was presented. There were no practice problems from the
course reader.
Lecture 8. September 27, 2005
Homework. Problem Set 2 all of Part I and Part II.
Practice Problems. Course Reader: 2A1, 2A4, 2A9, 2A11, 2A12.
1. Linear approximations. For a different... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
approximations occur so often, they should be committed
to memory. Each of the following is the linear approximation for x ≈ 0, together with the terms in
the quadratic and higher approximations.
1
1−x
≈
1 + + x2 + x3 + . . . ,
x
(1 + x)r
≈
1 + rx
+
x2 +
x3 + . . . ,
� �
r
2
� �
r
3
sin(x)
≈
x − x3/3! ... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
Fall 2005
(iii) The linear approximation of cf (x) for x ≈ a is c times the linear approximation of f (x) for
x ≈ a,
cf (x) ≈ cf (a
) +
cf �(a)(x − a) .
This is different than the previous rule. Also, the linear approximation of f (x) + g(x) for x ≈ a is
the sum of the linear approximations of f (x) and g(x),
(f +... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
x,
f (x)
g(x)
≈
f (a) + f �(a)(x − a)
g(a) + g�(a)(x − a)
= (f (a) + f �(a)(x − a))
1
1
g(a) 1 − (−g�(a)(x − a)/g(a))
≈
(f (a) + f �(a)(x − a))
(1 − g�(a)(x − a)/g(a)) =
1
g(a)
1
g(a)2
(f (a) + f �(a)(x − a))(g(a) − g�(a)(x − a)).
This simplifies to,
f (x)/g(x) ≈ f (a)/g(a
) + (1
/g(a)2)(f �(a)g(a) − f ... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
This simplifes to,
g(f (x)) ≈ g(f (a
)) +
g�(f (a))f �(a)(x − a).
This is equivalent to the chain rule,
d
dx
(g(f (x))) =
dg
dx
(f (x))
(x).
df
dx
Together, these 6 rules account for all the general rules we have regarding differentiation. So every
rule of differentiation has an equivalent formulation in term... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
function f (x) is
differentiable on the interval having a and b as endpoints, then there is a point c strictly between a
and b so that the slope of the tangent line to y = f (x) at x = c equals the slope of the secant line
to y = f (x) containing (a, f (a)) and (b, f (b)),
f �(c) =
f (b) − f (a)
.
b − a
21
18.... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
derivative f �(x) is,
f (x) − f (a)
x − a
= f �(c),
f �(x) =
−(3x2 + 2x + 1)
.
(1 + x + x2 + x3)2
For −1/2 ≤ x ≤ 1/2, this is bounded by,
|f �(x)
| ≤
3(1/2)2 + 2(1/2) + 1
[1 + (−1/2) + (−1/2)2 + (−1/2)3]2
= 7.04.
Thus the Mean Value Theorem gives,
f (x) − f (a) = f �(c
|
|
|
)||x − a|
≤ 7.04 x − a ≤ 7.04h... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
resp. nonincreasing. If f (x) is increasing, the graph rises to the
right. If f (x) is decreasing, the graph rises to the left.
If f �(a) is positive, the First Derivative Test guarantees that f (x) is increasing for all x sufficiently
close to a. If f �(a) is negative, the First Derivative Test guarantees that f (x)... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
the left of a, the graph of f rises to the left of a. Thus
a is not a local maximum. In particular, if f is defined to both the right and left of a, if f �(a) is
defined, and if a is a local maximum, then f �(a) equals 0. Similarly, if f is defined to both the right
and left of a, if f �(a) is defined, and if a is a loc... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
is a local maximum and x = 1/3 is a local
minimum.
2
The plurals of “maximum” and “minimum” are “maxima” and “minima”. Together, local maxima
and local minima are called extremal points, or extrema. These are points where f takes on an
23
18.01 Calculus
Jason Starr
Fall 2005
extreme value, either positive or ne... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
(a))/(c − a) ≥ (f (b) − f (a))/(b − a) whenever a < c < b.
For a differentiable function f , this equation is close to,
f �(c) ≤ f �(b) whenever a < c < b,
resp. f �(c) ≥ f �(b) whenever a > c > b.
This precisely says that f � is nondecreasing, resp. f � is nonincreasing. If f � is nondecreasing,
resp. nonincre... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
minimum.
2
5. Inflection points. If f is differentiable, but for every neighborhood of a, f is neither concave
up nor concave down on the entire neighborhood, then a is an inflection point. If f ��(a) is defined,
the Second Derivative Test says that f ��(a) must equal 0. Except in pathological cases, an inflection
point... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
f (x) = −∞, lim f (x) = +∞, lim f (x) = −∞.
x a−
→
x a−
→
x a+
→
x a+
→
In each case, the graph of y = f (x) becomes unbounded, and becomes arbitrarily close to the line
x = a. If x = a is a vertical asymptote, then f (x) has an infinite discontinuity at x = a.
The function f has a horizontal asymptote y = b if a... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
fence to form the other 2 sides of a rectangle, what is the largest area that can be enclosed in this
corner?
Step 1. Identify parameters. A parameter is a constant or variable. The constant in this
problem is 10 meters. Two variables are the length l of one side of the rectangle, and the width w
of the remaining s... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
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