limite-et-continuité.pdf
p.f(a) + q.f(b)=(p + q).f(c). Exercice 19 [ 01808 ] [Correction]. Notre objectif dans cet exercice est d'établir la proposition :.
f f(t)dt^pf(a) + qf(b) - JSTOR
2° Let A+ y
31Image formation by Mirrors and Lenses - Physics Courses
pqf = 1?1 qf p Real =fp =(10)(30) ?=15cmpf30 ?10 =? q=? 15=?0 5p30 InvertedReduced Example An object is placed 30 cm in front of a diverginglens with a focal length of -10 cm Find the image distance and magnification + 1=1 pqf = 1?1F qf p10 cm fp( ?10)(30)30 cm == =?7 5cmp?f30 ??( 10)Virtual image =? =? ?=q7 5M 0 25 30 Upright imagereduced
Generalized Fibonacci Sequences and Its Properties - Journal
F pF qF k with k k k t 12 F a F b 012 This was introduced by Gupta Panwar and Sikhwal We shall use the Induction method and Binet’s formula for derivation 1 Introduction It is well-known that the Fibonacci sequence is most prominent examples of recursive sequence The Fibonacci sequence is famous for possessing wonderful
d p x q (notation means f = (pf qf - University of Arizona
around using integration by parts For any two functions f;g in C1 0 [a;b] hLf;gi= Z b a d dx p(x) df dx g(x) + q(x)f(x)g(x)dx = Z b a p(x) df dx dg dx + q(x)f(x)g(x)dx = Z b a d dx p(x) dg dx f(x) + q(x)f(x)g(x)dx = hf;Lgi: Thus S-L operators are self-adjoint on C1 0 [a;b]
Rt The iff - Purdue University
Probability measures P and Q on S coincide if Pf Qf t f e BCS Roof of Theorem 1 2 Suppose that for any closed AES we are able to find a function f e BC S such that I
Notes on Convergence of Probability Measures by Billingsly
gives Qf QF Stringing these together gives PF QF akingT to zero gives PF QF(like in the last theorem QF !QF) Of course the argument works equally well to show QF PF so we get the desired result De nition 7 We say that a probability measure P on S is tight for every >0 there exists a compact set Kso that PK>1 This notion of tight is a
3.1 Images formed by Mirrors and Lenses
• Images • Image formation by mirrors • Images formed by lensesObject-Image
• A physical object is usually observed by reflected light that diverges from the object. • An optical system (mirrors or lenses) can produce an image of the object by redirecting the light. - Real Image - Virtual ImageReal ImageObject
real ImageOptical System
diverging converging divergingLight passes through the real image
Film at the position of the real image is exposed.Virtual ImageObject
virtual ImageOptical System diverging di ve r g in gLight appears to come from the virtual image but does not pass through the virtual image Film at the position of the virtual image is not exposed.Image formed by a plane mirror.
The virtual image is formed
directly behind the mirror.Light does not
pass through the imageObject ImageEach point on the image can be determined
by tracing 2 rays from the object. mirrorB B' AA'object
image p q A virtual image is formed by a plane mirror at a distance q behind the mirror. q = -p 2Parabolic Mirrors
Parallel rays reflected by a parabolic mirror are focused at a point, called theFocal Point located on the optic axis.
Optic Axis
Parabolic Reflector
Parabolic mirrors can be used to focus incoming parallel rays to a small area or to direct rays diverging from a small area into parallel rays.Spherical mirrors
•Spherical mirrors are much easier to fabricate than parabolic mirrors • A spherical mirror is an approximation of a parabolic mirror for small curvatures. (i.e. for paraxial rays -close to parallel to the optic axis. • Spherical mirrors can be convex or concave light light concave convexParallel beams focus at the focal point of
a Concave Mirror.Focal point
Ray tracing with a concave spherical mirrors
• A ray parallel to the mirror axis reflects through the focal point, F which is at a point half the radius distance from the mirror along the optic axis.
• A ray passing through the focal point reflects parallel to the mirror axis • A ray striking the center of the mirror reflects symmetrically around the mirror axis • A ray that passes through the center of curvature Creflects and passes back through itself FCMirror
axis RF2Law of
Reflection
The position of the image can be
determined from two rays from the object. F CThe image is real, inverted, reduced
When object distance > C
3A concave mirror can form real and
virtual imagesO > CC > O > FF > O
RealInverted
ReducedReal
Inverted
EnlargedVirtual
Upright
Enlarged
Simulation of image formation by a
mirror PHYSLETS were developed at Davidson University by Wolfgang Christian.Question
What image of yourself do you see when
you move toward a concave mirror?Far away
Real image
Inverted
Reduced
CInverted
Magnified
O~FMagnified Image
Real or Virtual?
4 OUpright
Enlarged
Convex Mirror
Image is virtual, upright, reduced
Ray parallel to the optic axis
reflects so that the reflected ray appears to pass through the focal point.Focal Point
A Convex Mirror always forms
virtual images virtual, upright, reduced virtual, upright, reducedQuestion
Describe how your image would appear as
you approach a convex mirror?Virtual Image
Upright
The image is
reduced in size and the field of view is larger.Virtual ImageUpright
5Virtual Image
Upright
Mirror Equation
f O I p q 111pqf p is positivefor real objects. f is positive if the light from infinity goes through the focal point. f positivefor concave mirrors, f negativefor convex mirrors q is positiveif the light goes through the image -real image q is negativeif light does not go through image -virtual imagep - object distance q - image distance f - focal length
Magnification
f O I p q 'hqMhp h h' q -positive - image is realM is negative - the image is inverted.
Magnification
f O I p q hqMhp h h' q is negative - the image is virtualM is positive - the image is upright.
Question
A boy stands 2.0 m in front of a concave mirror with a focal length of 0.50 m. Find the position of the image. Find the magnification. Is the image real or virtual? Is the image inverted or erect?
p O I q 111pqf
111qfp
fpqpf0.5(2.0)0.672.0 0.5m
qmp0.670.332.0
Real image
invertedImage formed by refraction
• Light rays are deflected by refraction through media with different refractive indexes. • An image is formed by refraction across flat or curved interfaces and by passage through lenses. 6Image formed by Refraction
ooImage of the tip
Image formed by refraction through
a refracting surface.Real image formed by refraction.Light is
caused toConverge.
Rotation
of the ray at the interfaceConverging Lenses
Fatter in the middle.
Cause light to converge toward the optic axis
Diverging Lenses
Thinner in the middle
Cause light to diverge away from the optic axis
Parallel light though a converging
lens is focused at the focal point.A real image is formed
Ray tracing for lenses
• A line parallel to the lens axis passes through the focal point • A line through the center of the lens passes through undeflected. f 7Ray diagram for a converging
lensesObjectImage
A converging lens can form real
and virtual images converging light converging light diverging lightRealInverted
reduced RealInverted
Enlarged
Virtual
Upright
Enlarged
At the focal point the image changes
from real to virtualQuestion
How will an object viewed through a
converging lens appear as the lens is brought closer to the object?Real Image
Inverted
Real Image
Inverted
Magnified
Virtual Image
Upright
Magnified
8Virtual image
Upright
Parallel light though a diverging
lens appears to go through the focal point.A virtual image is formed.
Image formed by a Diverging lens
Virtual
Upright
Reduced
A Diverging lens always forms a
virtual imageQuestion
How will the image of an object formed by a
diverging lens change as the lens is brought closer to the object?Virtual Image
Upright
Reduced
9Virtual image
Upright
Reduced
Virtual image
Upright
Reduced
Thin lens equation.
111pqf p is positive for real objects f is positive for converging lenses f is negative for diverging lenses q is positive for real images q is negative for virtual images.p and q are positive if light passes through
Magnification
h' qMhp for real image q is positive - image is inverted for virtual image q is negative - image is uprightM positive- uprightM negative- inverted
Example
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