In this chapter we will study aspects of analytic geometry that are important in application of calculus ,we will begin by introducing polar coordinates systems ,which are used for example in tracking the motion of plants and satellites in identifying the locations of objects from information on radar screen and in the desigen of antenna polar coordinates we will then discuss relation shipes between curves in polar coordinates and parmetric cuvers in rectangular coordinates to satefied many purpoises it does not whther polar angles are measured in degrees or radians however in proplemes that involve derivatives or integrals they must be measured in radibns since the derivative of the trigonometric functions were derived under this assumptions, and we will discuss method for finding areas in polar coordinates and tangents lines to curves given in polar coordinates we will derive the formulas required to find slops tangents tines and arc lengths of parametic and polar coordinates curves we will then review the basic properties of parapolas ,ellipses and hyperbolas and discuss some of the basic geometric properties of parapolas thes curves play importants role in calculus and also arise naturally in a board range of applicationsin suvh as planetary motion,designof telescopes and antennas geodetic positioning ,and medicine ,to name a few these curves in the cntext of polar coordinates hyperbolas and derive their standard equations. These curves are called conic sections, or
conics, and model the paths traveled by planets, satellites, and other bodies whose motions are driven by inverse square forces. we will see that once the path of a moving
body is known to be a conic, we immediately have information about the body’s velocity and the force that drives it. Planetary motion is best described with the help of polar coordinates, so we also investigate curves, derivatives, and integrals in this new coordinate system
An ellipse is the set of points in a plane whose distances from two fixed points
in the plane have a constant sum. The two fixed points are the foci of the
ellipse The line through the foci of an ellipse is the ellipse’s focal axis. The point on the
axis halfway between the foci is the center. The points where the focal axis andAAtheir use as reflectors of light and radio
waves. Rays originating at a parabola’s focus are reflected out of the parabola parallel to
the parabola’s axis Moreover, the time any ray takes from
the focus to a line parallel to the parabola’s directrix (thus perpendicular to its axis) is the
same for each of the rays. These properties are used by flashlight, headlight, and spotlight
reflectors and by microwave broadcast antennas. A hyperbola is the set of points in a plane whose distances from two fixed points
in the plane have a constant difference. The two fixed points are the foci of the
hyperbolaA The chief applications of parabolas involve their use as reflectors of light and radio
waves. Rays originating at a parabola’s focus are reflected out of the parabola parallel to
the parabola’s axis Moreover, the time any ray takes from
the focus to a line parallel to the parabola’s directrix (thus perpendicular to its axis) is the
same for each of the rays. These properties are used by flashlight, headlight, and spotlight reflectors and by microwave broadcast antennas
المادة المعروضة اعلاه هي مدخل الى المحاضرة المرفوعة بواسطة استاذ(ة) المادة . وقد تبدو لك غير متكاملة . حيث يضع استاذ المادة في بعض الاحيان فقط الجزء الاول من المحاضرة من اجل الاطلاع على ما ستقوم بتحميله لاحقا . في نظام التعليم الالكتروني نوفر هذه الخدمة لكي نبقيك على اطلاع حول محتوى الملف الذي ستقوم بتحميله .