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** Boğaziçi Üniversitesi,İtanbul TEKNİK ÜNİVERSİTESİ, bilkent üniversitesi , orta doğu teknik üniversitesi, Başkent üniversitesi ,Bilgi Üniversitesi,Koç Üniversitesi,Işık Üniversitesi, İstanbul Üniversitesi,Yıldız Üniversitesi ,Yeditepe Üniversitesi Haliç Üniversitesi Beykent Üniversitesi ,Doğuş Üniversitesi,Kadir HAS , Okan UNİVERSİTESİ,Özyeğin ÜNİVERSİTESİ, Maltepe ÜNİVERSİTESİ, Yeni YÜZYİL ÜN.,ve Kültür Üniversitelerinin ekonomi işletme ,bilgisayar mühendisliği,ekonomi bölümü,insaat mühendisliği ,uluslararasi iliskiler ve çalışma ekonomisi gibi bölümlerde okuyan onlarca üniversite öğrencisine özel dersler verdim. **

**not pandemi döneminde türkiyenin hemen hemen bir çok üniversitersinde ve yurtdışında okuyan bir çok öğrenciye özel dersler verdim **

Course Content:

**Introduction****Real numbers and the completeness axiom****Mathematical induction****Some inequalities**

**Integral Calculus****Functions****Area as a set function and step functions****Integrals of step functions****Integrals of general functions****Integrals of monotonic functions****Integrals of powers and polynomials****Properties of the integral**

**Applications of Integration****Area between two graphs****Integrating trig functions****Integrals in polar coordinates****Calculation of volume****Work and average value****Indefinite integrals**

**Continuous Functions****Limits of functions****Continuity****Properties of limits****Composite functions and continuity****Intermediate value theorem****Inverse functions and their properties****Extrema and uniform continuity****Integrability of continuous functions**

**Differential Calculus****Differentiating a function****Algebra of derivatives****The derivative as a slope****The chain rule****Applications of the chain rule****Extreme values of functions****The mean value theorem****First and second derivative tests****Curve sketching**

**Integration and Differentiation****The derivative of an indefinite integral****Primitive functions****Integration by substitution****Integration by parts**

**The Logarithm, The Exponential, and Inverse Trigonometric Functions****Definition of logarithm as an integral****Differentiation and integration formulas****The exponential function****Differentiation and integration formulas****Inverse trigonometric functions**

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**Contents1 Integrals 11.1 Areas and Distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.2 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.3 The Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.4 Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.5 The Substitution Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 Applications of Integration 352.1 Areas Between Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2 Areas in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3 Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.4 Volumes by Cylindrical Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 Techniques of Integration 553.1 Integration By Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2 Trigonometric Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.3 Trigonometric Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.4 Integration of Rational Functions by Partial Fractions . . . . . . . . . . . . . . . . . . . . . .3.5 Strategy for Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.6 Approximate Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.7 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 Further Applications of Integration 954.1 Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iii CONTENTS4.2 Area of a Surface of Revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.3 Calculus with Parametric Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 Infinite Sequences and Series 1095.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.2 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.3 The Integral Test and Estimates of Sums . . . . . . . . . . . . . . . . . . . . . . . . . .5.4 The Comparison Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.5 Alternating Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.6 Absolute Convergence and the Ratio and Root Test . . . . . . . . . . . . . . . . . . . . . . . .5.7 Strategy for Testing Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.8 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.9 Representation of Functions as Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.10 Taylor and Maclaurin Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.11 Applications of Taylor Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 A First Look at Differential Equations 1636.1 Modeling with Differential Equations, Direction Fields . . . . . . . . . . . . . . . . . . . . . .6.2 Separable Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.3 Models for Population Growth**

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**3-Dimensional Space – In this chapter we will start looking at three dimensional space. This chapter is generally prep work for Calculus III and so we will cover the standard 3D coordinate system as well as a couple of alternative coordinate systems. We will also discuss how to find the equations of lines and planes in three dimensional space. We will look at some standard 3D surfaces and their equations. In addition we will introduce vector functions and some of their applications (tangent and normal vectors, arc length, curvature and velocity and acceleration).The 3-D Coordinate System – In this section we will introduce the standard three dimensional coordinate system as well as some common notation and concepts needed to work in three dimensions.Equations of Lines – In this section we will derive the vector form and parametric form for the equation of lines in three dimensional space. We will also give the symmetric equations of lines in three dimensional space. Note as well that while these forms can also be useful for lines in two dimensional space.Equations of Planes – In this section we will derive the vector and scalar equation of a plane. We also show how to write the equation of a plane from three points that lie in the plane.Quadric Surfaces – In this section we will be looking at some examples of quadric surfaces. Some examples of quadric surfaces are cones, cylinders, ellipsoids, and elliptic paraboloids.Functions of Several Variables – In this section we will give a quick review of some important topics about functions of several variables. In particular we will discuss finding the domain of a function of several variables as well as level curves, level surfaces and traces.Vector Functions – In this section we introduce the concept of vector functions concentrating primarily on curves in three dimensional space. We will however, touch briefly on surfaces as well. We will illustrate how to find the domain of a vector function and how to graph a vector function. We will also show a simple relationship between vector functions and parametric equations that will be very useful at times.Calculus with Vector Functions – In this section here we discuss how to do basic calculus, i.e. limits, derivatives and integrals, with vector functions.Tangent, Normal and Binormal Vectors – In this section we will define the tangent, normal and binormal vectors.Arc Length with Vector Functions – In this section we will extend the arc length formula we used early in the material to include finding the arc length of a vector function. As we will see the new formula really is just an almost natural extension of one we’ve already seen.Curvature – In this section we give two formulas for computing the curvature (**

**not: Üniversitede okuyan öğrenciler dersleri düzenli takip etmekte zorlanır. Ayrıca liseden de çok fazla çalışmadan üniversitede bir bölüme girmiş olabilir, fakat her ne kadar bazı özel üniversitelerde dersleri geçmek kolay olsa da genelde bazı ana dersleri iyi bir ortalama ile geçmek çok kolay değil. Özel ders, öğrencilerin iyi bir notla geçmelerini ve takip edemedikleri konuları da kısa sürede öğrenmelerini sağlar. Ders verdiğim öğrenciler bu tür problemleri yaşamadan ve yüksek notlarla dersini geçmiş oluyorlar**

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