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Ayman - Jeddah$66
$66Claculus I, II , III for university students studying engineering and math
Trusted teacher: Chapter 1: Relationships The central question of this introductory chapter – which contains no calculus – is “What is a function?” The objective is to help students separate this concept from other relationships between varying quantities, and especially to separate the idea of function from such ideas as formula and equation. The concept of function is the basic building block of mathematics. A deep understanding of function will facilitate your future study of mathematics and computer science. Throughout this course, we will be working with multiple representations of functions. The authors of our text present functions verbally, numerically, and visually as well as algebraically. Chapter 2: Models of Growth: Rates of Change In this chapter, we will investigate some basic reasons for studying calculus. In particular we will investigate problem situations which can be modeled using differential equations. Topics introduced in this chapter include difference quotients, derivatives, slope fields, initial value problems whose solutions are functions and families of functions. The primary example of this chapter is natural population growth, the simplest ODE (ordinary differential equation) to solve. This example provides an immediate reason for moving beyond polynomials to other families of functions (e.g., to exponential and logarithmic functions). We will conclude this chapter by using tools of calculus to analyze the spread of the AIDS virus. Chapter 3: Initial Value Problems This short chapter builds on Chapter 2, introducing Newton’s Law of Cooling (exponential decay) to solve a murder mystery, then studying falling objects without air resistance (polynomial solutions). Chapter 4: Differential Calculus and Its Uses This chapter is the heart of first-semester calculus, consolidating what has been learned about derivatives to take up problems involving optimization, concavity, Newton’s Method (as an exercise in local linearity), and the basic formulas for differentiation. The product rule is introduced to study the growth rate of energy consumption, the chain rule to study reflection and refraction, and implicit differentiation to calculate derivatives of logarithmic functions and general powers. The process of zooming in on a graph is related to differentials and Leibniz notation. The chapter concludes with an interesting application of calculus to a problem in air-traffic control. Chapter 5: Modeling with Differential Equations This chapter builds on the problems introduced in Chapter 3, introducing air resistance to problems of falling bodies (e.g., raindrops and skydivers). The authors introduce problems of periodic motion, which are modeled using trigonometric functions and their derivatives.
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