tailieunhanh - Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 70

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 70. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | A Brief Introduction to Vectors 671 A BRIEF INTRODUCTION TO VECTORS Questions of displacement and velocity have figured prominently in our studies. When we talk about displacement we ask how far and in what direction When we talk about velocity we ask how fast and in what direction We ve been answering the question in what direction by using a plus or a minus sign. This has restricted the type of motion we describe. Our objects are going up or down our boats are headed east or west our cars travel on long straight highways our trains are either inbound or outbound. If we wander along a meandering path we can report our velocity only with reference to some benchmark point on the path we re either heading toward it or away from it. We can free ourselves from this world of dichotomy by using an arrow to indicate direction. In fact we can have the arrow answer both the amount and direction questions by having the length of the arrow indicate how far or how fast and the direction the arrow is pointing indicate the direction of the displacement or motion. An arrow used in this way is called a vector. Defining Vectors Definition A vector has two defining characteristics length or magnitude and direction. A vector is often denoted by V a boldface letter with an arrow above it and represented by an arrow. The length of V is denoted by V . Vectors can be used in a large variety of situations for example vectors can be used to model velocity force and displacement. A velocity vector is a vector whose length represents speed and whose direction indicates the direction of motion. A force vector is a vector whose magnitude corresponds to the magnitude of the force and whose direction indicates the direction of application of the force. We can give a similar interpretation for a displacement vector. We refer to the head of the vector the tip of the arrow and the tail of the vector the other end . A vector represented by an arrow whose tail is at A a1 a2 and whose head .