Chap2_Sec1 - LIMITS LIMITS 2 LIMITS The idea of a limit...

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LIMITS LIMITS 2
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LIMITS The idea of a limit underlies the various branches of calculus. It is therefore appropriate to begin our study of calculus by investigating limits and their properties.
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In this section, we will learn: How limits arise when we attempt to find the tangent to a curve or the velocity of an object. 2.1 The Tangent and Velocity Problems LIMITS
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THE TANGENT PROBLEM The word tangent is derived from the Latin word tangens, which means ‘touching.’ Thus, a tangent to a curve is a line that touches the curve. In other words, a tangent line should have the same direction as the curve at the point of contact. How can this idea be made precise?
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For a circle, we could simply follow Euclid and say that a tangent is a line that intersects the circle once and only once. THE TANGENT PROBLEM
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THE TANGENT PROBLEM For more complicated curves, that definition is inadequate. The figure displays two lines l and t passing through a point P on a curve. The line l intersects only once, but it certainly does not look like what is thought of as a tangent.
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In contrast, the line t looks like a tangent, but it intersects twice. THE TANGENT PROBLEM
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To be specific, let’s look at the problem of trying to find a tangent line to the parabola y = x 2 in the following example. THE TANGENT PROBLEM
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THE TANGENT PROBLEM Example 1 Find an equation of the tangent line to the parabola y = x 2 at the point P (1,1). We will be able to find an equation of the
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Chap2_Sec1 - LIMITS LIMITS 2 LIMITS The idea of a limit...

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