CH2 Part1 - LIMITS LIMITS The idea of a limit underlies the...

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LIMITS
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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 tangent line as soon as we know its slope m. ± The difficulty is that we know only one point, P, on t, whereas we need two points to compute the slope.
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THE TANGENT PROBLEM However, we can compute an approximation to m by choosing a nearby point Q(x, x 2 ) on the parabola and computing the slope m PQ of the secant line PQ. Example 1
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We choose so that . ± Then, ± For instance, for the point Q(1.5, 2.25), we have: THE TANGENT PROBLEM 1 x Q P 2 1 1 PQ x m x = 2.25 1 1.25 2.5 1.5 1 0.5 PQ m == = Example 1
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THE TANGENT PROBLEM The tables below the values of m PQ for several values of x close to 1. The closer Q is to P, the closer x is to 1 and, it appears from the tables, the closer m is to 2. ± This suggests that the slope of the tangent line tshould be m= 2. Example 1
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The slope of the tangent line is said to be the limit of the slopes of the secant lines. This is expressed symbolically as follows. lim PQ QP mm = 2 1 1 2 1 x x x = THE TANGENT PROBLEM Example 1
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THE TANGENT PROBLEM Assuming that the slope of the tangent line is indeed 2, we can use the point- slope form of the equation of a line to write the equation of the tangent line through (1, 1) as: or 12 ( 1 ) yx −= 21 y x =− Example 1
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THE TANGENT PROBLEM The figure illustrates the limiting process that occurs in this example.
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CH2 Part1 - LIMITS LIMITS The idea of a limit underlies the...

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