THE MEAN-VALUE THEOREM AND THE SIGN OF THE DERIVATIVE in .NET framework

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THE MEAN-VALUE THEOREM AND THE SIGN OF THE DERIVATIVE
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17.23 Prove that the equation x 4 + x = 1 has at least one solution in the interval [0, 1]. 17.24 Find a point on the graph of y = x 2 + x + 3, between x = 1 and x = 2, where the tangent line is parallel to the line connecting (1, 5) and (2, 9). 17.25 (a) Show that f (x) = x 5 + x 1 has exactly one real zero. (b) GC Locate the real zero of x 5 + x 1 correct to the rst decimal place. 17.26 (a) GC Use a graphing calculator to estimate the intervals in which the function f (x) = x 4 3x 2 + x 4 is increasing and the intervals in which it is decreasing. (b) As in part (a), but for the function f (x) = x 3 2x 2 + x 2.
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Rectilinear Motion and Instantaneous Velocity
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Rectilinear motion is motion along a straight line. Consider, for instance, an automobile moving along a straight road. We can imagine a coordinate system imposed on the line containing the road (see Fig. 18-1). (On many highways there actually is such a coordinate system, with markers along the side of the road indicating the distance from one end of the highway.) If s designates the coordinate of the automobile and t denotes the time, then the motion of the automobile is speci ed by expressing s, its position, as a function of t: s = f (t).
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Fig. 18-1 The speedometer indicates how fast the automobile is moving. Since the speedometer reading often varies continuously, it is obvious that the speedometer indicates how fast the car is moving at the moment when it is read. Let us analyze this notion in order to nd the mathematical concept that lies behind it. If the automobile moves according to the equation s = f (t), its position at time t is f (t), and at time t + h, very close to time t, its position is f (t + h). The distance1 between its position at time t and its position at time t + h is f (t + h) f (t) (which can be negative). The time elapsed between t and t + h is h. Hence, the average velocity2 during this time interval is f (t + h) f (t) h (Average velocity = displacement time.) Now as the elapsed time h gets closer to 0, the average velocity approaches what we intuitively think of as the instantaneous velocity v at time t. Thus, v = lim
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f (t + h) f (t) h
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In other words, the instantaneous velocity v is the derivative f (t).
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precisely, the displacement, since it can be positive, negative, or zero. use the term velocity rather than speed because the quantity referred to can be negative. Speed is de ned as the magnitude of the velocity and is never negative.
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Copyright 2008, 1997, 1985 by The McGraw-Hill Companies, Inc. Click here for terms of use.
RECTILINEAR MOTION AND INSTANTANEOUS VELOCITY
[CHAP. 18
EXAMPLES
(a) The height s of a water column is observed to follow the law s = f (t) = 3t + 2. Thus, the instantaneous velocity v of the top surface is f (t) = 3. (b) The position s of an automobile along a highway is given by s = f (t) = t 2 2t. Hence, its instantaneous velocity is v = f (t) = 2t 2. At time t = 3, its velocity v is 2(3) 2 = 4.
The sign of the instantaneous velocity v indicates the direction in which the object is moving. If v = ds/dt > 0 over a time interval, Theorem 17.3 tells us that s is increasing in that interval. Thus, if the s-axis is horizontal and directed to the right, as in Fig.18-2(a), then the object is moving to the right; but if the s-axis is vertical and directed upward, as in Fig. 18-2(b), then the object is moving upward. On the other hand if v = ds/dt < 0 over a time interval, then s must be decreasing in that interval. In Fig. 18-2(a), the object would be moving to the left (in the direction of decreasing s); in Fig. 18-2(b), the object would be moving downward.
Fig. 18-2 A consequence of these facts is that at an instant t when a continuously moving object reverses direction, its instantaneous velocity v must be 0. For if v were, say, positive at t, it would be positive in a small interval of time surrounding t; the object would therefore be moving in the same direction just before and just after t. Or, to say the same thing in a slightly different way, a reversal in direction means a relative extremum of s, which in turn implies (by Theorem 14.1) ds/dt = 0. EXAMPLE An object moves along a straight line as indicated in Fig. 18-3(a). In functional form,
s = f (t) = (t + 2)2 [s in meters, t in seconds]
as graphed in Fig. 18-3(b). The object s instantaneous velocity is v = f (t) = 2(t + 2) [meters per second]
For t + 2 < 0, or t < 2, v is negative and the object is moving to the left; for t + 2 > 0, or t > 2, v is positive and the object is moving to the right. The object reverses direction at t = 2, and at that instant v = 0. [Note that f (t) has a relative minimum at t = 2.]
Free Fall Consider an object that has been thrown straight up or down, or has been dropped from rest, and which is acted upon solely by the gravitational pull of the earth. The ensuing rectilinear motion is called free fall. Let us put a coordinate system on the vertical line along which the object moves, such that the s-axis is directed upward, away from the earth, with s = 0 located at the surface of the earth (Fig. 18-4). Then, according to physics, the equation of free fall is s = s0 + v0 t 16t 2 (18.1)
CHAP. 18]
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