# THE MEAN-VALUE THEOREM AND THE SIGN OF THE DERIVATIVE in .NET framework Generating QR Code JIS X 0510 in .NET framework THE MEAN-VALUE THEOREM AND THE SIGN OF THE DERIVATIVE

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 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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