RECTILINEAR MOTION AND INSTANTANEOUS VELOCITY in Visual Studio .NET

Encoder QR Code in Visual Studio .NET RECTILINEAR MOTION AND INSTANTANEOUS VELOCITY

RECTILINEAR MOTION AND INSTANTANEOUS VELOCITY
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18.10 The position of a moving object on a line is given by the formula s = (t 1)3 (t 5). (a) When is the object moving to the right (b) When is it moving to the left (c) When is it changing direction (d) When is it at rest (e) What is the farthest to the left of the origin that it moves 18.11 A particle moves on a straight line so that its position s (miles) at time t (hours) is given by s = (4t 1)(t 1)2 . (a) When is the particle moving to the right (b) When is the particle moving to the left (c) When does it change direction (d) When the particle is moving to the left, what is the maximum speed that it achieves (The speed is the absolute value of the velocity.) 18.12 A particle moves along the x-axis according to the equation x = 10t 2t 2 . What is the total distance covered by the particle between t = 0 and t = 3 18.13 A rocket was shot straight up from the ground. What must have been its initial velocity if it returned to earth in 20 seconds 18.14 Two particles move along the x-axis. Their positions f (t) and g(t) are given by f (t) = 6t t 2 and g(t) = t 2 4t. (a) When do they have the same position (b) When do they have the same velocity (c) When they have the same position, are they moving in the same direction 18.15 A rock is dropped and strikes the ground with a velocity of 49 meters per second. (a) How long did it fall (b) Find the height from which it was dropped. 18.16 A ball is thrown vertically upward from the top of a 96-foot tower. Two seconds later, the velocity of the ball is 16 feet per second. Find: (a) the maximum height that the ball reaches; (b) the speed of the ball when it hits the ground.
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One quantity, y, may be related to another quantity, x, by a function f : y = f (x). A change in the value of x usually induces a corresponding change in the value of y. EXAMPLE Let x be the length of the side of a cube, and let y be the volume of the cube. Then y = x3 . In the case where the side has length x = 2 units, consider a small change x in the length.
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notation x (read delta-ex ) is the traditional symbol in calculus for a small change in x. a product of and x. In earlier chapters, the role of x often was taken by the symbol h. x is considered a single symbol, not
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The new volume will be (2 + x)3 , and so the change in the value of the volume y is (2 + x)3 23 . This change in y is denoted traditionally by y, y = (2 + x)3 23
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Now the natural way to compare the change y in y to the change x in x is to calculate the ratio y/ x. This ratio depends of course on x, but if we let x approach 0, then the limit of y/ x will de ne the instantaneous rate of change of y compared to x, when x = 2. We have (algebra, Problem 11.2) y = (2 + x)3 23 = [(2)3 + 3(2)2 ( x)1 + 3(2)1 ( x)2 + ( x)3 ] 23
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= 12 x + 6( x)2 + ( x)3 = ( x)(12 + 6 x + ( x)2 ) Hence, and lim y = 12 + 6 x + ( x)2 x
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y = lim (12 + 6 x + ( x)2 ) = 12 x 0 x
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Therefore, when the side is 2, the rate of change of the volume with respect to the side is 12. This means that, for sides close to 2, the change y in the volume is approximately 12 times the change x in the side (since y/ x is close to 12). Let us look at a few numerical cases.
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