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Related Rates

This guide explores the application of related rates in calculus, focusing on differentiable functions and their rates of change. We will examine various scenarios, including the distance of a moving point on a curve, changes in spherical volume with increasing radius, and how moving edges of a cube affect its volume over time. Additionally, we will assess the motion of a plank being lifted to find how quickly it moves along the ground. Each example illustrates fundamental principles of related rates, ideal for students and anyone looking to understand dynamic changes in geometry.

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Related Rates

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  1. Related Rates

  2. 1. Assume that x and y are both differentiable functions of t and find the required values of dy/dt and dx/dt.(similar to p.187 #4)

  3. 2. Find the rate of change of the distance between the origin and a moving point on the graph of y = cos x if dx/dt = 5 centimeters per second.(similar to p.187 #12)

  4. 3. The radius r of a sphere is increasing at a rate of 5 inches per minute. (Formula: V = 4/3πr3)Find the rates of change of the volume when r = 10 inches and r = 20 inches(similar to p.187 #16)

  5. 4. All edges of a cube are expanding at a rate of 4 centimeters per second. How fast is the volume changing when each edge is (a) 2 centimeters and (b) 20 centimeters(similar to p.187 #18) NEXT TIME SPRING 2013

  6. 5. The formula for the volume of a cone is V = 1/3πr2h. Find the rates of change of the volume if dr/dt is 4 inches per minute and h = 6r when (a) r = 3 inches and (b) r = 12 inches(similar to p.187 #20)

  7. 6. A 10 m plank (that is originally laying flat on the ground) is pulled up the side of a building by means of a rope tied to the end closest to the building. It is pulled up at a rate of 0.25 meters per second. How fast is the end of the plank on the ground moving along the ground when it is 3 meters from the wall of the building?(similar to p.188 #26)

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