MTE 494 04 20 10 Average Speed Rates of Change 1 State the meaning of constant speed that we have publicly agreed to use An object is moving at a constant speed distance with respect to time if and only if 2 State the meaning of average speed that we have publicly agreed to use 3 Extend the above meaning of average speed to average rate of change of one quantity with respect to another 4 A spherical tank in a microbrewery is being filled with water The supervisor records the amount volume of water in the tank and the water s depth in the tank at two moments in time At 2 72 feet there were 99 35 gallons of water in the tank At 4 35 feet there were 175 72 gallons of water in the tank What was the average rate of change of volume with respect to height over this interval of height Explain what this number means 1 MTE 494 04 20 10 5 The idea of average rate of change involves thinking of a hypothetical or fictitious scenario What role such a scenario play in making the idea of average rate of change sensible 6 When an object falls from a resting start the distance in feet it has fallen x seconds after being released is given by the function d x 16x2 assuming no air resistance a Consider the function g x d x 1 2 d x for d as given above i Graph the function g ii What does 1 2 represent What does the point 0 7 g 0 7 represent iii What does each point on the graph of g represent iv What question would the graph of g answer b Consider the function h x d x 01 d x for d as given above i Graph the function h ii What does 01 represent iii What does the point 0 7 h 0 7 represent iv What does each point on the graph of h represent for d as given above c Consider the function i Graph the function k ii What does 1 5 represent iii What does each point on the graph of k represent Give an example where h is a parameter that varies over d Consider the function the interval 0 1 10 i Anticipate what the graph of m will be ii What does h represent iii What does each point on the graph of m represent Give an example to explain why the graph of m x is e Expand and simplify the expression always a straight line regardless of the value of h 2 MTE 494 04 20 10 f What does this simplified expression tell you about changes in the object s average speeds over time 3
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