SM2H Average Rate of Change

Transcription

1 Name Period Core Alignment: F.IF., F.IF., F.IF.9 SMH 5. - Average Rate of Change The average rate of change between two points is essentiall the slope of the line that connects the two points. For eample, if the function f(t) represents the distance in miles a car travels at an time t 0 (hours), then finding the slope between t = and t = will give the average speed (mph) the car was traveling during those three hours. Recall that slope is =. In this eample is the distance traveled and is the time interval that elapses. The slope or rate of change shows the change in distance divided b the change in time which results in the average speed. Below is the graph of two runners as the run a 00 meter hurdles race. Runner A is the dashed line and Runner B is the solid line. Distance (meters) Time Runner A s Distance Runner B s Distance Time (seconds) Which runner has a faster average speed for the first 9 seconds? Which runner has a faster average speed for 9 to 0 seconds? Which runner has a faster average speed for 0 to seconds? Which runner has a faster average speed for 9 to seconds? Which runner wins the race? Use mathematical evidence to support our answer.

2 For each of the following, draw the line that connects the two points. Write the coordinates of the two points then calculate the average rate of change on the specified interval.. f( ) = + 9 on [, ]. f( ) = + on [-, ]. + f on [0, ] ( ) = f ( ) = + 5 on [0, ] 5. f( ) = + on [-, -]. f( ) = on [-, 0] f( ) = on [0, ] 8. f ( ) = on [0, ] 9. f( ) ( ) = on [, 5]

3 0. Population Growth: Suppose 5 flour beetles are left undisturbed in a warehouse bin. The beetle population doubles in size ever week. The equation P( ) = 5 can be used to determine the number of beetles after weeks. Complete the table below. Week 0 5 Population Calculate the average growth rate between weeks and. Calculate the average growth rate for the first five weeks [0, 5]. Which average growth rate is higher? Wh do ou think it is higher?. Gravit and the Moon: The gravitational constant on the moon is ver different from the gravitational constant on the earth. If a rocket were launched from the moon s surface with an initial velocit of meters per second, then the equation ht ( ) = 0.8t + t would model the rocket s height, in meters, at an time t, in seconds. Complete the table below. Time Height Calculate the average velocit of the rocket for the first seconds of flight. Calculate the average velocit of the rocket for to 9 seconds of flight. Calculate the average velocit of the rocket for 9 to 5 seconds. Which interval had the greatest average rate of the change? Wh do ou think this interval had the greatest average rate of change?

4 Write the coordinates of the endpoints of the interval then find the average rate of change over the specified interval.. f = + on [-, ] + ( ). f( ) = + on [-,]. f( ) = + 7 on [0, ] 5. f( ) = + 5+ on [-, - ] f( ) = + 5 on. ( ) [-, 0] 7. f( ) = on [-, 0] 8. A rocket is launched with an initial velocit of 7 meters per second from a platform that is meter above the earth s surface. The height of the rocket was recorded at 0.5 second intervals. Scientists wondered how this launch would compare to a similar launch on the planets of Saturn and Venus. Based on the gravitational constants of each planet the came up with the following models. Time (Seconds) Earth Saturn Venus ( ) Height (Meters) Height (meters) Time (seconds) ht =.5t + 7t+

5 a. What is the domain for each scenario? b. What is the range for each scenario? c. On which planet would the rocket reach the greatest height? d. On which planet would the rocket reach its maimum height the fastest? e. On which planet would the rocket have the greatest average rate of change for the first three seconds? f. On which planet would the rocket have the greatest average rate of change between and seconds? g. On which planet would the rocket be in the air the longest? h. On which planet would the rocket hit the ground in the shortest amount of time?