Some of these prompts don’t ask for an exact (true) answer, but instead an approximation; just approximate as well as you can. If you have control over the precision of your approximation, be accurate to within one-thousandth of the exact figure.
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Morpheus is wearing a pedometer to track the steps he’s taken today.
Time 6am 8am 10am Noon 3pm 5pm 8pm 10pm Steps 0 531 3441 5590 6013 6778 11340 12019 Briskly sketch a scatter plot of this data.
- On average, how many steps-per-hour did Morpheus take today?
- On average, how many steps-per-hour did Morpheus take this morning between 6am and noon?
- On your scatter plot, what is the slope of the secant line between the points corresponding to 8am and 3pm? What is the meaning of this slope in context of the situation?
- If Morpheus asks you to look at this data and to tell him you how fast he was walking (steps/hr) at exactly 5pm, how would you estimate this?
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A particle moves back and forth along a straight line. After designating a location on the line as
position zero
, you begin observing the particle at time \(t=0\) and surmise that the distance, in inches, the particle is from theposition zero
after \(t\) seconds is given by the function \(f(t) = \cos\bigl(t^2-3t+1\bigr) \,.\) (The cosine function expects an argument in radian measure.)- Approximately what is the average velocity of the particle between \(t=0\) and \(t=1?\) What about between \(t=0\) and \(t=0.5?\) What about between \(t=0\) and \(t=0.1?\) What about between \(t=0\) and \(t=0.01?\)
- Continuing in this manner, approximate the velocity of the particle the instant you began observing it.
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What is an equation for the line secant to the graph of \(f(x) = \tfrac{1}{2}(x-3)^2-1\) between its \(y\)-intercept and its vertex?
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A child on the roof of the Empire State Building hurls a penny towards the city streets below. The height of the penny, in feet, after \(t\) seconds, (ignoring air resistance) can be modelled by the function \({f(t) = 1250 - 42t - 5t^2\,.}\)
- According to this model, approximately how long does it take for the penny to hit the ground?
- According to this model, approximately what is the speed of the penny when it hits the ground?
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Suppose we have a model for how far \(d\) a dragster has travelled from the starting line for any time \(t\) between zero and ten seconds after it started a pass, and that model for the distance is given by a function \(f\) of time as
\(\displaystyle d \;=\; f(t) = t^3+32t\,.\)Approximately how fast was the dragster going at the \(t=10\) seconds mark, the moment it crossed the finish line?
- Two brothers just ran a 100-yard race. The older brother won by 7 yards. In other words, when the older brother reached the finish line the younger brother had run 93 yards. They decide to race again, this time with the older brother starting 7 yards behind the starting line. Assuming that the brothers both run the second race at the same speed as first race, will it be a tie? or does one them win?
- Stewart Suppose a man drives to work at a speed of 50 mph. On the return trip home he drives at more leisurely speed of 30 mph. What is the man’s average speed for the round trip?
- Stewart Two runners on a circular track start running laps at the same time in the same direction from the same starting point. One runner runs at a pace of a lap every 50 seconds, and the other runs at a pace of a lap every 30 seconds. How long before the runners first pass each other?
- Stewart Two trains are on same track racing toward each other on a collision course. The speed of first train is 50 mph and the speed of second train is 70 mph. At the moment the trains are 100 miles apart, a bee starts flying between the trains at 80 mph (fast bee!), starting on the nose of the first train, flying towards and touching the nose of the second, then returning to and touching the nose of the first, and so on, like an Olympic swimmer doing laps. What is the total distance travelled by the bee before the trains collide?