Note that some of the prompts on this page don’t ask for an exact (true) answer, but instead an approximation. Try to keep your approximations accurate to within one-thousandth of the true answer.
-
Morpheus is wearing a pedometer
that tracks the number of steps he’s taken today.
First, briskly sketch a scatterplot of this data with your independent variable \(t\) measured in hours since 6am.
Time 6am 8am 10am Noon 3pm 5pm 8pm 10pm Steps 0 531 3441 5590 6013 6778 11340 12019 - 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?
- In your 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 this person asked you to look at this data and to tell them you how fast they were walking (steps/hr) at exactly 5pm, how would you estimate this? Note: we can’t know this only from this data, so your task is only to make an informed estimation.
-
A particle moves back and forth along a straight line. After designating a point along its path as the origin, declaring that location to be position zero, you begin observing the particle at time \(t=0\) and surmise that the distance, in miles, the particle is from the origin after \(t\) seconds is given by the function
\(\displaystyle f(t) = \cos\bigl(t^2-3t+1\bigr) \,.\)- 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 \(t=0.01?\)
- Continue what you were doing in the previous part to approximate the speed of the particle the moment you began observing it.
- Approximately, what is the slope of the line tangent to the curve \(y = \sqrt{x}\tan(x)\) at the point where \(x=3?\)
-
A small child on the roof of the Empire State Building
hurls a penny towards the city streets below.
The height of the penny after \(t\) seconds,
willfully ignoring air resistance,
can be fairly accurately modelled by the function
\(\displaystyle f(t) = 1250 - 42t - 5t^2\,.\)
- 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?
-
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?
- The average velocity of the dragster for the entire pass was 132ft/s. Conceivably, since the dragster starts out at 0ft/s and ends the pass going much faster than 132ft/s, at some time during the pass it was going exactly 132ft/s. Can you estimate at what time during the pass the dragster’s velocity was 132ft/s?
- What’s an equation for the line tangent to the graph \(y= f(x)\) of the function \(f(x) = \sqrt{3x^2-8}+2\) at the point \((2,f(2))?\)