A rock falls m down a perpendicular rock face. The distance covered by the rock (in m) is given by the formula where is the time in seconds, while the rock is still falling and hasn't reached the ground yet.
Calculate the average velocity of the rock in the first seconds.
Calculate the velocity of the rock after exactly seconds. (Use sequential difference quotients to make this calculation, but check your result with the graphing calculator.)
Calculate the velocity of the rock at the moment that it hits the ground.
Use your graphing calculator to look at the graph of the function .
Calculate the slope at using a sequence of difference quotients.
You can already tell from the graph if the slope should be positive or negative. What feature of the graph provides that information?
Determine the equation describing the tangent at for the graph of .
For the interval you are given the function rule .
Calculate the rate of change of at .
There is one other point in the graph of where the slope of the curve is the same as at point . Which point is that? Give an explanation for your answer.
At point the function does not have a value. What then is the slope of the graph at that point? And what happens to the graph?
The concentration of a certain substance that has been dissolved in water is decreasing over time according to the formula . In the formula, is in g/L (gram per liter) and in hours.
The amount of substance disappearing from the water is not the same every hour. Why is that?
How much of the substance on average disappears every hour during the first hours? (Give your answer rounded to two decimals.)
The rate of decay of this substance at time is not equal to the average amount that has disappeared every hour up to that point. Calculate this rate of decay and give your answer rounded to two decimals.
Here you see a graph of the growth of a tree (length in meters) over time (in years).
On average, how much longer does the tree get every year during the first years?
What is the speed of growth after exactly years? Give an estimate that is as precise as possible.
At which point do you see the fastest growth? Give an explanation.
What is the final speed of growth, presuming the tree remains healthy?
A firework follows a parabolic trajectory until is explodes. This trajectory is given by the formula with both and in meters.
What is the slope of the trajectory when the firework is first launched?
Using the slope found in part a) you can use trigonometry to calculate at what angle the firework has been launched. Calculate an exact value of this angle in degrees.
At what point of the trajectory do you see a slope of ?
The firework explodes after is has covered meter of horizontal distance. How high up in the air is it at that point, and what is the slope of the trajectory?