- Suppose that the measure of quantity A changes at a non-zero constant
rate with respect to the measure of quantity B. Is it therefore true
that the measure of quantity B changes at a constant rate with respect
to the measure of quantity A? Explain.
Let c represent the speed of light (299,792.458 km/sec), let
d represent the distance a light particle has traveled, and let
t represent elapsed time, in seconds.
- Write the meanings of “dd” and “dt” in words. (Do not
write “dee dee” or “dee tee”!)
- Note that “d” and “d” have different meanings in “dd”.
What are their respective meanings?
- What is dd when dt = 0.000 025 sec? Be sure to
include the unit of dd.
- What is dd when dt = 3.25 sec? Be sure to include
the unit of dd.
- e) What is dt when dd = 2 km?
- What is dd when dt = -0.000 025 sec? What does it
mean for dt or dd to have a negative value?
A train traveled at a constant speed of 105 km/hr for 0.15
hours. Let s represent the number of kilometers the train
traveled and let t represent the number of hours it has
- Write the meanings of “ds” and “dt” in words. (Do not
write “dee ess” or “dee tee”!)
- What is ds when dt = 0.001 sec?
- What is dt when ds is 1.5 km?
- What is ds when dt = -0.001 sec? What does it mean
for dt or ds to have a negative value?
A cylindrical tank is 3 m tall with radius 1.5 m. It fills from its
bottom by a pipe over a period of 5 minutes. The tank begins partially
filled, fills rapidly at first and then slowly. Let V
represent the water’s volume in $m^3$, and let h represent the
water’s height in meters.
- The water’s volume does not change at a constant rate with
respect to time. Does the water’s volume change at a constant rate
with respect to its height in the tank? Explain.
- Explain what dV and dh mean in this context.
- Express dV in terms of dh.
- When you express dV in terms of dh, why is it
important that the water’s volume changes at a constant rate with
respect to its height?
- In this situation, how is the meaning of V different from
the meaning of dV?
One common form of constant rate of change is
constant speed, which occurs when changes in a
quantity are proportional to changes in time.
For Questions 5 - 7: A car is traveling at a constant speed of 65
miles/hour since entering the highway. Therefore, the number of miles it
travels over any period of time at this constant speed is 65 times the
number of hours in that period of time.
What quantities are y and x in this statement?
What are dy and dx? State the relationship between dy
and dx symbolically.
If after driving all day a car travels for another 0.3 hours at a
constant speed of 65 mi/hr, during that period of time the number of
miles the car has traveled that day increases by (65)(0.3) miles. What
are y and x in this statement? What are dy
and dx? What is m?
Is there a difference between stating that y = mx
and that dy = m dx? Explain in terms of
this car’s motion.
A different car left Phoenix headed toward Los Angeles. It is now
traveling at a constant speed of 57 mi/hr. Let y be the
car’s number of miles from Los Angeles (not Phoenix) and x
be the number of hours driven that day. Which statement about the
relationship between y and x is true? Explain.
- y = mx
- dy = m dx
A barrel is being filled with water at the constant rate of 2.5
gal/min. The barrel already had 7 gallons when the spigot opened. Let
y be the number of gallons in the barrel and let x
be the number of minutes since the spigot opened. Which statement
about the relationship between y and x is true?
- y = mx
- dy = m dx
Suppose that the value of r changes at a constant rate of
3.75 with respect to p.
- If the value of p changes by 6, how much will the
value of r change?
- If the value of p is 28.3, what is the value of r?
- If the value of p changes by $-3.1$, how much does the
value of r change?
- If the value of r changed by $-70.3$, how much did the
value of p change?
- How much does the value of r change when the value of p
changes by $\Delta p$?
- How much does the value of p change when the value of r
changes by $\Delta r$?
The value of quantity B changes at a constant rate of 2/5
with respect to changes in the value of quantity A.
- Determine the values of dA and dB when the value
of A changes from:
i) 7 to 13 ii) 258 to
264 iii) -2.5 to 3.5
iv) 4 to 2.3 v) -49 to
-50.7 vi) 0.5 to -1.2
- Exercise 11.a was designed to highlight an important aspect of
constant rate of change that was not stated explicitly in this
section. What was that aspect of constant rate of change?
For each of the scenarios below, express your answer both in words
and using mathematical symbols (be sure to define your variables
- Joanne is purchasing fabric. The total cost of the fabric she
buys increases at a constant rate of $7.25 per yard of fabric
purchased. If she decides to purchase an additional amount of
fabric, what can you conclude about the resulting total cost?
- Between 1990 to 2003, the concentration of carbon monoxide in the
atmosphere decreased at a constant rate of 0.248 parts per million per
year. What does this mean for the carbon monoxide concentration
over a given interval of time in years between 1990 and 2003?
- Between 1980 and 2004, the number of Medicare enrollees increased at
a constant rate of 0.554 million people per year. Given some
duration of time in years between 1980 and 2004, what will the result
be for the number of Medicare enrollees?
A cyclist travels a long stretch of road at a constant speed of 18.4
- What two changing quantities are related by constant rate of change
in this situation? Define these quantities precisely and
thoroughly, including units. (One-word responses are
unacceptable, as they will be vague and incomplete!)
- What two quantities are proportional in this situation?
- Using the quantities you defined in part a. and/or b , explain the
meaning of 18.4 mph as …
ii) a relative size
- How far will the cyclist travel in 37 minutes? How long will
it take the cyclist to go 25 miles? Solve with rate of change
reasoning from this section.
Determine which of the following tables represent a constant rate of
change of x with respect to y. For those
tables that do, state any assumptions you must make to claim that x
changes at a constant rate with respect to y.
planes took off simultaneously from Phoenix to fly 1870 miles
to New York City, flying parallel paths. One plane flew at a
constant speed of 510 miles/hour. The second plane at first
flew at a constant speed of 420 mi/hr for 3.2 hours.
- At what constant speed must the second plane fly the remainder of
its trip so that the two planes arrive in New York at essentially the
- Sketch graphs of each plane’s distance from Phoenix versus number of
hours flown (assuming they take off and land at the same time). Sketch
both graphs on the same axes.
Jim can mow a rectangular lawn in 3 hours. Tom can mow the same lawn
in 2 hours. They decide to collaborate: They will start at opposite
ends and meet somewhere in between.
- What quantities are involved in this situation?
- Consider the quantity “fraction of the lawn mowed”. Does this
quantity change at a constant rate with respect to time when Jim and
Tom work together? If so, at what rate?
- Use your responses to (a) and (b) to determine the number of hours
that Jim and Tom take to mow the lawn when they mow it together.
The definition of constant rate of change given above
says that two quantities change at a constant rate with respect to
each other if changes in their measures are proportional. This is
actually vague. An object can have a speed of 0 mi/hr. Are changes in
the number of miles it travels proportional to changes in the number
of measured hours? Are changes in the number of hours measured
proportional to the distance it travels in that number of hours?
Refine the definition of constant rate of change so that it
accommodates this example.