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Section 4.1 Constant Rate of Change


The concept of rate of change is foundational to all of calculus. We are unable to give an overview that would convey its significance adequately because this whole book is about rate of change. So we’ll just start.

Two quantities change at a constant rate with respect to each other if, and only if, changes in their measures are proportional.

If two quantities change at a constant rate with respect to each other, and their measures are represented by x and y, then we will use dx to represent a change in x and dy to represent a change in y. The change dx is called a differential in x. The change dy is called a differential in y.

We speak of differentials in two quantities’ measures only when their measures change at a constant rate with respect to each other.

The letter “d” in “dy” means “a little change in”. So dy, a differential in y, just means “a little change in y”. Similarly, “dx” means “a little change in x”. If k represents the constant rate of change of y with respect to x then $\mathrm{d}y = k dx$. In words, “$\mathrm{d}y = k dx$” means “a little change in y is k times as large as the little change in x during which dy happened”. Again, we speak of differentials in two quantities’ measures only when the quantities change at a constant rate with respect to each other over some interval of the independent variable.

Suppose the mass m of a bacterial culture, measured in grams, changed at a constant rate of 5 grams/hr with respect to time, measured in hours, for a period of t hours. This means that within this period, any change in the culture’s number of grams will be 5 times as large as the change in the number of hours since measurements began that produced it.

Let dm represent a differential in the culture’s mass during this period, and let dt represent the differential in the number of hours during which dm happened. Then $\mathrm{d}m = 5\mathrm{d}t$, regardless of the value of dt. Notice: If dt is negative, we are representing a movement of t within intervals prior to the current time.
Important: All of a-d are necessarily true only under the condition that values of dt are within the period of time that the mass changed at a constant rate of 5 grams/hr.

Rate of change always involves two quantities’ measures. The measure of one quantity changes with respect to changes in the measure of another quantity. Any time you speak of a rate of change, you must say, or think, what the two quantities are and how they are measured — even if they are abstract measures like x and y. The measure of one quantity changes with respect to changes in the measure of another.

You will find your thinking clarified when you speak and think in terms of numbers rather than just in physical dimensions. For example, compare these two statements about a car’s speed:
The second phrase makes it clear to yourself and to others how you are measuring distance and time and that you are comparing changes in those measures. The first just says that distance and time are involved. If you do say the first phrase, you should be thinking the second phrase.

Exercise Set 4.1

  1. 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.
  2. 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.

    1. Write the meanings of “dd” and “dt” in words. (Do not write “dee dee” or “dee tee”!)
    2. Note that “d” and “d” have different meanings in “dd”. What are their respective meanings?
    3. What is dd when dt = 0.000 025 sec? Be sure to include the unit of dd.
    4. What is dd when dt = 3.25 sec? Be sure to include the unit of dd.
    5. e) What is dt when dd = 2 km?
    6. What is dd when dt = -0.000 025 sec? What does it mean for dt or dd to have a negative value?
  3. 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 traveled.

    1. Write the meanings of “ds” and “dt” in words. (Do not write “dee ess” or “dee tee”!)
    2. What is ds when dt = 0.001 sec?
    3. What is dt when ds is 1.5 km?
    4. What is ds when dt = -0.001 sec? What does it mean for dt or ds to have a negative value?
  4. 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.

    1. 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.
    2. Explain what dV and dh mean in this context.
    3. Express dV in terms of dh.
    4. 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?
    5. 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.
  5. What quantities are y and x in this statement? What are dy and dx? State the relationship between dy and dx symbolically.

  6. 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?

  7. Is there a difference between stating that y = mx and that dy = m dx? Explain in terms of this car’s motion.

  8. 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.

    1. y = mx
    2. dy = m dx
  9. 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? Explain.

    1. y = mx
    2. dy = m dx
  10. Suppose that the value of r changes at a constant rate of 3.75 with respect to p.

    1. If the value of  p changes by 6, how much will the value of r change?
    2. If the value of p is 28.3, what is the value of r?
    3. If the value of p changes by $-3.1$, how much does the value of r change?
    4. If the value of r changed by $-70.3$, how much did the value of p change?
    5. How much does the value of r change when the value of p changes by $\Delta p$?
    6. How much does the value of p change when the value of r changes by $\Delta r$?
  11. The value of quantity B changes at a constant rate of 2/5 with respect to changes in the value of quantity A.

    1. 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
    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?
  12. For each of the scenarios below, express your answer both in words and using mathematical symbols (be sure to define your variables first).

    1. 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?
    2. 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?
    3. 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?
  13. A cyclist travels a long stretch of road at a constant speed of 18.4 mph.

    1. 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!)
    2. What two quantities are proportional in this situation?
    3. Using the quantities you defined in part a. and/or b , explain the meaning of 18.4 mph as …
                                    i) a multiplier            ii) a relative size
    4. 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.
  14. 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.

  15. Two 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.

    1. 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 same time?
    2. 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.
  16. 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.

    1. What quantities are involved in this situation?
    2. 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?
    3. 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.
  17. 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.