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We will use the letter "d" preceding a variable to mean that the variable's value "varies a little bit".
The way to envision a differential is to first keep in mind that a variable represents the value of a quantity whose value varies. Variables vary, always.
Another key idea in understanding differentials is that they are variables whose values vary "a little bit". To vary a little bit, they must vary from somewhere. We therefore think of differentials of an independent variable as varying through fixed intervals in the variable's domain.
Finally, a differential is a variable by which another variable varies. This surely sounds confusing. Figure 4.1.1 illustrates this idea. The term "left(x)" in Figure 4.1.1 means the left end of the interval containing the current value of x.
Figure 4.1.1. Variables and Differentials. The value of x varies smoothly through its domain. Differentials in x (dx) are small variations in the variable's value. The symbols "dx" and "$\Delta x$" have differnt meanings.
Reflection 4.1.1 (1) Where is $\Delta x$ in Figure 4.1.1? (2) The caption to Figure 4.1.1 says that dx and $\Delta x$ have different meanings. According to Figure 4.1.1, what are their meanings and how do they differ?
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.
Constant rate of change always involves two quantities varying together smoothly and continuously.
Two quantities vary at a
constant rate with respect to each other if, and only if,
variations in one are proportional to variations in the other.
Suppose that y varies at a constant rate with respect to x. Then $dx$ is a variation (large or tiny) in x as x varies, and $dy$ is a variation in y as the value of y varies in relation to the variation $dx$.
The relationship between variations is $dy=m\cdot dx$ for some number $m,\, m \neq 0$. If the variation in the value of x is a variation from 0, then $dx=x$ and $dy=m\cdot x$.
The value of y when $x=0$ is represented by the letter b, thus $y=b$ when $x=0$.
Therefore, $y=mx+b$ is the general form of the relationship between values of two quantities that vary at a constant rate with respect to each other.
The letter m represents the value of that constant rate of change.
We speak of differentials in two quantities’ measures only when their measures vary at a constant rate with respect to each other.
The letter “d” in “$dy$” means “a little variation in the value of y”. Similarly, “$dx$” means “a little variation in the value of x”. If k represents the constant rate of change of y with respect to x then $dy=k\cdot dx$. In words, “$dy=k\cdot dx$” means “a little variation in y is k times as large as the variation in x during which dy happened”.
Again, we speak of differentials in two quantities’ measures only when the quantities vary 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, varied at a constant rate of 5 grams/hr with respect to time, measured in hours, for t hours. This means that within this period, any variation in the culture’s number of grams will be 5 times as large as the variation in the number of hours since measurements began that produced it.
Let $dm$ represent a differential (a varying change) in the culture’s mass during this period, and let $dt$ represent the differential (a varying change) in the number of hours during which $dm$ happens. Then $dm = 5dt$ as the value of $dt$ varies.
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 varied at a constant rate of 5 grams/hr.
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.
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.
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.
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.
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.
Suppose that the value of r varies at a constant rate of 3.75 with respect to p.
The value of quantity B varies at a constant rate of 2/5 with respect to variations in the value of quantity A.
For each of the scenarios below, express your answer both in words and using mathematical symbols (be sure to define your variables first).
A cyclist travels a long stretch of road at a constant speed of 18.4 mph.
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 varies at a constant rate with respect to y.
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.
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.
The definition of constant rate of change given above says that two quantities vary at a constant rate with respect to each other if variations in their measures are proportional. This is actually vague. An object can have a speed of 0 mi/hr. Are variations in the number of miles it travels proportional to variations in the number of measured hours? Are variations 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.