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Section 12.0
From Functions of an Argument to
Functions In Relation to Each Other

Coordinating values of two functions of the same variable.

In prior chapters we defined fumctions that gave information about the value of one quantity with respect to the value of another directly.

We asked, for example, how far an object has traveled after some number of seconds have elapsed when given the object's rate of change of distance with respect to time.

The accumulation function we defined in terms of the object's rate of change with respect to time allowed us to answer questions about the object's distance traveled at any moment in time.

There are other questions that arise naturally in such situations that cannot be answered so directly. For example:

We have a function for a car's velocity at each moment in time since it began moving. What was the car's velocity at every distance it traveled?

The difficulty we meet in answering this question is that we have distance as a function of time and velocity as a function of time, but we do not have velocity as a function of distance.

A "Concrete" Example

Cement is being poured into a building's foundation at the rate of $r_f(t)=t^4e^{-t/2}\text{ m$^3$/min}$ $t$ minutes after the pouring chute opened. At what rate is cement being poured when $V\text{ m$^3$}$ of cement has been poured?

The cement's rate of change of volume with respect to time after $t$ minutes is $r_f(t)=t^4e^{-t/2}\text{ m$^3$/min}$; its volume after $t$ minutes is $\displaystyle{f(t)=\int_0^t r_f(u)du}\text{ m$^3$}$.

At first thought it seems we must solve for $t$ in terms of $V$ in $\displaystyle{\int_0^t r_f(u)du=V}$ and then substitute that expression for $t$ in $r_f(t)$.

Using integration by parts repeatedly, we get $$\begin{align} f(t)&=\int_0^t u^4e^{-u/2}du\\[1ex]&=-2e^{-t/2}\left(t^4+8t^3+48t^2+192t+384\right)+768.\end{align}$$

Solving for $t$ in terms of $V$ in $-2e^{-t/2}\left(t^4+8t^3+48t^2+192t+384\right)+768=V$ will be daunting.

However, we can use GC to take a graphical approach. We can command GC to produce a graph of $r_f(t)$, the rate of change of poured concrete's volume with respect to time, in relation to $f(t)$, the concrete's volume with respect to time. The idea will be to put values of $f(t)$ on the horizontal axis and values of $r_f(t)$ on the vertical axis. We can then inspect the graph for values of $r_f(t)$ that correspond to values of $f(t)$.

How to Envision Graphing $r_f(t)$ in relation to $f(t)$

Figures 12.0.1, 12.0.2, and 12.0.3 show three stages in envisioning a graph of values of one function in relation to values of another function when they share a common argument.

Figure 12.0.1 focuses on a way to imagine coordinating values of $r_f(t)$ and $f(t)$ to make points on the graph of $r_f$ in relation to $f$.

Figure 12.0.2 summarizes 12.0.1 by collecting values of both functions in a table and then plotting points from the table.

Figure 12.0.3 illustrates a way to think of coordinating values of $r_f(t)$ and $f(t)$ as the value of $t$ varies smoothly.

Coordinating values of $r_f(t)$ and $f(t)$

Figure 12.0.1. To sketch a graph of $r_f(t)$ in relation to $f(t)$, do this repeatedly:
(1) Select a value of $t$;
(2) Determine values of $r_f(t)$ and $f(t)$;
(3) Place the value of $f(t)$ on the horizontal axis;
(4) Place the value of $r_f(t)$ on the vertical axis;
(5) Plot the correspondence point $\left(f(t),r_f(t)\right)$.

Collecting values in a table

Figure 12.0.2. Collect corresponding values of $r_f(t)$ and $f(t)$ for values of $t$. Then plot the correspondence points $\left(f(t),r_f(t)\right)$.

Coordinating values of $r_f(t)$ and $f(t)$ as $t$ varies smoothly

Figure 12.0.3. Values of $r_f(t)$ and $f(t)$ vary as $t$ varies. We coordinate values of $r_f(t)$ and $f(t)$ by imagining the
value of $r_f(t)$ varying on the vertical axis as the value of $f(t)$ varies on the horizontal axis.

Reflection 12.0.1. Describe how Figures 12.0.1, 12.0.2, and 12.0.3 give complementary information.

Answering questions from graphs

The graph of $r(t)$ in relation to $f(t)$ allows us to approach the question of how fast concrete is being poured when $V$ $\text{ m$^3$}$ of concrete has been poured. Figure 12.0.4 shows the point $(547.187,51.7134)$ highlighted on the graph of $r_f$ in relation to $f$. This means that when the volume is 547.187 $\text{ m$^3$}$ the volume is changing at the rate of 51.7134 $\text{ m$^3$/min}$.

Figure 12.0.4. The concrete's volume is changing at 51.7134 $\text{ m$^3$/min}$ when its volume is 547.187 $\text{ m$^3$}$.

Using GC to graph values of one function in relation to values of another

Figure 12.0.5 shows the GC file that generated Figure 12.0.4.

Figure 12.0.5. The GC file that created Figure 12.0.4.
Define $r_f$ and $f$. Then press ctrl 2 to get a 2-row by 1-column array.
Then type = ctrl 2. In the second array, replace x by $f(30t)$ and y by $r_f(30t)$.
Note that, in GC, the value of t varies from 0 to 1 by default.

One aspect of GC's conventions is worth noticing. You can change the domain of $t$ (bottom right of equations pane in Figure 12.0.5) by typing new numbers for lower and upper values of its range.

Changing the range of values for $t$ is advisable only when you are graphing a single relationship, or when all relationships rely on $t$ having the same range of values. When different relationships rely on $t$ having different ranges of values it is better to leave the range of $t$ as $[0,1]$ and adjust the arguments accordingly of functions in the different relationships.

For example, in Figure 12.0.5 the arguments in $f(30t)$ and $r_f(30t)$ vary from 0 to 30 when the value of $t$ varies from 0 to 1. You could change the upper limit for $t$ to 30 (bottom right of equations pane) and use "$f(t)$" and "$r_f(t)$" in the statement that generates the graph.

Exercise Set 12.0.1

  1. Print a copy of this file. Then print a copy of this file and use it to sketch a graph of $x^2$ in relation to $x^3$.
  2. Print a copy of this file. Then print a copy of this file and use it to sketch a graph of $x^3$ in relation to $x^2$.
  3. Download this file and open it in GC.
    Note: GC is designed so that you must use "t" in statements that define a curve in terms of coordinates of points on it.
    1. Function $f$ is an exponential function with base $b$. Interpret values of $r(x)$ in relation to the function $f$.
    2. What is the purpose of the function $p$? Hint: Examine how $p(x)$ varies as $x$ varies from 0 to 1.
    3. Why does the graph of $r$ relative to $f$ not extend into the 2nd or 3rd quadrant for any value of $b$?
    4. The graph of $r$ relative to $f$ is a straight line for all values of $b>0$. What does this tell you about the relationship between the functions $r$ and $f$ for any value of $b>0$?
    5. What value of $b$ (approximately) makes the graph of $r$ in relation to $f$ coincide with the graph of $y=x$? What does this tell you about $r$ and $f$ when $b$ has this value? Do you recognize this value of $b$?

    Section 12.1 will refer back to Exercises 4 and 5, so attend to them closely.

  4. Download this file and open it in GC.
    1. What is the purpose of the argument "$2\pi t$" in $\cos()$ and $\sin()$?
    2. Why does the statement$$\left[\array{x \cr y}\right]=\left[\array{\cos(2\pi t)\cr \sin(2\pi t)}\right]$$produce a circle of radius 1 centered at the origin?

      Hint: For each value of t, $\cos(2\pi t)$ is a point's x-coordinate and $\sin(2\pi t)$ is that point's y-coordinate. Show this in a diagram and then use Pythagoras' Theorem and the identity $(\sin\theta)^2+(\cos\theta)^2=1$ to conclude that every point with coordinates $[\cos(2\pi t),\sin(2\pi t)]$ is 1 unit from $[0,0]$.

    3. How does the statement$$\left[\array{x \cr y}\right]=\left[\array{\cos(2\pi n)\cr \sin(2\pi n)}\right]$$make the highlighted point move around the circle when you play the $n$ slider?
    4. Modify the statement$$\left[\array{x \cr y}\right]=\left[\array{\cos(2\pi t)\cr \sin(2\pi t)}\right]$$so that it can produce a circle of radius $m$ for values of $m>0$.
    5. The notation $t\mapsto [x(t),y(t)]$ is commonly used to denote a relationship between x and y parametrically. It means "values of t are mapped to ordered pairs $[x(t),y(t)]$".

      Does the relationship $t\mapsto [\cos(2\pi t),\sin(2\pi t)],\,0\le t\le 1$ make the circle a function of $t$?

      Put another way: Is any value of $t$ mapped to more than one point on the circle? Does every point on the circle have a value of $t$ mapped to it? If your answer is "no" to either question, the circle is not a graph of a function. If both answers are "yes", then the circle is the graph of a function from values of t to points on the unit circle.

  5. Download this file and open it in GC.
    1. Show the graphs of $$\left[\array{x \cr y}\right]=\left[\array{\cos(2\pi t) \cr \sin(2\pi t)}\right]\text{ and }\left[\array{x \cr y}\right]=\left[\array{\sin(2\pi t) \cr \cos(2\pi t)}\right].$$Are the graphs the same?
    2. Show the points$$\left[\array{x \cr y}\right]=\left[\array{\cos(2\pi n) \cr \sin(2\pi n)}\right]\text{ and }\left[\array{x \cr y}\right]=\left[\array{\sin(2\pi n) \cr \cos(2\pi n)}\right].$$Now play the $n$ slider. Are the graphs the same? Explain.
    3. Circles, when graphed parametrically, have orientations. The orientation of a circle is determined by the direction of a point $[x(n),y(n)]$ that moves along it as the value of n increases through the parameter's domain.

      If the point moves clockwise as n increases, the circle has positive orientation. If the point moves counter-clockwise, the circle has negative orientation.

      What are the orientations of the circles created in Part (a) of this exercise? Explain.

  6. Download this file and open it in GC.
    1. The graph appears to be an ellipse. Is it? Hint: Do points on the graph satisfy the equation $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=c^2$ for all values of $a,\ b,\, c\gt 0$?
    2. Vary the value of $c$ for fixed values of $a$ and $b$. Any graph produced by varying the value of $c$ appears to be similar to every other graph that the statement produces with these values of $a$ and $b$. Is this true?
  7. The mass of a bacterial culture $x$ hours after measurements began changed at the rate of $r_m(x)=10\dfrac{e^{(x-5)}}{\left(e^{(x-5)}+1\right)^2}$ grams/hour. The culture's mass was 7.5 grams at $x=0$.
    1. At what rate was the culture's mass changing when its mass was 16 grams?
    2. What was the culture's mass at the moment its rate of change was greatest?
    3. After how many hours was the culture's mass 9 grams? How fast was the culture's mass changing at that moment?
  8. Let $r$ be the radius of s a spherical balloon as air is pumped into it. The balloon's volume is $V(r)=\frac{4}{3}\pi r^3$. Its surface area is $S(r)=4\pi r^2$.
    1. Use GC to produce a graph of a balloon's surface area in relation to its volume.
    2. Define S in terms of V symbolically. Compare the graph of surface area as a function of volume with the graph you generated in part (a).
    3. What advantage do you gain when defining surface area as a function of volume instead of just graphing the relationship parametrically?
  9. It is well known in physics that horizontal motion and vertical motion are independent in a gravitational field.

    This can be illustrated by an experiment where a bean bag is shot horizontally at the same moment that an object is released from the same height as the bean bag.

    The two will collide if they don't hit the ground first -- regardless of the bean bag's horizontal speed and regardless of the distance between object and bean bag. (See animation below.)

    1. Create a GC file that simulates the above animation. Use points for the bean bag and the cat.
    2. Modify your file so that bag and cat always meet the moment they reach the ground, regardless of their initial height or the distance between them.