For much of your study of algebra, precalculus, and calculus you have been focused on working with functions that have a scalar input and a scalar output, like \(y=f(x)\text{.}\) In Chapter 10, we used vector-valued functions of one variable as our first case of multivariable functions. Vector valued functions of one variable have a single scalar input and an output corresponding to multiple variables that we either organized as a vector output, \(\vr(t)=\langle x(t),y(t),z(t) \rangle\) or as parametric functions (\(x(t)\text{,}\)\(y(t)\text{,}\) and \(z(t)\)) . Our work in Chapter 10 was very focused on paths in space and the motion of an object on these paths. This narrow focus was because we had just one direction to move (while staying on the path). We worked with vector-valued functions of one variable as our first new class of functions because the calculus of these object was relatively easy. We applied limits, derivatives, and integrals to these functions componentwise, but we saw how useful a combination of vector tools and calculus measurements was for describing many features of vector-valued functions of one variable and their graphs as paths in space.
Many problems (both theoretical and applied) involve a much larger space of inputs and outputs. For instance, when studying weather patterns and behavior it is useful to measure temperature or atmospheric pressure. Both temperature and pressure are scalar measurements (measured by a single number) that will vary over three dimensions (north/south, east/west, and over elevation). Temperature can be given by a function with a location in three dimensions as the input and the temperature at that location (a scalar) as an output. Wind direction and strength are also very important when working with weather patterns. Wind is measured with a vector (magnitude and direction) that varies with location in a three dimensional space. So wind would be given by a function with a location in space as the input and a vector as the output. We would call both the temperature and the wind functions functions of several variables because each of these functions has multiple scalar inputs. We will look at the calculus of multivariable functions with scalar outputs (like temperature and pressure) in the next couple of chapters and will look at studying functions with multivariable inputs and outputs in Chapter 13.
In Chapter 10, all of the applications and analogies were centered around an object moving along a curve in space. When working with functions of several variables, we will want to use a few different types of functions for our applications and examples because of the possible conceptual relationships between inputs and outputs. For instance, the first type of example we will consider is thinking of the elevation or height of land as being a function of map coordinates. In this case, the input and output are connected to make the surface and have the same units. In other words the input and outputs are each the same type of measurement and would be easily understood by the plot of a surface.
The example of temperature as a multivariable function as described above gives a different kind of interpretation where the output is a different kind of measurement than the inputs. For instance, think of a function whose input is the (x,y) location on a heated metal plate and the output is the temperature of the metal plate at this location. In this case, the input is thought of as a point or location and the output is a measure of thermal energy at that location. The inputs and outputs are different kinds of measurements, but a plot of the different temperature values across the different (x,y) locations would still visually make sense.
We can also consider a function that describes the horizontal distance traveled by a arrow before the arrow hits the ground as determined by the angle at which the arrow is shot and the initial speed of the arrow. In this case, the inputs to our function are an angle and a speed where as the output is a distance (horizontal displacement). In this case, the units and types of measurements for the inputs and outputs are all quite different and a plot of the corresponding surface of displacement, angle, and speed values would be a more abstract visual representation as compared to the temperature and elevation examples.
In the next few chapters, we will use these types of examples, as well as applications to economics, to motivate our study of functions of several variables. Our preview activity will use a basic finance example to get you used to the notation and descriptions of different aspects of functions of several variables.
Suppose you invest money in an account that pays 5% interest compounded continuously. If you have an initial investment of \(P\) dollars in the account, then \(A\text{,}\) the amount of money in the account after \(t\) years is given by
Now consider only the situation where the amount invested is fixed at 1000 dollars. Calculate the amount of money in the account after \(t\) years as indicated in Table 11.1.1. Round payments to the nearest penny.
Table11.1.1.Amount of money in an account with an initial investment of 1000 dollars.
Now consider the situation where we want to know the amount of money in the account after 10 years given various initial investments. Calculate the amount of money in the account as indicated in Table 11.1.3. Round payments to the nearest penny.
Table11.1.3.Amount of money in an account after 10 years
Up to this Chapter, we have primarily been concerned with functions of a single variable. Remember that a function is a rule that assigns exactly one output for each allowed input. For instance, the rule that assigns a student ID number to each student at your school is a function because each student (input) gets assigned one and only one ID number. The rule that assigns the classrooms for your courses this semester is NOT a function because you likely have more than one classroom (multiple outputs) for each student (a single input).
We saw the behavior of a function in Preview Activity 11.1.1, where each pair of inputs, \((P,t)\text{,}\) produces a unique output \(A(P,t)\text{.}\) Additionally, the two variables \(P\) and \(t\) had no real relation to each other. That is, we could choose any value of \(P\) without limiting what value \(t\) might have, and we could select any value of \(t\) to use without regard to what value \(P\) might have. For that reason we say that the variables \(t\) and \(P\) are independent of each other. Thus, we call \(A = A(P,t)\) a function of the two independent variables \(P\) and \(t\text{.}\) This is the key idea in defining a function of two independent variables.
A function \(f\) of two independent variables is a rule that assigns to each ordered pair \((x,y)\) in some set \(D\) exactly one real number \(f(x,y)\text{.}\)
defines \(f\) as a function of the three variables \(x\text{,}\)\(y\text{,}\) and \(z\text{.}\) In general, a function of \(n\) independent variables is a rule that assigns to an ordered \(n\)-tuple \((x_1, x_2, \ldots, x_n)\) in some set \(D\) exactly one real number. 1
You may notice that we use variable names like \(x,y,z\) when using three variables but when there are more than three variables, it will be convienent to index the variables like \(x_i\) so that we can refer to the variables by index. For example the fifth variable in \((x_1, x_2, \ldots, x_n)\) will be \(x_5\) and the last two variables will be \(x_{n-1}\) and \(x_n\text{.}\)
In this example, we will look at the domain and range of a couple of multivariable functions. Our first function will be given by \(f(x,y)=\sqrt{xy}\text{.}\) For the domain of \(f\) we can only use \((x,y)\)-inputs where the square root of \(xy\) is non-negative. We will use set-builder notation to describe this domain. The domain of \(f\) is \(\{(x,y)\in\mathbb{R}^2|xy \geq 0\}\text{.}\) This notation is read as “the set of \((x,y)\) coordinates in the cartesian plane such that x times y is greater than or equal to zero”. Set builder notation is very flexible because this follows the form \(\{\text{kind of objects}|\text{conditions on objects}\}\text{.}\) The symbol \(|\) is read as “such that”. Note that the domain of \(f\) does not say that \(x\) and \(y\) are both non-negative (quadrant I), but rather that the product of \(x\) and \(y\) needs to be non-negative (quadrants I and III).
When we consider the range of \(f(x,y)=\sqrt{xy}\) we need to think about what values can actually be outputs of \(f\text{.}\) You may be tempted to say that outputs are scalars, so the range should be all real numbers, \(\mathbb{R}\text{.}\) There is no way to get a negative output for \(f\text{,}\) so not all real numbers are actual outputs of \(f\text{.}\) Instead, the range of \(f\) will be the interval \([0,\infty)\) because those are the values you can actually get as an output from \(f\text{.}\)
For our second example, we will look at the domain and range of \(g(x,y)=\frac{\arcsin(y)}{2x}\text{.}\) First we need to recall the domain and range of the arcsin function are \([-1,1]\) and \([-\frac{\pi}{2},\frac{\pi}{2}]\text{,}\) respectively. So we will only be allowed to use inputs with \(y\)-values in the interval \([-1,1]\) and \(x\neq0\) (because we cannot divide by zero). Thus the domain of \(g\) is \(\{(x,y)\in\mathbb{R}^2 | x \neq 0 \text{ and } -1 \leq y \leq 1\}\text{.}\)
The range of arcsin is \([-\frac{\pi}{2},\frac{\pi}{2}]\) but since the output of \(g\) will be \(\frac{\arcsin(y)}{2x}\text{,}\) the range of \(g\) will be \(\mathbb{R}\text{,}\) all real numbers. For example, the input \((\frac{\pi}{2},\frac{1}{2a})\) will be sent to \(a\) for any value of \(a\neq 0\) and \(g(0,1)=0\text{.}\)
Subsection11.1.3Representations for Functions of Two Variables
Algebraic Rules: You have already seen one representation of a function of several variables, specifically, the algebraic notation like \(f(x,y)=\sqrt{xy}\) which shows an algebraic rule to get the output of the function \(f\) for given input values \(x\) and \(y\text{.}\) These kinds of representations are very convenient to use with the algebraic rules for derivatives or for computing the output of the function given particular inputs. The drawback to this representation is that you need a lot of intuition about the type of function being used to understand how the output is changing over a range of values.
Tables: One of the techniques we use to study functions of one variable is to create a table of values. We can do the same for functions of two variables, except that our tables will have to allow us to keep track of both input variables. We can do this with a 2-dimensional table, where we list the \(x\)-values down the first column and the \(y\)-values across the first row.
As an example, suppose we launch a projectile, using a golf club, a cannon, or some other device, from ground level. Under ideal conditions (ignoring wind resistance, spin, or any other forces except the force of gravity) the horizontal distance the object will travel before hitting the ground depends on the initial velocity \(x\) the object is given, and the angle \(y\) at which it is launched. If we let \(f\) represent the horizontal distance the object travels, then \(f\) is a function of the two variables \(x\) and \(y\text{,}\) and we represent \(f\) in functional notation by
where \(g\) is the acceleration due to gravity. (Note that \(g\) is constant, 9.8 meters per second squared or 32 feet per second squared.) To create a table of values for \(f\text{,}\) we list the \(x\)-values down the first column and the \(y\)-values across the first row. The value \(f(x,y)\) is then displayed in the location where the \(x\) row intersects the \(y\) column, as shown in Table 11.1.10 (where we measure \(x\) in feet per second and \(y\) in radians). For example, \(f(75,0.8)=175.7\) is shown in the table by looking at the value in the row with \(x=75\) and column corresponding to \(y=0.8\text{.}\)
Representing a function with a table should reinforce the idea that each of the two independent values for the input variables will have a corresponding output value, but you probably struggled to gain insight into the relationship described by the function when looking at a large table of values. Additionally, the table is only useful for the input values selected in the rows and columns. While this may seem like a very limited use case, many applied problems involve tables like this because an algebraic function can be extremely difficult to find for every case and it can be very expensive to test many input values to obtain the corresponding output values. You likely have already used spreadsheets to organize information in a multivariable setting like this.
Graphs: If \(f\) is a function of a single variable \(x\text{,}\) then we define the graph of \(f\) to be the set of points of the form \((x,f(x))\text{,}\) where \(x\) is in the domain of \(f\text{.}\) We then plot these points using the coordinate axes in order to visualize the graph. We can do a similar thing with functions of several variables. Table 11.1.10 identifies points of the form \((x,y,f(x,y))\text{,}\) and we define the graph of \(f\) to be the set of these points in \(\R^\text{.}\)
The graph of a function \(f = f(x,y)\) is the set of points of the form \((x,y,f(x,y))\text{,}\) where the point \((x,y)\) is in the domain of \(f\text{.}\)
We also often refer to the graph of a function \(f\) of two variables as the surface generated by \(f\text{.}\) Points of the form \((x,y,f(x,y))\) are in three dimensions. If we consider the graph of the distance function defined by \(f(x,y) = \frac{x^2 \sin(2y)}{g}\text{,}\) then the function \(f\) is continuous in both variables. So, when these points are plotted in the right hand coordinate system, the points connect to form a surface in 3-space. The graph of the distance function \(f\) is shown in Figure 11.1.12.
There are many graphing tools available for drawing three-dimensional surfaces as indicated in the Preface (see Links to interactive graphics in Features of the Text) and almost all of the in-text three dimensional graphics are generated by Sage. Since we will be able to visualize graphs of functions of two independent variables (using plots in three dimensions), but not functions of more than two variables, we will primarily deal with functions of two variables in this chapter. It is important to note, however, that the techniques we develop apply to functions of any number of variables and we will highlight these extensions to higher dimensions in Section 11.8.
You may have noticed that we used the notation \(z=f(x,y)\) in the caption to Figure 11.1.12, which is explicitly saying that we are viewing the \(z\)-coordinate in our plot as the output of our function \(f\text{.}\) In other words only \((x,y,z)\) points of the form \((x,y,f(x,y))\) will be a part of the graph plotted. In some situations it may be advantageous to express a graph of interest using a function with some input and output roles of in a different configuration of coordinates. For instance, the surface shown in Figure 11.1.13 will not be able to be expressed with the \(z\)-coordinate at a function of the \(x\)- and \(y\)-coordinates because there will need to be more than one \(z\) coordinate associated to inputs like \((x,y)=(0,1)\) (as highlighted by the red points in the figure). The points on this surface can be expressed with \(y\) as a function of \(x\) an \(y\text{,}\) specifically \(y=g(x,z) = \frac{x^2}{5}+z^2\text{.}\) We will typically look at examples where \(z\) is expressed as a function of \(x\) and \(y\text{,}\) but you should think carefully if it will be advantageous to consider a different coordinate view.
When we study functions of several variables we are often interested in how each individual variable affects the function in and of itself. In Preview Activity 11.1.1, we saw that the amount of money in an account depends on the amount initially invested and the duration of the investment. However, if we fix the initial investment, the amount of money in the account depends only on the duration of the investment, and if we set the duration of the investment constant, then the amount of money in the account depends only on the initial investment. This idea of keeping one variable constant while we allow the other to change will be an important tool for us when studying functions of several variables. 2
This will be the first of many times we will employ the following approach
where \(x\) is the initial velocity of an object in feet per second, \(y\) is the launch angle in radians, and \(g\) is the acceleration due to gravity (32 feet per second squared). If we hold the launch angle constant at \(y=0.6\) radians, we can consider \(f\) a function of the initial velocity alone. In this case we have
We can plot this curve on the surface by tracing out the points on the surface when \(y = 0.6\text{,}\) as shown at left in Figure 11.1.15. The formula for \(f(x,0.6)\) shows that \(f\) is quadratic in the \(x\)-direction. More descriptively, as we increase the launch velocity while keeping the launch angle constant, the horizontal distance the object travels increases proportional to the square of the initial velocity.
Similarly, if we fix the initial velocity at 150 feet per second, we can consider the distance as a function of the launch angle only. In this case we have
We can again plot this curve on the surface by tracing out the points on the surface when \(x=150\text{,}\) as shown at right in Figure 11.1.15. The formula for \(f(150,y)\) shows that \(f\) is sinusoidal in the \(y\)-direction. More descriptively, as we increase the launch angle while keeping the initial velocity constant, the horizontal distance traveled by the object is proportional to the sine of twice the launch angle.
The curves we define when we fix one of the independent variables in our two variable function are called traces. In Figure 11.1.15, we have made a copy of the trace off to the side of the figure so you can see each trace as a curve on the proper 2D slice
A trace in the \(x\) direction of a function \(f\) of two independent variables (\(x\) and \(y\)) is a curve of the form \(z = f(x,c)\text{,}\) where \(c\) is a constant. Similarly, a trace in the \(y\) direction of a function \(f\) of two independent variables (\(x\) and \(y\)) is a curve of the form \(z = f(c,y)\text{,}\) where \(c\) is a constant.
Understanding trends in the behavior of functions of two variables can be challenging, as can sketching their graphs; traces help us with both of these tasks.
Identify the \(y = 0.6\) trace for the distance function \(f\) defined by \(f(x,y) = \frac{x^2 \sin(2y)}{g}\) by highlighting or circling the appropriate cells in Table 11.1.10. Write a sentence to describe the behavior of the function along this trace.
Identify the \(x = 150\) trace for the distance function by highlighting or circling the appropriate cells in Table 11.1.10. Write a sentence to describe the behavior of the function along this trace.
In the next several tasks, we will be looking at using traces to help us draw an accurate plot of the surface given by \(z=g(x,y)=yx^2\text{.}\) For our first task, find the equation for the \(y=1\) trace of \(z=g(x,y)=yx^2\) and draw a graph of the \(y=1\) trace on the corresponding face in Figure 11.1.17.
Draw a few traces (at least three more) that correspond to values in the middle of the plot to fill in a plot of the surface given by \(z=x^2y\text{.}\)
The previous activity shows how drawing traces on a plot of a surface can give the viewer more information on the orientation of the graph. In Figure 11.1.18 you can see a plot of the surface generated in our projectile motion example. Without any of the traces drawn on the surface, it is difficult to understand the shape of the surface (and would be even harder if not for the shading on the computer generated graphics). You can click on the check box at the top of the figure to add traces to the plot and notice how much easier it is to distinguish the shape of the surface at different points.
When drawing plots by hand, it is very important to use parallel structure to make it easier for the viewer to orient the traces as being in a direction parallel to one of the coordinate axes. It may take time for you to feel comfortable drawing these kinds of features by hand but good visual representations of surfaces and their features will be very valuable in our work for the next several chapters.
As you saw in the previous subsection, traces give important information about how a slice of a surface looks like when holding one of the inputs constant. This corresponds to looking at the intersection of a fundamental plane of the form \(x=a\) or \(y=b\) with the surface given by \(z=f(x,y)\text{.}\) These fundamental planes are oriented vertically when we consider a conventional right handed coordinate system. In this section, we will explore the intersection of a surface given by \(z=f(x,y)\) with a fundamental plane of the form \(z=c\text{.}\) This will correspond to looking at the points on the surface with a fixed height (specifically \(z=c\)).
You may have seen topographic maps such as the one of the Porcupine Mountains in the upper peninsula of Michigan shown in Figure 11.1.19. 3
Map source: Michigan Department of Natural Resources, with permission of the Michigan DNR and Bob Wild.
The curves on these maps show the locations with a particular altitude (as labeled on the curve). The amount of space between these curves also depicts the rate of change in altitude: curves on the topographic map that are close together signify steep ascents or descents, while curves that are far apart indicate slower changes in elevation. Thus, these topographic maps with curves of constant elevation can tell us a lot about three-dimensional surfaces. Mathematically, if \(f(x,y)\) represents the altitude at the point \((x,y)\text{,}\) then each of the curves with constant elevation is the graph of an equation of the form \(f(x,y) = k\text{,}\) for some constant \(k\text{.}\)
Use the topographical map of the Porcupine Mountains in Figure 11.1.19 to answer the following questions. Note that points of interest are sometimes marked with an X and have their altitude listed.
Describe how your elevation would change if you walked in a straight line from the lowest to the highest elevation points. You may want to sketch a plot of the elevation along your path.
If you walk along the Big Carp River Trail (in the top left part of Figure 11.1.19), describe which parts of the trail will have steep increases in elevation and which parts you think will be the most like level ground.
Curves on a surface that describe points at the same height or level are called level curves and as you saw in the topographic map above, a plot with level curves can be useful for representing information on a surface using just a two dimensional plot.
A level curve (or contour) of a function \(f\) of two independent variables \(x\) and \(y\) is a curve of the form \(k = f(x,y)\text{,}\) where \(k\) is a constant.
Topographical maps can be used to create a three-dimensional surface from the two-dimensional contours or level curves. For example, level curves of the distance function defined by \(f(x,y) = \frac{x^2 \sin(2y)}{32}\) plotted in the \(xy\)-plane are shown at in Figure 11.1.21. You can change the number of contours in the plot by moving the slider at the top of Figure 11.1.21. Additionally, you will sometimes see a legend for contours shown when different colors are used or a contour plot with different colors and shading filling in the region between contours. You can toggle these options in the contour plot below by clicking on the checkbox next to the different options.
If we lift these contours and plot them at their respective heights, then we get a picture of the surface itself, as illustrated at right in Figure 11.1.22.
The use of level curves and traces can help us construct the graph of a function of two variables. For the surface definded by \(z=f(x,y) = \frac{x^2 \sin(2y)}{32}\) we can plot a grid of \(x\)-traces, \(y\)-traces, and contours to get a good idea of what the surface looks like as shown in Figure 11.1.23. If you compare the paths plotted in Figure 11.1.23 to Figure 11.1.12, you will see that the mesh shown on the surfaces in most plots IS a grid of traces of contours and that is how the orientation of surfaces is typically drawn by computer generated plots.
Isotherms are another example of where you are likely to encounter level curves. Isotherms are a plot of locations that have the same temperature and a plot with lots of isotherms will allow you to see how the temperature changes over a region.
In this activity, we will be making contour plots by hand and looking at how the spacing of contours in your plot should give an idea about how different surface shapes can be distinguished.
Let \(f(x,y) = x^2+y^2\text{.}\) Draw the level curves \(f(x,y) = k\) for \(k=1\text{,}\)\(k=2\text{,}\)\(k=3\text{,}\) and \(k=4\) on the left set of axes given in Figure 11.1.25. (You decide on the scale of the axes.) Explain what the surface defined by \(f\) looks like.
Let \(g(x,y) = \sqrt{x^2+y^2}\text{.}\) Draw the level curves \(g(x,y) = k\) for \(k=1\text{,}\)\(k=2\text{,}\)\(k=3\text{,}\) and \(k=4\) on the right set of axes given in Figure 11.1.25. You should use the same scale on these axes as in the previous task. Explain what the surface defined by \(g\) looks like
Compare and contrast the graphs of \(f\) and \(g\text{.}\) How are they alike? How are they different? Use traces for each function to help answer these questions.
The traces and level curves of a function of two variables are curves in space. Traces are much easier to parameterize as a curve in space because one input variable is fixed and the other input variable can act as the parameter. For example, the trace with \(y=b\) of the surface given by \(z=f(x,y)\) will be parameterized by \(\vr(t)=\langle t,b,f(t,b)\rangle\) because the \(z\)-coordinate can be expressed in terms of the fixed input value and the other input variable (as the parameter). Parameterizing contours can be quite difficult and often relies on knowing the particular shape of the contours. Remember that it may not be possible to express a curve of the form \(k=f(x,y)\) with one coordinate as a function of the other.
A function \(f\) of several variables is a rule that assigns a unique number to an ordered collection of independent inputs. The domain of a function of several variables is the set of all inputs for which the function is defined.
A trace of a function \(f\) of two independent variables \(x\) and \(y\) is a curve of the form \(z = f(x,c)\) (for a x-trace) or \(z = f(c,y)\) (for a y-trace), where \(c\) is a constant. A trace tells us how the function depends on a single independent variable if we treat the other independent variable as a constant.
A level curve of a function \(f\) of two independent variables \(x\) and \(y\) is a curve of the form \(k = f(x,y)\text{,}\) where \(k\) is a constant. A level curve describes the set of inputs that lead to a specific output of the function.
Match the surfaces with the verbal description of the level curves by placing the letter of the verbal description to the left of the number of the surface.
(a) Write a formula for the cost, \(C\text{,}\) of renting a car as a function of the number of days, \(d\text{,}\) and the number of miles driven, \(m\text{.}\)
A store sells CDs at one price and DVDs at another price. The figure below shows the revenue (in dollars) of the music store as a function of the number, \(c\text{,}\) of CDs and the number, \(d\text{,}\) of DVDs that it sells. The values of the revenue are shown on each line.
(Hint: for this problem there are many possible ways to estimate the requisite values; you should be able to find information from the figure that allows you to give an answer that is essentially exact.)
Consider the concentration, C, (in mg/liter) of a drug in the blood as a function of the amount of drug given, x, and the time since injection, t. For \(0 \leq x \leq 5\) mg and \(t \geq 0\) hours, we have
\begin{equation*}
C = f(x,t) = 26te^{-\left(5-x\right)t}
\end{equation*}
A manufacturer sells aardvark masks at a price of $180 per mask and butterfly masks at a price of $440 per mask. A quantity of a aardvark masks and b butterfly masks is sold at a total cost of $550 to the manufacturer.
Consider the concentration, \(C\text{,}\) in mg per liter (L), of a drug in the blood as a function of \(x\text{,}\) the amount, in mg, of the drug given and \(t\text{,}\) the time in hours since the injection. For \(0 \leq x \leq 4\) and \(t \geq 0\text{,}\) we have \(C = f(x,t) = t e^{-t(5-x)}\text{.}\)
Graph the following two single variable functions on a separate page, being sure that you can explain their significance in terms of drug concentration.
The Ideal Gas Law, \(PV = RT\text{,}\) relates the pressure (\(P\text{,}\) in pascals), temperature (\(T\text{,}\) in Kelvin), and volume (\(V\text{,}\) in cubic meters) of 1 mole of a gas (\(R = 8.314 \ \frac{\text{J} }{\text{ mol } \ \text{K} }\) is the universal gas constant), and describes the behavior of gases that do not liquefy easily, such as oxygen and hydrogen. We can solve the ideal gas law for the volume and hence treat the volume as a function of the pressure and temperature:
Use 2 or three additional traces in each direction to make a rough sketch of the surface over the domain of \(V\) where \(P\) and \(T\) are each nonnegative. Write at least one sentence that describes the way the surface looks.
When people buy a large ticket item like a car or a house, they often take out a loan to make the purchase. The loan is paid back in monthly installments until the entire amount of the loan, plus interest, is paid. The monthly payment that the borrower has to make depends on the amount \(P\) of money borrowed (called the principal), the duration \(t\) of the loan in years, and the interest rate \(r\text{.}\) For example, if we borrow $18,000 to buy a car, the monthly payment \(M\) that we need to make to pay off the loan is given by the formula
Create a table of values that illustrates the trace of \(M\) with \(r\) fixed at 5%. Use yearly values of \(t\) from 2 to 6. Round payments to the nearest penny. Explain in detail in words what this trace tells us about \(M\text{.}\)
Create a table of values that illustrates the trace of \(M\) with \(t\) fixed at 3 years. Use rates from 3% to 11% in increments of 2%. Round payments to the nearest penny. Explain in detail what this trace tells us about \(M\text{.}\)
Consider the combinations of interest rates and durations of loans that result in a monthly payment of $200. Solve the equation \(M(r,t) = 200\) for \(t\) to write the duration of the loan in terms of the interest rate. Graph this level curve and explain as best you can the relationship between \(t\) and \(r\text{.}\)
The range of a function is the set of all outputs the function generates. Given that the range of the square root function \(g(t) = \sqrt{t}\) is the set of all nonnegative real numbers, what do you think is the range of \(h\text{?}\) Why?
Sketch an overall picture of the surface generated by \(h\) and write at least one sentence to describe how the surface appears visually. Does the surface remind you of a familiar physical structure in nature?
