Homework One
Section 11.1
#4 and #7. The original table on page 3 of the text book shows the quantity of beef (in pounds of beef per week) that a family with an income of I (in thousands of dollars per year) would buy if the beef had a price of p (in dollars per pound.) Thus, the original table is a numerical representation of the function q = q(I,p). In problem 4, we want to define a new function that gives the total amount of money, M, that a family with an income of I would spend on beef if it had a price of p. This new function should give M in units of dollars per week. Analyzing the units, we see that multiplying q(I,p) by the price p will produce units of (dollars/ pound)*(pounds/ week) = (dollars/ week). Thus, the function M(I,p) can be written as M = q(I,p) * p. A table representation of this is shown below. Likewise, in problem 7, we seek a function which gives the proportion of a family’s income that is spent on beef in a week. A proportion should be dimensionless. If we take the amount of money spent per week (M) and divide by the family’s weekly income (that is, I divided by 52) then we get units of (dollars/ week)/ [(dollars/ year) * (weeks/ year)] = 1. The final function we want is P = P(I,p) = M(I,p)/(I/52) = 52pq(I,p)/I. A table of this information is presented below.
Price of beef, p ($/pound) |
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3.00 |
3.50 |
4.00 |
4.50 |
||
Income, I |
20 |
7.95 |
9.07 |
10.04 |
10.94 |
in $1000 |
40 |
12.42 |
14.18 |
15.76 |
17.46 |
per year |
60 |
15.33 |
17.50 |
19.88 |
21.78 |
80 |
16.05 |
18.52 |
20.76 |
22.82 |
|
100 |
17.37 |
20.20 |
22.40 |
24.89 |
|
The function M(I,p) as a table of values. |
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Price of beef, p ($/pound) |
|||||
3.00 |
3.50 |
4.00 |
4.50 |
||
Income, I |
20 |
0.0207 |
0.0236 |
0.0261 |
0.0284 |
in $1000 |
40 |
0.0161 |
0.0184 |
0.0205 |
0.0227 |
per year |
60 |
0.0133 |
0.0152 |
0.0172 |
0.0189 |
80 |
0.0104 |
0.0120 |
0.0135 |
0.0148 |
|
100 |
0.0090 |
0.0105 |
0.0116 |
0.0129 |
|
The function P(I,p) as a table of values. |
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What would these three functions be useful for? The original table of quantities would help a grocer place orders for meat to be sure that he or she would have enough on hand. The second table of money spent, would make pricing decisions and determine profit/ cost ratios. The final information would show how much money a typical family would have left to buy other things (like chicken, rice, etc.)
#11. If we consider the acceleration due to gravity, g, as a function of both the size of the planet, m (its mass,) and the distance we are from the surface, h, then we can determine the dependence of g on both h and m by considering these variables separately.
#12. We are given a table of values of wind chill (ie. The temperature, T, "it feels like") as a function of the actual temperature, t, (in degrees Fahrenheit) and the wind speed, w, (in mph.) One can view T as a function of the two variables t and w.
#18. The functions f(x,0) and f(x,1) represent two cross sections of the function f(x,t) with t fixed. Since f(x,t) represents the height above equilibrium that a piece of the vibrating string at position x at time t occupies, we note that fixing t is like taking a photo of the string at that time. It shows the height of each piece of string at that instant of time. Thus, f(x,0) is a picture of the string’s displacement right as the string is released. The function f(x,1) shows the same string at t = 1, when it has relaxed a bit and started to sink towards equilibrium.
Section 11.2
#3. Starting at the point (-1,-3,-3) and facing the yz plane with our head upright (toward +z) would place us looking toward the positive x axis (see diagram.) Walking forward 2 units would bring us to (-1+2,-3,-3) = (1,-3,-3). Turning left would face us toward the positive y axis so that walking forward 2 units brings us to (1,-3+2,-3) = (1,-1,-3). The observer on page 9 would see us in front of the yz plane (positive x coordinate,) left of the xz plane (negative y coordinate,) and below the xy plane (negative z coordinate.)#8. To find a formula for the shortest distance between the point (a,b,c) and the y axis we will need two things: the coordinates of a point on the y axis that is closest to (a,b,c) and the distance formula. In general, a point on the y axis has x = 0 and z = 0, but y can take any value. What value of y puts the point (0,y,0) closest to (a,b,c)? Draw a picture and remember that the shortest distance will create a line that would be perpendicular to the y axis. Thus, the closest point on the y axis is (0,b,0). The distance formula then gives the distance as . An alternative to this method would be to simply write down the distance between (0,y,0) and (a,b,c) as a function of y and then minimize this function over the possible y values. You should get the same end result.
#9. The set of all points whose distance from the x axis is 2 would form a cylinder of radius 2 which is centered so that its axis is the x axis.
#10. The set of all points (x,y,z) whose distance from the x axis (a point on the x axis would have coordinates (x,0,0) ) equals the distance to the yz plane (a point on the yz plane has coordinates (0,y,z) ) would be the set of all points (x,y,z) such that
. This can be simplified to the form which is the equation of a cone centered along the x axis. See the illustrations on page 48 of the text for a picture.
#14. The center of this rectangle is at (1,1,-2). If the length of 13 is parallel to the y axis, then the front and back of the solid are at y = (1+13/2) = 7.5 and y = (1-13/2) = -5.5, respectively. The top and bottom are at z = (-2 + 5/2) = 0.5 and z = (-2 – 5/2) = -4.5, since the height of 5 is parallel to the z axis. Lastly, the sides are at x = (1 + 6/2) = 4 and x = (1-6/2) = -2. These six equations are the equations of the planes which make up the six faces of the object. Each corner is a point that lies on three of these faces. Drawing a simple picture in 3D of the solid will give the corners coordinates (-2,-5.5,0.5), (4,-5.5,0.5), (4,-5.5,-4.5), (-2,-5.5,-4.5), (-2,7.5,0.5), (4,7.5,0.5), (4,7.5,-4.5) and (-2,7.5,-4.5).
#15. Using the distance formula (given on page 12) gives us the following (we denote the distance from point P to point PI by d(P, i).) We find: d(P,1) = , d(P,2) = , d(P,3) = and d(P,4) = . Thus, point 4 is closest to P.
#16. A line through the origin, lying in the xz plane, and such that as x increases, z decreases, would look much like the diagram to the right. Note the use of the blue lines and the green lines to help indicate perspective. Use this technique in your own drawings in order to help add The Third Dimension.#20. The equation given represents a sphere of radius 2 centered at the point (1,-3,2). Why is this? The definition of a sphere is the set of all points (x,y,z) such that each is the same distance (called the radius) from the center of the sphere. Thus, we can use the distance formula to write down that the distance between the point (x,y,z) and the point (1,-3,2) is 2. Squaring both sides results in the equation shown in problem 20 on page 14.
Section 11.3
#3. I encourage you to use WinPlot to graph each of these functions as a surface. This will help you in making a decision.
#5. Note that (a) and (b) have circular symmetry (since the contours are circles and the only appearance of the independent variables x and y is of the form ). Thus, only graphs (I) and (V) could match. Since (b) is always negative (e to a power is always positive, but the negative sign in front reverses this) we conclude that (b) must be (V). This leaves (a) to be matched with (I). Equation (c) represents a linear function, whose graph is a plane, so that it must be (IV). Equation (d) is missing any x dependence, indicating that it is a "cylinder" (see page 18-19). It must look the same for any value of x. Only graph (II) matches this (note that its cross-sections for fixed x are all upside down parabolas in a plane parallel to the yz plane.) That leaves (e) to be matched with (III).
#6. This problem deals with a verbal description of a surface. The independent variables are pizza and cola and the dependent variable is happiness. To answer the questions, try looking at the surfaces (I) –(IV) in terms of sections. Fix either the number of pizzas or the amount of cola and compare the graphs.
#8. In order to answer this question, look at each of the surfaces as a series of sections. Figure 11.37 shows what the sections should look like. At y = 0, the surface should be level at z = 0. For positive y values (like f(x,1) and f(x,2)) we see that the surface should be concave upward, have reflective symmetry in the x direction across the z axis, and should get steeper as y increases. The same is true for the negative y values, except that the surface should be concave downward. Thus, only graph (IV) works, since the others lack the reflective symmetry in the x direction (among other things.) One could also use the fact that f is negative for negative y and positive for positive y to answer this question. To get a better feel for the surface, use WinPlot to graph .
#9. To answer this question, build a quick (and easy) visual aid. Take a sheet of paper and fold it in half along the diagonal from the upper left to the lower right corner. Now fold each half up so that you can place the paper down and have a tent-like structure. Label the lower edge of the paper with t increasing to the right and the left edge with x increasing down the page. The origin should be on the hump in the center. This represents the surface shown in figure 11.39.