Functions of Two Variables

Transcription

1 Unit #17 - Functions of Two Variables Some problems and solutions selected or adapted from Hughes-Hallett Calculus. Functions of More Than One Variable 1. The balance, B, in dollars, in a bank account depends on the amount deposited, A dollars, the annual interest rate, r%, and the time, t, in months since the deposit, so B = f(a, r, t). (a) Is f an increasing or decreasing function of A? Of r? Of t? (b) Interpret the statement f(1250, 1, 25) Give units. (a) As the amount deposited increases, so too does the bank balance f increases with increases to A. As the interest rate increases, so does the amount earned and therefore the final bank balance f increases with increases to r. As the length of time increases, more interest is earned f increases with increases to t. (b) f(1250, 1, 25) 1276 means that an initial deposit of $1250, put in an account at 1% interest for 25 months, will grow to approximately $ The temperature adjusted for wind-chill is a temperature which tells you how cold it feels, as a result of the combination of wind and temperature. Some wind-chill values are shown below. Wind Speed (km/h) Temperature ( o C) Source: Environment Canada (a) If the temperature is -5 o C and the wind speed is 20 km/h, how cold does it feel? (b) If the temperature is -10 o C, what wind speed makes it feel like -20 o C? (c) If the temperature is -15 o C, what wind speed makes it feel like -25 o C? (d) If the wind is blowing at 20 km/h, what temperature feels like -25 o C? (a) According to the wind-chill table, it feels like -12 o C. (b) A wind of 30 km/h, according to the table. (c) About 25 km/h, halfway between the table entries 20 km/h (-24 o C) and 30 km/h (-26 o C). (d) About -16 o C. One sixth of the way from the table entries 15 o C (feels like 24 o C) and 20 o C (feels like 30 o C). 3. A car rental company charges $40 a day and 15 cents a km for its cars. (a) Write a formula for the cost, C, of renting a car as a function, f, of the number of days, d, and the number of km driven, m. (b) If C = f(d, m), find f(5, 300) and interpret it. (a) C = f(d, m) = 40d m (b) f(5, 300) = 40(5) (300) = $245. It costs $245 to a rent a car for five days and drive it for 300 km. 1

2 4. The concentration, C, in mg per liter, of a drug in the blood as a function of x, the amount, in mg, of the drug given and t, the time in hours since the injection. For 0 x 4 and t 0, we have C = f(x, t) = te t(5 x) (a) f( 3 (b) (a) Find f(3, 2). Give units and interpret in terms of drug concentration. (b) Graph the following two single variable functions and explain their significance in terms of drug concentration. }{{} x (i) f(4, t) (ii) f(x, 1), 2 }{{} t ) = 2e ( 2)(5 3) mg/liter. f(3, 2) gives the concentration of drug (mg/liter) in the blood after a dose of x = 3 mg and t = 2 hours since the injection. (i) If x is fixed at 4, we have C(t) = te t(5 4) = te t. We graphed this function a few weeks ago. For t > 0, C is always positive, and it starts at t = 0, C = 0, and as t, C 0. This graph shows the drug concentration over time given an initial injection of x = 4 mg. (ii) If t is fixed at 1, we have C(x) = 1e 1(5 x) = e x 5. This represents the amount of drug in the body after 1 hour, depending in the original dosage. Spatial Reasoning 5. A cube is located such that its top four corners have the coordinates (-1, -2, 2), (-1, 3, 2), (4, -2, 2) and (4, 3, 2). Give the coordinates of the center of the cube. By drawing the top four corners, we find that the length of the edge of the cube is 5. We also notice that the edges of the cube are parallel to the coordinate axes. 2

3 The x-coordinate of the the center is halfway between two x coordinates = Similarly, the y-coordinate of the the center equals = The z-coordinate of the the center are 2.5 unit below the top side, at = You are at the point (-1, -3, -3), standing upright and facing the yz-plane. You walk 2 units forward, turn left, and walk for another 2 units. (a) What is your final position? (b) From the point of view of an observer positioned far out along the positive x axis, are you in front of or behind the yz-plane? To the left or to the right of the xz-plane? Above or below the xy-plane? (a) You are walking in the positive x direction 2 units. When you take your left turn, you are pointing in the positive y direction, and move 2 more units. Your final position is therefore ( 1 + 2, 3 + 2, 3) = (1, 1, 3). (b) From the perspective of someone far out along the positive x axis, you would be: in front of the yz-plane (x > 0), to the left of the xz-plane (y < 0), and below the xy-plane (z < 0). 7. Sketch the graph of the equation x = 3 in 3-space. The graph is a plane parallel to the yz-plane, and passing through the point (-3, 0, 0). 3

4 8. Find a formula for the shortest distance between a point (a, b, c) and the y-axis. For any point along the y axis, the coordinates are (0, y 0, 0). I.e. x = 0 and z = 0, but y can be any value, which we represent by y 0. The distance between the given point (a, b, c) and (0, y 0, 0) can be computed with the regular distance formula, finding the distances along each dimension squared, then taking the square root of the total: Distance = (a 0) 2 + (b y 0 ) 2 + (c 0) 2 = a 2 + c 2 + (b y 0 ) 2 Our goal is make this distance as small as possible, and the only thing we can control is our selection of y 0. Looking at the form D = a 2 + c 2 + (b y 0 ) 2, choosing y 0 can only affect the last term under the square root, (b y 0 ) 2, and because that last term is squared, the smallest value we can make it is zero. We do this by choosing our y 0 = b so that (b y 0 ) 2 = (b b) 2 = 0. If we choose that point along the y axis, then the distance from our point on the y axis to the given point (a, b, c) will be given by: D = a 2 + c 2 + (b b) 2 = a 2 + c 2 9. Find the equations of planes that just touch the sphere (x 2) 2 + (y 3) 2 + (z 3) 2 = 16 and are parallel to (a) The xy-plane (b) The yz-plane (c) The xz-plane This is a sphere of radius 4, and with a center at (2, 3, 3). (a) Planes at z = = 7 and z = 3 4 = 1 will just touch the sphere. (b) Planes at x = = 6 and x = 2 4 = 2 will just touch the sphere. (c) Planes at y = = 7 and y = 3 4 = 1 will just touch the sphere. 10. Find an equation of the largest sphere contained in the cube determined by the planes x = 2, x = 6, y = 5, y = 9 and z = 1, z = 3. This is a cube with sides of length 6 2 = 4. Therefore the largest sphere we can fit inside will have diameter 4, or radius 2. The center of the sphere must be at the center of the cube. This is halfway along each of the edges, at x = 4, y = 7 and z = 1. Therefore, the equation for the largest sphere inside this cube is (x 4) 2 + (y 7) 2 + (z 1) 2 = 2 2 4

5 Graphs of Functions of Two Variables 11. Without a calculator or computer, match the functions with their graphs in the figure below. (a) z = 2 + x 2 + y 2 (b) z = 2 x 2 y 2 (c) z = 2(x 2 + y 2 ) (d) z = 2 + 2x y (e) z = 2 (a) (IV)- Has its minimum at (0, 0, 2) and is bowl-shaped upwards (b) (II)- Has its maximum at (0, 0, 2) and bowl-shaped downwards (c) (I) - Has its minimum at (0, 0, 0), and bowl-shaped upwards (d) (V) - Is linear with changes in x and y (e) (III) - Has a constant z value for every x, y. In Problems 12-15, sketch a graph of the surface and briefly describe it in words. 12. z = 3 A flat plane parallel to the xy plane, but at height z = 3. 5

6 13. z = x 2 + y A radially symmetric bowl shape, opening upwards, with its minimum at (0, 0, 4). 14. z = y 2 A parabolic shaped trough. constant. The parabolas are parallel to yz plane, while moving out along the x axis, the shape is 15. x 2 + y 2 = 4 z does not appear in this equation, so its value is arbitrary. Since the formula would be for a circle in 2 dimensions, and we can place that circle at any z value, we get an infinitely long cylinder. 6

7 16. Without a calculator or computer, match the functions with their graphs in the figure below. (a) z = 1 x 2 + y 2 (b) z = e x2 y 2 (c) z = x + 2y + 3 (d) z = y 2 (e) z = x 3 sin y. (a) (I) - z always positive, has an asymptote at (x, y) = (0, 0) (b) (V) - z Always negative, has point (0, 0, -1) (height=-1 at the (x,y) origin), and z goes to zero as (x 2 + y 2 ), or as you move away from the origin. (c) (IV) - Function is linear in x and y. (d) (II) - z is always negative. Function is the same as x changes. (e) (III) - Process of elimination 7

8 17. Without a computer or calculator, match the equations (a)-(i) with the graphs (I)-(IX). (a) z = xye (x2 +y 2 ) (b) z = cos( x 2 + y 2 ) (c) z = sin y (d) z = 1 x 2 + y 2 (e) z = cos 2 x cos 2 y (f) z = sin(x2 + y 2 ) x 2 + y 2 (g) z = cos(xy) (h) z = x y (i) z = (2x 2 + y 2 )e 1 x2 y 2 This matching problem requires a combination of intuition, analysis of different properties, and elimination. The key ingredients we used to identify the graphs are indicated in our selection. 8

9 (a) (IV) - Function is positive in opposite quadrants (xy > 0), negative in other two opposite quadrants (xy < 0). Graph has value z = 0 at (0, 0), at heads towards z = 0 as we move away from the origin. Along each axis, z = 0. (b) (IX) - radially symmetric, has value z = 1 at (0, 0), and the peaks are the same height as you move away from (0, 0). (c) (VII) - looks like a sin wave when we cut along y, function doesn t change with x (d) (I) - z always negative, asymptotic at (0, 0). (e) (VIII) - z always positive and oscillating. Along x = 0 or y = 0, we should get something like cos graph (really cos 2 ). (f) (II) - radially symmetric, z 1 as (x, y) (0, 0), like sin(x)/x, z 0 as we move away from the origin (g) (VI) - oscillates. Along x = 0 and y = 0, the function always has value z = cos(0) = 1. (h) (III) - z always positive, is linear in each quadrant (i) (V) - mostly process of elimination, though the fact that z is always positive, and it has some shapes like e x2 from 1D functions is reassuring. 18. For each of the following functions, decide whether it could be a bowl, a plate, or neither. If it is a bowl, indicate whether it would hold water. Consider a plate to be any fairly flat surface; for a bowl, identify whether it could hold water or not based on the positive z-axis being up, and gravity acting down. (a) z = x 2 + y 2 (b) z = 1 x 2 y 2 (c) x + y + z = 1 (d) z = 5 x 2 y 2 (e) z = 3 (a) Could be a bowl. It is radially symmetric, and z increases as we move away from (0, 0). It would hold water, as the opening in the bowl is towards the positive z direction. (b) Is a bowl shape, but upside-down so it would not hold water. It is radially symmetric, but z decreases as we move away from (0, 0). (c) This is a plane (or could be called a plate-shape). (d) This represents part of the sphere z 2 + x 2 + y 2 = 5. Since we have the negative root, it represents the bottom of the sphere, and so could be called bowl-shaped. It opens upwards, so it would hold water. (e) This is the completely flat plane, which could be a plate. 19. By setting one variable constant, find a plane that intersects the graph of z = 4x 2 y in a: (a) Parabola opening upward (b) Parabola opening downward (c) Pair of intersecting straight lines This problem requires some experimentation. Just try selecting some values of x or y, and see what you get. (a) Select the plane y = 0. Then z = 4x 2 + 1, which is a parabola which opens up (b) Select the plane x = 0. Then z = y = 1 y 2, which is a parabola that opens down (c) Select the plane z = 1. Then 1 = 4x 2 y 2 + 1, or y 2 = 4x 2. This has the solutions y = ±2x, which makes two straight lines. 20. By setting one variable constant, find a plane that intersects the graph of z = (x 2 + 1) sin(y) + xy 2 in a: (a) Parabola (b) Straight line (c) Sine curve 9

10 Again, experimentation is key for solving a problem like this. Try simple values for x, y, or z, and see what happens to relationship. (a) Select the plane y = π 2. It s tempting to try y = 0, but since sin(0) = 0 and 02 = 0, we don t have anything left (z always equals zero) if y = 0. Selecting y = π 2 keeps the sin function in play: z = (x2 + 1) + x π2. This is a parabola. 22 (b) We need to get rid of the x 2 or y 2. If we pick a y so that sin(y) = 0, we ll get that. Select the plane y = π, so z = (x 2 + 1) sin(π) + xπ 2 = xπ 2. This is a straight line with slope π 2 9. Alternatively, a simpler answer is simply setting y = 0, which makes z = 0. This is about as simple a straight line as you can get. (c) Select the plane x = 0. Then z = sin(y) + 0y 2 = sin(y). 10