2 (x+2) 2 +(y 1) 2 a) What is the gradient of the potential at (1, 2)? 2 j at (1, 2)?

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1 Word Problem Review This handout gives you some experience in solving word problems. This is important, because as engineers and scientists, your mathematics will be worth nothing if you cannot look at a situation from your field and see what the relevant mathematics is for the situation. In each problem read the problem over and ask, What s the question? Have I seen a problem like this before? If you have done all of the syllabus problems, the computer labs, and looked over the back exams and review materials, then every problem on the final will be like some problem you have seen before. Read over the next problem, and ask yourself what kind of a problem is this, and where have I seen a similar one? (Space for your answers below) 1) A charge distribution on a plane is creating an electric field. The electrical potential P (x, y) measures the potential energy of a unit point charge due to its position in the field. The function is given by P (x, y) = 2 (x+2) 2 +(y 1) 2 a) What is the gradient of the potential at (1, 2)? b) What is the rate of change of P in the direction u = 1 2 i j at (1, 2)? c) An equipotential line is a curve on our plate along which the potential is constant. What is an equation for the tangent line of the equipotential passing through (1, 2)? d) The electric field vector is orthogonal to the equipotential lines, and points in the direction of decreasing potential. What direction does it point at (1, 2)? (Your answer should be a unit vector.) Read over the next problem, and ask yourself what kind of a problem is this, and where have I seen a similar one? (Space for your answers below) 2) (10 points) A current of 34 amperes branches into currents x, y, and z through resistors with resistances 1, 4, and 6 ohms as shown. (The current I1 through the top wire is x, while the resistor in the top wire has resistance R1 equal to 1, etc.) It is known that the current splits in such a way that the sum of the currents through the three resistors equals the

2 initial current. The energy E generated in a resistor of resistance R by current I is given by E = I 2 R. It turns out that nature always splits the currents so that the total energy is minimized. Find the current in each branch. Read over the next problem, and ask yourself what kind of a problem is this, and where have I seen a similar one? (Space for your answers below) 3) The temperature z in degrees centigrade at a point on a road, x meters East and y meters North from a fixed location (the origin), is given as z = f(x, y). If a bug at a point on the road moves East, the temperature increases at the rate of 0.8 degrees/meter If it moves North, the temperature decreases at the rate of 1.4 degrees/meter. At time t = 0, the bug decides to move along a path (x(t), y(t)) from its current location with velocity (x (0), y (0)) = (0.8, 1.4) meters/minute. What is the rate of change in temperature along the bug s path at t = 0? Read over the next problem, and ask yourself what kind of a problem is this, and where have I seen a similar one? (Space for your answers below) 4) (10 points) Marine biologists have determined that when a shark detects the presence of blood in the water, it will swim in the direction in which the concentration of the blood increases most rapidly. Based on certain tests, the concentration of blood (in parts per million) at a point P (x, y) on the surface of seawater is approximated by C(x, y) = 1 (2x 2 + 2y 2 )/10 4 where x and y are measured in meters in a rectangular coordinate system with the blood source at the origin. The positive x-axis points east and the positive y-axis due north. The approximation is considered acceptable within 70m radius. There s a shark sighted at P (50, 25). Starting from the shark s location: a) Is concentration increasing or decreasing due north? Find the rate of increase/decrease. b) How about due southwest? What is the rate? c) In what direction the concentration doesn t change? d) In what direction will the shark start swimming upon sensing the blood at the origin? What is the rate of change in that direction? e) In what direction does the concentration drop off the fastest? Read over the next problem, and ask yourself what kind of a problem is this, and where have I seen a similar one? (Space for your answers below) 5.) (10 points) An engineer wants to measure n the number of moles of an ideal gas in a container with a moveable piston. If the percent error in measuring volume is ±.5% and the percent error in measuring temperature and pressure is ±1%, what is the percent error in measuring n? (Recall the ideal gas equation is P V = nrt with R = ) Read over the next problem, and ask yourself what kind of a problem is this, and where have I seen a similar one? (Space for your answers below) 6.) (8 points) Now the piston in problem 4 is moving down. Suppose n is constant and n = 1 mole. At time t = 0, V =.1 liter, V is decreasing at a rate of.01 liters/sec, P is 2 atmospheres and is increasing at a rate of 82.2 atmospheres/sec. What is the rate of change of temperature with respect to time at this instant? Read over the next problem, and ask yourself what kind of a problem is this, and where have I seen a similar one? (Space for your answers below)

3 7.) (10 points) You want to put together a jewelry box with a volume of 12 cubic inches such that the top and bottom are made of mahogany and the remaining sides are made of cherry. If mahogany costs $3 per square inch and cherry costs $2 per square inch, what is the least you can spend to make such a jewelry box? Read over the next problem, and ask yourself what kind of a problem is this, and where have I seen a similar one? (Space for your answers below) 8.) The OPERA experiment at Gran Sasso Laboratory near Rome has recently reported a measurement of superluminal ie. faster than light! velocities for neutrinos from the particle accelerator at CERN in Geneva. The neutrino velocity v is computed from the usual formula v = l t, where l is the distance traveled by the neutrino, and t is the total time for the trip. (a.) [8 points] Compute the differential dv in terms of dl and dt. (b.) [6 points] The OPERA scientists have measured the distance from CERN to Gran Sasso to be l = m and the neutrino time-of-flight to be t = s. If the errors on these measurements are l = ±0.2 m and t = ±10 5 s, estimate the maximal error on the neutrino velocity v. Include three significant figures in your answer, and be sure to give units. (c.) [6 points] The OPERA scientists have reported the percent difference between the neutrino velocity v and the speed of light c to be δ = v c = %. c If we assume the percent error on l to be l l 0.2 m = ± m = ± %, and we want the corresponding percent error on v to be at most ±10 3 % (so that the reported result for δ is trustworthy), what is the largest percent error that we can tolerate in the measurement of the time t? Again, report your answer to three significant figures.

4 Lines and Planes Review 1) Find the three forms of the equation of the line through the given point with the given direction: A) Point= (1, 2, 3), Direction=< 4, 5, 2 > B) Point= (1, 2, 3), Direction=< 4, 5, 0 > C) Point= (0, 2, 3), Direction=< 2, 0, 3 > 2) Find the three forms of the equation of the line determined by the points A) (1, 2, 3), (7, 8, 9) B) ( 2, 3, 4), (6, 0, 3) 3) Find the direction of the line with the given equation and find a point on the line. A) r(t) =< 3, 5, 7 > +t < 2, 4, 3 > B) r(t) =< 2 t, 4, 6 + 8t > 4) Are the lines with equations r(t) =< 3, 5, 7 > +t < 2, 4, 3 > parallel? r(t) =< 6 + 3t, 10 6t, 7 + 9/2t > 5) Show that the lines given below meet in a common point and find the angle between the lines. r(t) =< 3, 5, 7 > +t < 2, 4, 3 > Planes r(t) =< 3, 5, 7 > +t < 3, 3, 2 > 1) Find the equation of the plane through the given point with the given normal: A) Point= (1, 2, 3), Normal=< 4, 5, 0 > B) Point= (0, 2, 3), Normal=< 2, 0, 3 > C) Find another normal for A) and give a new equation for the plane using your normal. 2) Find the equation of the plane passing through the point (0, 0, 0) and perpendicular to the line with equation r(t) =< 3, 5, 7 > +t < 2, 4, 3 >. 1

5 3) Find the equation of the plane with x-slope 7 and y-slope 2, passing through < 3, 5, 7 >. Also find a normal to the plane. 4) Find the equation of the plane passing through the point (0, 1, 0) and parallel to the plane with equation. 2x 4y + 3z = 7. 5) Find the angle between the two planes with equations x + y + z = 1 x y z = 1 6) Find the equation of the line of intersection of the two planes of 5) (Hint: a common point is (1, 0, 0).) 7) What is the angle between the line with equation r(t) =< 1, 0, 0 > +t < 1, 1, 1 > and the plane with equation x y z = 1. 2

6 Linearization and Differentials. 1) The following table gives the Wind Chill f(t, w) for various values of Temperature ( F ), and wind velocity w. T = 35 T = 30 T = 25 T = 20 w = w = w = a) Use the table to find approximate values for f f x (30, 15), y (30, 15). b) Give an interpretation of the values of the partial derivatives you found in a). c) Using the table and your answers to a) give the local linearization for T at (30, 15). d) Use the local linearization to approximate f(31, 17). 2) The following table gives the Volume (in cubic feet) at various temperatures and pressures of one pound of steam. Pressure, p, is measured in (lb/in 2 ), while temperature, T is measured in ( F ). p = 20 p = 22 p = 24 p = 26 T = T = T = a) Use the table to find approximate values for V p, V T when T = 520 and p = 24. 1

7 b) Using the table and your answers to a) give the local linearization for V when T = 520 and p = 24. c) Use the local linearization to approximate V when T = 530 and p = 25. d) Use the differential to approximate the change in V if the temperature increases by 2, and pressure decreases by.5lb/in 2 3) One way to figure out the moment of inertia I of a complicated object is to suspend it, set it swinging and then measure its period T. I can be calculated from the formula I(T, m, d) = T 2 mgd 4π 2 where m is the mass of the object, g is the acceleration due to gravity, and d is the distance from the pivot to the center of gravity of the object. Use the differential to approximate the percent error in measuring the moment of inertia, if the percent error in measuring the mass is plus or minus 1%, while the error in measuring d is plus or minus 2%, and the error in measuring T is 2%. 4) The speed of sound v in an ideal gas is given by v = γp ρ where p is the pressure, ρ is the density and γ a constant related to the heat capacity of the gas. 2

8 a) What is the differential of v(p, ρ, γ)? b) An engineer measures the density and pressure of a gas that behaves like an ideal gas, and gets the value of γ from a handbook. Use the differential of v to estimate the percent error in the calculation of v, if the percent error in the measurement of pressure and density is plus or minus 1% and the error in the measurement of γ is plus or minus.5%. 5) (10 points) If the radius of a cylinder is increased by 2%, and the height of a cylinder is decreased by 3%, use the differential to approximate the percent change in volume. (Remember that the percent change in height is dh h expressed as a percentage). 6) You want to measure the height of a tower. You fix a starting point, and measure the distance d from your starting point to the tower as 100ft. You measure the angle θ between the ground and a line running from your starting point to the top of the tower as π/4 radians. Given the accuracy of your tools, you figure that the error d in measuring d is.1 ft, and the error θ in measuring θ is π/400. Find h, the height of the tower, and use the formula for the differential of h to estimate the error in your estimate of h. (Helpful formulas d dθ tan θ = d sec2 θ, dθ cot θ = csc2 θ.) 3

9 Gradients and Chain rule problems 1) The following table gives the altitude readings h at various positions on a hill, all measurements in meters, positive y axis points North. y = 20 y = 0 y = 20 y = 40 x = x = x = a) Use the table to find approximate values for the gradient of h at (0,20). b) What is the slope of the hill at (0, 20)? c) What is the rate of change of h in the Northeast direction? d) In which direction is the hill level? e) Which is the direction in which elevation falls the fastest? 2) Suppose a block of insulation occupies the region 0 < x < 20, 0 < y < 2, 0 < z < 3y 3, where the distances are in centimeters. The temperature T at the point (x, y, z) in this solid is given by T (x, y, z) = x 2 + y 2 2z 2 (Temperature measuered in degrees Celsius.) a) Compute the gradient of T (x, y, z) at the point (1, 1, 1). b) Find the rate of change of the temperature in the direction i + j k at (1, 1, 1). 1

10 c) From the point (1, 1, 1), in what direction (unit vector) should one move so that the temperature decreases as quickly as possible? 2

11 noindent 3.) Suppose you are given two infinite wires which are parallel to the z-axis, passing through the points (1, 0) and ( 1, 0) in the xy plane which are homogeneously and oppositely charged. The potenial energy P of a unit charge in the xy plane is given by P = (1/2) ln((x 1) 2 + y 2 ) (1/2) ln((x + 1) 2 + y 2 ) a) What is the gradient of the potential energy at (3, 3)? b) What is the rate of change of P in the direction u = 1/2 i 3/2 j at (3, 3)? c) An equipotential line is a curve along which the electrical potential is constant. What is the direction of the equipotential line passing through (3, 3)? 3

12 Review Problems for Linearization and Differential 1) Suppose T (x, y) is the temperatrure of a plate at position (x, y). Suppose T is measured in degrees Celsius, x, y measured in centimeters. Suppose T x (1, 4) =.5, T y (1, 4) = 2. a) What are the units of T x (1, 4) and T y (1, 4)? b) If y is fixed and the change in x is.3cm., what is the approximate change in T? c) If x is fixed and the change in y is.2cm., what is the approximate change in T? d) What is the differential of T at (1, 4)? What are the units of dt? Explain why from the formula for dt. e) Suppose x increases by.3cm. and y decreases by.2cm, what is the approximate change in T? Here is one way to think of the relation between the differential of T and its partial derivatives. T x is a rate of change of T with y fixed, so T x dx is the approximate change in T as x changes by dx. In problem 1) T depended on two variables, so if both x and y changed, dt is the sum of two terms, one for the change due to the change in x and one due to the change in y. 2) Suppose G depends on A, v and θ. How many terms will the differential of G have? What will the fomula be? 3a) (5 points) A solid steel cylinder is manufactured with a radius of 2 cm ±0.1 cm, and a height of 8 cm ±0.2 cm. Approximate the maximum error in the volume of this cylinder, using differentials. 3b) (5 points) The manufacturers now want an error formula that works for every cylinder. So, if a solid steel cylinder is manufactured with at most p% error in the radius and q% error in the height, approximate the maximum percent error in the volume of this cylinder. 1

13 Finding Critical Points 1.) Find the critical points of f(x, y) = y 3 yx 2 y 2 + x 2 = (y 1)(y x)(y + x) Hint: f x = 2x 2xy,f y = 3y 2 x 2 2y, so factor the first equation and use a tree to keep track of the possibilities. 2.) Use the second derivative test to decide whether the critical points of f(x, y) = y 3 yx 2 y 2 + x 2 are local maxima, local minima or saddle points. 3.) Draw the level curves of f.(hint: All of the saddle points are on level 0. So draw this level first, then try to fill in.) 1

14 Integration in Rectangular, Cylindrical and Spherical Coordinates These problems are designed to deepen your understanding about which coordinates to use when calculating triple integrals. In deciding which method to use, you need to understand the shape of the solid you are integrating over. If the solid is symmetric around an axis (usually the z-axis), like cylinders or cones, use integration in cylindrical coordinates. If it is symmetric around the origin like spheres are, then use spherical coordinates. In some problems you can use either spherical or cylindrical coordinates check to see which gives the easier integration. If the solid is not symmetric, then use rectangular coordinates. The method of rays and shadows works in all three cases, so it is worth mastering. Once you decide on the coordinates you want to use, label each piece of the boundary of the solid with its equation in the chosen coordinate system. This will help you apply the method of rays and shadows. Sometimes, a volume problem can be made easier by adding on another solid of known volume, so that the limits of the new solid are easier to find. The volume you are looking for is then the difference of the two volumes. 1) Suppose the density of a tetrahedron at (x, y, z) is given by the height of the point above the xy plane. Find the mass of the tetrahedron if the base is the plane defined by z = 2, and walls defined by y = 0, x = 0, z + 2x + 3y = 14. A) Draw the solid, and label each smooth boundary with its equation. B) Draw the shadow, and label each smooth boundary with its equation. C) Setup the integral. 2) Calculate the volume of the solid bounded above by the sphere of radius 6, centered at (0, 0, 0), and below by the plane z = 3 (Answer 45π.) 3) Suppose the density of a solid is given by the distance to the xz plane. Find the mass if the solid has as floor the plane defined by z = 4, and walls defined by y = x, x = 0, and, z + 6x + 3y = 14. A) Draw the solid, and label each smooth boundary with its equation. B) Draw the shadow, and label each smooth boundary with its equation. C) Setup the integral. Set up as many integrals as you can; do not evaluate. 4) Calculate the volume of the sphere of radius R centered at the origin, from which a cylinder of radius a, symmetric around the z axis has been drilled out, a < R.(Answer: (4π/3)(R 2 a 2 ) 3/2.) 5) Find the center of mass of a hemisphere of radius R with center at the origin, of density proportional to the distance to the xy plane, bounded below by the xy plane. 6) Find the center of mass of an orange slice of radius R of constant density with center at at the origin formed from the planes x = 0 and the plane which makes a 30 angle with the plane y = 0. (Volume of a sphere of radius R is (4/3)πR 3.) 7) Find the center of mass of the hemispherical shell of constant density, if the outer radius is 10 meters and the inner radius is 9 meters, both spheres centered at the origin, and the hemispheres are the part of the spheres above the xy plane.

15 Line integral Practice 1) < 3x, 3y > nds where C is the rectangle with corners at (2, 1), (3, 1), (3, 2) and (2, 2) >. C 2) F (x, y) =< x 10 2y, y x >, the curve consists of the 4 line segments from (2, 0) to (0, 2), from (0, 2) to ( 2, 0), from ( 2, 0) to (0, 2) and from (0, 2) to (2, 0). 3) F (x, y) =< 3y, 2x >, C the line segment from (1, 0) to (0, 5), then from (0, 5) to ( 4, 0). 1

16 4) F (x, y) =< x 4 2xy, y 5 +cos(y) x 2 >, C the curve which consists of the part of the curve y = x 5 x 4 6x from (0, 0) to (2, 4), and the part of the circle from (2, 4) to (4, 0) which also goes through (0, 0). 5) Suppose F =< 2e x + y 3, 3xy cos(y) > and C is the spiral with parameterization r(t) =< ln(t) cos(πt), ln(t) sin(πt)) > with 1 t 4. 6) F (x, y) =< 7y/(x 2 + y 2 ), 7x/(x 2 + y 2 >, C the circle (x 2) 2 + (y 2) 2 = 1 oriented counterclockwise. 7) Same F as E) but the curve is the oval (x 1) 6 + (y 1) 6 = 4. 8) F (x, y) =< 3y, 4x + e y sin(y) >, C the semi circle starting at (3, 0) ending at ( 3, 0). 2

17 Practice Parameterizing Surfaces 1) Paramterize the part of the plane with equation 4x + 3y + 2z = 12 in the first quadrant. 2) Paramterize the plane containing the point (0, 0, 6) and the vectors < 3, 0 6 >, and < 0, 4, 6 >. 3) Find the parameterization of the cone with tip at (0, 0, 10), z axis for axis, radius of the base is 2, base lies in the plane z = 0. Make sure you give the domain of your parameterization. (You can do the problem without this, but if you get stuck, the equation of the cone is (10 z) 2 (25x y 2 ) = 0, 10 z 0.) 4) A parameterization of the lateral surface of the cylinder of height 5, with radius 2, base on the xy plane, with z axis as axis is given by r(θ, z) =< 2 cos θ, 2 sin θ, z > where 0 z 5, 0 θ 2π Notice that r(π/4, 2) =< 2, 2, 2 >. Use the given parameterization to find a parameterization of the tangent plane to the cylinder at the point (< 2, 2, 2 >. 1

18 5) Find the parameterization of the cone with tip at the origin, z axis for axis, radius of the base is 2 and height is 10, base lies in the plane z = 10. Make sure you give the domain of your parameterization. 6A) Find the parameterization of the hemisphere with radius 10, center at the origin, z 0. Make sure you give the domain of your parameterization. 6B) Find the parameterization of the part of the hemisphere with radius 10, center at the origin, 5 z 0 in the first octant. Make sure you give the domain of your parameterization. 7) Find two parameterizations of the portion of the surface with equation z = 16 x 2 y 2 which lies above or on the xy plane. One parameterization should use rectangular coordinates, the other cylindrical. Remember to say what the domain is in each case. 2

19 Surface and integral practice 1. (10 points) The electric field E due to a point charge at the origin is given by E = kq < x, y, z > (x 2 + y 2 + z 2 ) 3/2 where q is the charge, and k is the constant in Coulomb s law. Find the flux integral of E over the sphere of radius R centered at (0, 0, 0). This is the electric flux due to E. 2. (10 points) Given F =< 3y + 4x 6, 3x 2z 6, y 6 > If S is the surface defined by z = 36 (x 2 + y 2 ), with z 0 and with normal pointing away from the origin, find S curl( F ) ds. 3. Calculate the total outward flux of the vector field x 3, y 3, 1, across the boundary of the solid bounded by the paraboloid z = 1 x 2 y 2 and the xy-plane. 4. Calculate the integral C F d r where F = z + x 2, x + y 2, z 2 and C is the boundary of the surface S defined by z = 1 x y and x 2 + y 2 9. The surface S is oriented upward and C is positively orientated with respect to the orientation of S. 1

20 The Meaning of Curl F Today we have seen that we can think of the vector field R z =< y/2, x/2, 0 > as a rotation field around the z-axis, and the curl of CR z is C k. So the direction of curl R z is the axis of rotation and the magnitude of curl R z which is C tells how big the rotational part is. In this exercise you will show that this is true in general. Given any linear vector field G, we can break G into the sum of a gradient part and a rotational part just as we did for linear vectorfields in the plane. Here the rotational part has three parts, one for rotation around each axis. We can write the rotational part as R 1 Rx + R 2 Ry + R 3 Rz. Here R x =< 0, z/2, y/2 > R y =< z/2, 0, x/2 > R z =< y/2, x/2, 0 >, R 1, R 2, R 3 are constants which depend on G. 1A) Show that curl G = curl (R 1 Rx + R 2 Ry + R 3 Rz ). 1B) Show that curl (R 1 Rx + R 2 Ry + R 3 Rz ) =< R 1, R 2, R 3 >. This implies that curl G = < R 1, R 2, R 3 >. 2) (Challenge Problem) Show that curl (R 1 Rx + R 2 Ry + R 3 Rz ) is orthogonal to (R 1 Rx + R 2 Ry + R 3 Rz ). Putting these two exercises together shows that the rotational part of a linear vector field G is a rotation field, whose axis is the curl of G, and the magnitude of curl G measures the size of the rotational part of G. 1

21 Stokes Theorem problems drawn from past exams 1) Compute S curl F d S where F = x 3 +zx+y, ln(3+z +y), cos(x+y +z) and S is the part of paraboloid z = 4 x 2 y 2, z 0 oriented downward. i) What is a good cap for this problem? ii) How does the vector field simplify? iii) What is the important part of the curl of the new simpler vector field? (Only need the part in the direction of the normal of the new cap.) iv) What is the flux integral of the curl of the new vector field? 2) Let S be the northern hemisphere of the sphere of radius 1, centered at the origin, with normal vectors pointing away from the origin. Let F = y 2 + sin(zy), x + z/(x 2 + 3x), x + y + z. Use Stokes Theorem to calculate S curl F d S. i) What is a good cap for this problem? ii) How does the vector field simplify? iii) What is the important part of the curl of the new simpler vector field? (Only need the part in the direction of the normal of the new cap.) iv) What is the flux integral of the curl of the new vector field? 3) ) Consider the vector field F = y, 3xz, y 2 e z Let S be surface of paraboloid z = x 2 + y 2, 0 z 1, with outwards orientation. Compute the flux integral of curl( F) over S. 1

22 4) Calculate the integral C F d r where F = z + x 2, x + y 2, z 2 and C is the boundary of the surface S defined by z = 1 x y and x 2 + y 2 9. The surface S is oriented upward and C is positively orientated with respect to the orientation of S. 5) Let F (x, y, z) = (z y) i + (z + x) j + xyz k and S be the hemisphere z = 1 x 2 y 2 oriented upward. Use Stokes Theorem to calculate S curl F d S. i) What is a good cap for this problem? ii) How does the vector field simplify? iii) What is the important part of the curl of the new simpler vector field? (Only need the part in the direction of the normal of the new cap.) iv) What is the flux integral of the curl of the new vector field? 6) Consider the triangle T defined by the points (0, 0, 12), (6, 0, 0), and (0, 3, 0), orientation of the triangle defined by going around the triangle in the given order of the vertices. If F (x, y, z) =< x 3 + sin x 2, e y2 z, y + 1 (z 2 +1) >, find T F d r (Just set up)s 2

23 Verbal formulations of theorems and formulas: Sample problems Our ability to put the meaning of a formula or theorem into words is a good test of our understanding of it. With that in mind, here are some sample questions I could ask. 1) Give a verbal formulation of the point normal form of the equation of a plane. 2) Give a verbal formulation of the vector form of the equation of a line. 3) Give a verbal formulation of the equation of a sphere of radius R with center at (a, b, c). (You may use R in your answer.) 4) Give a verbal formulation of the equation of the tangent plane to the graph of a function of two variables. 5) Give a verbal formulation of the formula linking f and D u f. 6) List the important properties of the gradient of f. 1