alculus with Analytic Geometry 3 Fall 218 Practice Exercises for the Final Exam I present here a number of exercises that could serve as practice for the final exam. Some are easy and straightforward; others, I hope, will make you think a bit. The final exam will probably be more like the easier exercises in this list, but maybe not entirely. At least fifty percent of the final exam will consist of exercises from this list. Solutions to the exercises will be posted by December 1. Give the exercises a try. If you immediately give up and go for help, this course was (and all the preceding ones were) wasted on you. Every one of these exercises require NO knowledge past what is in the textbook, and was covered in class. Of course, if after making an honest effort you still are drawing a blank, by all means go for help. You may even discover that you basically had it figured out, were just missing a detail. 1. The points P (1, 2, 3), Q(1,, 6), and (4, 2, 4) are 3 vertices of the parallelogram spanned by the vectors P Q, P. (a) Find the coordinates of the fourth vertex. (b) A bug is crawling along the line of equation r(t) = (1 + t)i + (2 + t)j + (3 + 2t)k. Suppose time is measured in seconds, distance in meters. At time t = it is at the point of coordinates (1, 2, 3). It then moves in the positive t direction for a distance of exactly 1 meter along the line, and stops. Determine the coordinates of the point Z at which the bug has stopped. (c) Find the volume of the parallelepiped having the parallelogram of before as a base, and Z as a vertex. 2. onsider the curve given parametrically by x = a cos t, y = b sin t, z = c sin t, where a, b, c are positive real numbers. Show that it lies in a plane. Find the equation of the plane. 3. A curve is described by the vector function r(t) = t 2 i + t 3 j + 1 t k (t > ). Find the parametric equations of the tangent line at the point where t = 2. 4. A curve is described by the vector function r(t) = t 2 i + t 3 j + 1 t k (t > ). Find the equations of the normal plane at the point where t = 2. 5. A curve is given parametrically by x = t 2, y = t t 3, z =. Show that it self-intersects (goes twice through the same point) and find the equations of the two tangent lines at that point. 6. (a) Find the length of the curve described by the position vector r(t) = 12ti + 8t 3/2 j + 3t 2 k for t 1. It might help to realize that 144 = 4 36 and that (t + 2) 2 = t 2 + 4t + 4. (b) Suppose that the curve of part (a); that is, the curve of position vector r(t) = 12ti + 8t 3/2 j + 3t 2 k, gives the position of a particle at time t. Determine the time t h at which the particle has gone through half the length of the curve. That is, find t h such that the length of r(t), t t h equals the length of r(t), t h t 1. 7. Find the vectors T, N, and B for the curve given by r(t) = t 2, 2 3 t3, t at the point where t = 2. Note: omputations can be a bit nasty, so work carefully. There is a point where it could help to know that (2t + 1) 2 = 4t 2 + 4t + 1. 8. Find the velocity, acceleration, and speed (as functions of time t) of a particle with the position vector r(t) = t sin t, t cos t, t 2. 9. Find the position and velocity vectors of a particle if its acceleration and initial velocity and position are a(t) = i + 2j + 2tk, v() =, r() = i + k.
2 1. Let g(s, t) = f(u(s, t), v(s, t)) where f, u, v are differentiable, u(2, 1) = 1, v(2, 1) = 3, u s (2, 1) = 5, v s (2, 1) = 7, f(2, 1) =, f(1, 3) = 4, f(5, 7) = 11, f u (1, 3) = 2, f v (1, 3) = 2, f u (2, 1) = 1, f v (2, 1) = 11. Determine g s (2, 1). Some of the information provided is useless and only included to see if you can select what you need. 11. Find an equation of the tangent plane to the given surfaces at the specified point. (a) z = x 2 + y 3 at the point where (x, y) = (1, 2); that is at (1, 2, 9). (b) xy 2 + yz 2 + zx 2 = 3 at (1, 1, 1). (c) x + ze y + ye z = 4 at the point where (x, y) = (1, ). 12. Let f(x, y) = x 2 + 4xy y 2. (a) Find the unit vectors giving the direction of steepest ascent and steepest descent at (2, 1). (b) Find a vector that points in a direction of no change at (2, 1). 13. A mountain having an elliptical base can be described by the equation z = 25 x 2 4y 2. A climber will try to reach the top starting at the point of coordinates (3, 2, ). The climber wants to be always going in the direction of steepest ascent. The climber decides to follow the path that can be parameterized by x = 3e t, y = 2e 4t, z = 25 9e 2t 16e 8t, t. (a) Show that this is a path that is indeed on the mountain. (b) Show that if the climber follows it, at each point the climber will be moving in the direction of steepest ascent. 14. Find the critical points and use the second derivative test to classify them of f(x, y) = x 4 +y 4 4x 32y +1. 15. Find the critical points and use the second derivative test to classify them of f(x, y) = xye x y. 16. Find ALL critical points of f(x, y) = 3x 2 2x 2 y 4y 2 + 2y 3 and classify each one of them as local maximum, local minimum, or saddle point. 17. Show that the second derivative test is inconclusive when applied to f(x, y) = x 2 y 3 at (, ). Describe the behavior of the function at the critical point. 18. Show that the second derivative test is inconclusive when applied to f(x, y) = sin(x 2 y 2 ) at (, ). Describe the behavior of the function at the critical point. For the next few exercises things to know are: 1. In a closed and bounded region, a continuous function will assume a maximum value and it will assume a minimum value. 2. These values have to be assumed either at a critical interior point or on the boundary. They cannot be assumed anywhere else. Maybe I should add a third thing, if a problem asks for the maximum or minimum VALUE of a function, one might consider that a value is a number, not a pair of numbers or a point in the plane. Any answer that is not a number is essentially wrong. 19. Textbook, 12.8, Exercise # 48, p. 949: Find the absolute maximum and the absolute minimum values of the function f(x, y) = x 2 + y 2 2x 2y on the closed triangle of vertices (, ), (2, ) and (, 2).
3 2. Find the maximum and the minimum value of the following function in the indicated domain. If there is no maximum or minimum, say so. f(x, y) = 3 x 2 y 2 in the closed disc {(x, y) : x 2 + y 2 9}. 21. ompute xy da; is the region in the first quadrant bounded by x =, y = x2 and y = 8 x 2. 22. Write as an iterated integral in the order dx dy the integral of f over the region in quadrants 2 and 3 bounded by the semicircle of radius 3 centered at (, ). 23. Evaluate y2 da where is the region bounded by y = 1, y = 1 x, y = x 1. 24. Find the volume of the solid in the first octant bounded by the coordinate planes and the surface z = 1 y x 2. 25. Sketch the region of integration and evaluate the 2 4 x 2 xe 2y dy dx by reversing the order of integration. 4 y 26. Find the volume of the solid bounded by the paraboloids z = 2x 2 + y 2 and z = 27 x 2 2y 2. 27. Sketch the region inside both the cardioid r = 1 cos θ and the circle r = 1, and find its area. 28. Find the average distance of points within the cardioid r = 1 + cos θ and the origin. x y 29. ompute x 2 + y 2 da, is the unit disc. + 1 3. The figure shows the region of integration for the integral 1 1 x 1 x 2 f(x, y, z) dz dy dx. ewrite the integral as an iterated integral in the five other orders. That is, replace the question marks with appropriate expressions so the following five integrals are equal to the one given above. f(x, y, z) dy dz dx, f(x, y, z) dx dy dz, 31. Suppose f is continuous and a >. Show that f(x, y, z) dy dx dz, f(x, y, z) dx dz dy. f(x, y, z) dz, dx dy, a y z f(x) dx dz dy = 1 2 a (a x) 2 f(x) dx.
4 32. Let = {(x, y) : x π, y π}. Evaluate sin ( max(x 2, y 2 ) ) da. By max(a, b), if a, b are numbers, one understands a if a b; otherwise it is b. 33. Use cylindrical coordinates to evaluate 34. 3 9 x 2 2 3 1 1 + x 2 dz dy dx. + y2 Let D be the region in the first octant of 3 bounded by the paraboloid z = 4 (x 2 +y 2 ). ompute x dv. D The picture on the right shows the paraboloid and the plane z =. Hint: Use cylindrical coordinates. 35. Let D = {(x, y, z) : x, y, z, x 2 + y 2 + z 2 9}. ompute e z dv. using spherical coordinates. 36. Find the volume of the region common to the three cylinders D x 2 + y 2 1, y 2 + z 2 1, z 2 + x 2 1. There is a picture of the region on page 144 of the textbook. 37. The result of this exercise will be used in Exercise 38. Find the coordinates of the center of mass (centroid) of the portion of the cardioid r = 1 + cos θ above the x-axis; that is, the region described by = {(r, θ) : θ π, r 1 + cos θ}.
5 38. We are going to work here with the solid obtained by revolving a cardioid. We flip the cardioid of Exercise 37 so it looks like: We then rotate it about the z-axis. (a) Show, or at least convince yourself, that in spherical coordinates the solid so obtained is described by (b) Use spherical coordinates to find its volume. D = {(ρ, ϕ, θ) : θ 2π, ϕ π, ρ 1 + cos ϕ}. (c) A theorem due to Pappus (who lived some 1,7 years ago) states that the volume of a solid obtained by rotating a plane figure of area A about an axis equals A, where is the circumference of the circle described by the centroid of the figure. The solid of this problem is obtained by revolving about the z-axis the portion of the cardioid described in Exercise 37, except that what was the x-axis in Exercise 37 is now the z-axis. If the coordinates of the center of mass you found in Exercise 37 are (a, b), the center of mass of the figure rotated about the z-axis are at (b,, a). Think about it. Use this information to verify Pappus Theorem in this case. 39. A solid is bounded above by the paraboloid z = 4 x 2 y 2 and below by the cone z = x 2 + y 2. The density of the solid satisfies ρ(x, y, z) = z. Find the coordinates of the center of mass. Note: This exercise results into some really nasty integrals. While you will not have access (at least not legally) to any computer algebra for the final, you might consider using something like Wolfram alpha here. 4. Let be the region in the first quadrant bounded by the curves xy = 1, xy = 5, y = 3x 2, y = 3x + 2. Show that changing variables by u = xy, v = y 3x, changes the region into a rectangle {(u, v) : a u b, c v d}. Find the correct values of c and d and use the change of variables to compute (y + 3x) da.
6 ompute also the integral without changing variables and verify that you get the same result. ( ) 1 (x, y) (u, v) Here is a fact that could ease computations: (u, v) =. (x, y) 41. Evaluate xy da where is the region in the plane bounded by the curves y = x 2 /25, y = (x 1) 2 /25+1, y = x and y = x 4 (see picture below) by changing variables by x = u + 5v, y = u + v 2. In this, as in all change of variables exercises, you must clearly indicate what the transformation does; into what it changes the region of integration, and give at least a good reason why the new region is what you say it is. omputing the integral by any method except the indicated one does not count!. 42. D is the region bounded by the planes x + 2y + 3z =, x + 2y + 3z = 1, 4x y =, 4x y = 2, z = 3, z = 2. Use a change of variables to compute D (5x + y) dv. 43. The picture shows a number of level curves of the function z = x 2 +y or, as is equivalent, equipotential curves of the potential V (x, y) = x 2 + y (a) Show that at every point (x, y) the equipotential curve through that point has a tangent perpendicular to the gradient.
7 (b) Illustrate this by selecting several points (at least 3, no more than 5) and drawing at those points the gradient and the tangent to the curve. 44. A particle of mass m is moving under the action of a conservative force field F = V. Let r(t) = x(t)i + y(t)j + z(t)k be the position of the particle at time t and v(t) = r (t) its speed. Show that the energy E(t) = 1 2 mv(t)2 + V (x(t), y(t), z(t)) remains constant. 45. Escape Velocity. We compute the escape velocity, or rather the escape speed, from a planet of radius, surface acceleration of gravity g. In this model, only the planet exists in the whole universe. This model is not too bad as long as we don t get too far away from the planet. Infinity in physics may just be a large number. We assume a rocket is fired from the surface of the planet, in a direction perpendicular to the surface, with an initial speed v. The question to be answered is: What should v be so that the rocket does not return to the planet We assume the rocket is so small that its gravitational pull on the planet is negligible. The gravitational force field created by the planet is then, according to Newton s law of universal gravitation: where Here are your instructions: m = mass of the rocket, M = mass of the planet, F = GmM r 3 r, r = the radius vector with origin at the center of the planet, G = the gravitational constant. (a) Because G, M can be hard to compute, while g, are not, get rid of G, M by using the fact that on the planet surface F = mg; using this show that GM = g 2. (b) Show that F is conservative and find V such that F = V. V is only determined up to a constant; a convenient constant is one such that lim x 2 +y 2 +z 2 V (x, y, z) =. Using this constant will result in a negative V. (c) As seen in Exercise 44, the energy E(t) = V (r(t)) + 1 2 m r (t) 2 is constant. As the speed of the rocket decreases, V increases accordingly (becoming less negative). Once the speed of the rocket is, the rocket will start returning to the planet. The idea now is to figure out what v should be so that this only happens at infinity, where V is equal to. Equating the energy with the limit for r and the value when r = should produce the escape speed. (d) For the earth g 9.8m.s 2 and 6, 4, m. alculate the escape speed from earth. 46. Evaluate the following line integrals over the given curves (a) (1 points) xe yz ds where is the curve parameterized by r(t) = t, 2t, 4t, t 1. (b) (1 points) x ds, where is given by r(t) = t, 4t, t 2, t =, 2. It may help to know that x a + bx 2 dx = 1 3b (a + bx2 ) 3/2 +. (c) (x y + 2z) ds, where is a circle of radius 3 in the plane x = 1, centered at (1,, ). xy (d) ds where is the line segment from (1, 4, 1) to (3, 6, 3), followed by the arc of circle centered at z (,, 3) in the plane z = 3 from (3, 6, 3) to (6, 3, 3); clockwise.
8 (e) Evaluate z ds where is the curve r(t) = cos t sin t, cos t + sin t, 2 cos t, t 2π. 47. Find the work done by the force field F = y, x, z in moving a particle along the helix for t 2π. x = 2 cos t, y = 2 sin t, z = 48. (a) For what values of a, b, c and d is the field F = ax + by, cx + dy conservative (b) For what values of a, b, and c is the field F = ax 2 by 2, cxy conservative t 2π 49. onsider the vector field F(x, y, z) = yz + 1, xz + 2y, xy + 3z 2. (a) Show that F is conservative and determine a function f so that F = f. (b) Let be the curve given in vector notation by Evaluate r(t) = t 2 i + t 3 j + 2 k, t 1. 1 + t2 F dr. ealizing that these questions are related may save some time. 5. (a) Show that the vector field F = 2xe y + z 2 e x, x 2 e y + 2ye z, y 2 e z + 2ze x + 1 is conservative and find the function f such that f = F. (b) An object is moved along the path parameterized by x = t, t 2, z = t 3 from time t = to time t = 2. Find the work done by the force field of part (a). 51. Evaluate xy 2 dx + xy dy in two ways where is the positively (counterclockwise) oriented triangle of vertices (, ), (, 1), and (1, 1).: (a) Directly, using the definition. (b) Using Green s Theorem. 52. Let be the triangle in the (x, y)-plane of vertices (, ), (4, 2), and (8, 8). You are supposed to calculate x da in four different ways, which are parts (a),(b),(c), and (d) of this exercise. (a) alculate x da if is the triangle of vertices (, ), (4, 2), (8, 8) directly, as an iterated integral (or a sum of iterated integrals). As a help, the right boundary of the triangle lies on the line y = 3 2 x 4, the lower boundary on the line y = x/2. (b) alculate x da if is the triangle of vertices (, ), (4, 2), (8, 8) by Green s Theorem, using the fact that x = ( ) 1 x 2 x2. (c) alculate x da if is the triangle of vertices (, ), (4, 2), (8, 8) by Green s Theorem, using the fact that x = (xy) y.
(d) alculate x da if is the triangle of vertices (, ), (4, 2), (8, 8) by changing variables by u = 3x 2y v = x + 2y In this, as in all change of variables exercises, you must clearly indicate what the transformation does; into what it changes the region of integration, and give at least a good reason why the new region is what you say it is. 53. Use Green s Theorem to evaluate the following line integrals. (a) ( 3y + x 3/2 ) dx + (x y 2/3 ) dy where is the boundary of the halfdisc {(x, y) : x 2 + y 2 2, y } with positive (counterclockwise) orientation. (b) (x 3 y 2 ) dx (3x 2 y + xy) dy, where is the square of vertices (±1, ±1), with positive (counterclockwise) orientation. 54. (1 points) Use Green s Theorem to find the area of the region bounded by the counterclockwise circular arc from (1, ) to (, 1) of radius 1, centered at (, ), and the clockwise semicircle of radius 1/ 2 centered at (1/2, 1/2). See picture. 9 heck your answer computing the area using elementary school geometry. 55. Find the area of the shaded figure in two ways: (a) Using Green s Theorem. (b) Not using Green s Theorem, for example using double integrals and or elementary geometry
1 The boundary of the figure consists of 4 one quarter circle arcs as shown in the picture below ed: Quarter circle of radius 2 centered at (, 2) Blue: Quarter circle of radius 1 centered at (3, 2) Green: Quarter circle of radius 2 centered at (3, 1) Yellow: Quarter circle of radius 1 centered at (, 1) 56. The cycloid is a curve apparently first mentioned by Galileo. It became famous thanks to the Bernoulli brothers discovering that it was both the brachistochrone and the tautochrone (look it up!). Possibly the arches of the Ponte Vecchio in Florence Italy have a cycloidal shape. Ponte Vecchio over the Arno A parametrization of a single arc of the cycloid is given by x = a(t sin t), y = a(1 cos t), t 2π, where a > is a constant. Use Green s Theorem to find the area of the region bounded by the arc of a cycloid and the x axis. (The first person to compute this area, as well as finding the length of an arc of the cycloid, was Gilles Personne de oberval (162-1675); they were also computed only a little bit later, independently, by Evangelista Torricelli (168-1647), more famous for having invented the barometer.)
57. Find the area of the region bounded by the curves y = sin x, x = π + sin y, y = π and x =. 11