Exercises for Multivariable Differential Calculus XM521
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- Thomasina Moore
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1 This document lists all the exercises for XM521. The Type I (True/False) exercises will be given, and should be answered, online immediately following each lecture. The Type III exercises are to be done offline in the traditional paper and pencil format. Solutions to the Type III exercises will be provided. Exercises for Multivariable Differential Calculus XM521 TYPE I = True/False. TYPE III = Exercises to be done in the traditional paper and pencil format. Except where otherwise indicated, section ( ) numbers refer to the 6th edition of the textbook. For most sections, the corresponding chapter and section in the 7th and 8th editions can be found by subtracting one from the chapter number of the 6th edition; the corresponding chapter and section in the 9th editions can be found by subtracting two from the chapter number of the 6th edition. (E.g., section 13.4 of the 6th edition corresponds to section 12.4 of the 7th and 8th editions, and to section 11.4 of the 9th edition.) More information on the different editions is given in the Remarks on Editions listed under Supplemental Course Materials on the Course Page. Exercises marked with a star ( ) have a solution available on the Course Web Page. Those solutions use the exercise numbering of the 6th edition. If a 7th-, 8th-, or 9th-edition exercise is numbered differently from the corresponding exercise in the 6th edition, and if that exercise has a solution on the Course Web Page, then the corresponding 6th-edition number for that exercise is given in brackets after the 7th-, 8th- or 9th-edition number. In short, for the 7th-, 8th-, or 9th-edition listings of exercises, the notation x [y] means that problem number x in the 7th, 8th, or 9th edition does have a solution on the Course Web Page, and it corresponds to problem y in the 6th edition. Lecture 01 ( 13.1): 1) In the Cartesian coordinates for 3-space, the y-axis is the set {(0, y, 0) y is any real number}. 2) The xz-plane is perpendicular to the x-axis and to the z-axis. 3) The distance from the point (1, 2, 1) to (3, 0, 1) is 2 3. Lecture 02 ( 13.1): 1) If K < 0 then the graph of the equation x 2 + y 2 + z 2 + K = 0 is a sphere centered at the origin. 2) The graph (in 3-space) of the equation y = 4 is a plane parallel to the xz-plane, and passing through the point (9, 4, 1). 3) If f is a function of the single variable x, and (0, 1, 2) is on the graph (in 3-space) of z = f(x), then for every value of y, the point (0, y, 2) is also on the graph of z = f(x). TYPE III: 6 th Ed. Section 13.1, # 1, 6, 8, 9, 17, 18, 20, 21, 30, 33, 34, 37, 41, th Ed. Section 12.1, numbered as above. 1
2 8 th Ed. Section 12.1, # 1, 8, 10, 11, 21, 22, 24, 25, 34, 37, 38, 41 [37], 45 [41], 49 [43]. 9 th Ed. Section 11.1, # 1, 10, 12, 13, 27, 28, 30, 31, 40, 43, 44, 47 [37], 51 [41], 55 [43]. Lecture 03 ( 13.2): 1) If two arrows (or directed line segments) drawn in 3-space have different initial points then they must represent different vectors. 2) Given any two vectors v and w, the sum v + w is a vector with length equal to the sum of the lengths of v and w. 3) For any vector v, the vectors 2v and v + v are equal. Lecture 04 ( 13.2): 1) If the point P 1 has coordinates (1, 3) and P 2 has coordinates (0, 5), then P 1 P 2 = 1, 2. 2) If v = 2, 2 and w = 3, 4, then 2v 4w = 16, 20. 3) For any vectors v and w and any scalars k and l, (k + l)(v + w) = kv + lw. Lecture 05 ( 13.2): 1) If v = 5, then v = ) 3 1, 1, 4 is a unit vector pointing in a direction opposite to the vector 3, 3, ) 3i + 4j + 5k = 5 2. Lecture 06 ( 13.2): 1) If two forces F 1 and F 2 act on an object from opposite directions, then the effect is the same as that of a single force of magnitude F 2 F 1 acting in the same direction as the larger of the forces F 1 and F 2. 2) If two forces F 1 and F 2 each of magnitude 10N act on an object from perpendicular directions, then the effect is the same as that of a single force of 10N acting in a direction 45 degrees to each of the forces F 1 and F 2. 3) If F 1 and F 2 act on an object, and the angle between them is π/3, and F 1 = 10N, and the magnitude of the resultant is 10 3N, then F 2 = 10N. TYPE III (for lectures 3 6): 6 th Ed. Section 13.2, # 3, 4, 6, 7c, 9, 11, 14, 16, 17, 20, 24, 27, 28, 34, 36, 37, 38, 41, 44, 47, 49, 52, 53, 57, th Ed. Section 12.2, numbered as above. 8 th Ed. Section 12.2, numbered as above. 2
3 9 th Ed. Section 11.2, # 3, 4, 6, 7c, 9, 11, 14, 16, 21, 24, 28, 31, 32, 38, 40, 41, 42 [38], 45, 48, 51, 53 [49], 56 [52], 57 [53], 61 [57], 62 [58]. Lecture 07 ( 13.3): 1) The dot product of two vectors is a vector. 2) If u and v are nonzero and u v = (1/2) u v, then the angle between u and v is π/3. 3) If u v = u v, and u and v are nonzero, then there exists a scalar k < 0 such that u = kv. Lecture 08 ( 13.3): 1) If v 0 has direction angles α, β, and γ, then the vector (cosα)i + (cosβ)j + (cos γ)k is a unit vector pointing in the same direction as v. 2) In 2-space the vector 1, 1 makes 45 degree angles with the positive x and y axes, so in 3-space the vector 1, 1, 1 makes 45 degree angles with the x, y, and z axes. 3) If v is a unit vector with direction angles α, β and γ, and α = β = π/2, then either v = k or v = k. Lecture 09 ( 13.3): 1) For any vector v = v 1, v 2, proj i v = v 1, 0 and proj j v = 0, v 2. 2) If v = 2, 4 and e is the unit vector 1/2, 3/2, then the orthogonal projection of v on e is ( )e. 3) If e 1 and e 2 are orthogonal unit vectors, v is a nonzero vector in the plane of e 1 and e 2 such that the angle between v and e 1 is θ, then we can always write v = ( v cos θ)e 1 + ( v sin θ)e 2. TYPE III (for Lectures 7 9): 6 th Ed. Section 13.3, # 3, 4, 8, 9, 11, 13, 14, 15, 18, 20, 23, 25, 27, 28, 34, 37, 40, 41 7 th Ed. Section 12.3, numbered as above. 8 th Ed. Section 12.3, # 3, 4, 8, 9, 10 [14], 12, 13, 15, 18, 20, 27, 28, 30, 31 35, 39 [37], 42 [40], 43 [41]. 9 th Ed. Section 11.3, # 3, 4, 8, 9, 12, 13, 15, 18, 20, 27, 28 [14], 32, 34, 35, 37, 39 [37], 42 [40], 43 [41]. Lecture 10 ( 13.4): 1) For all vectors u and v, u v = (v u). 2) For all vectors u, v, and w, u (v w) = (u v) w. 3) 1, 2, 3 4, 5, 6 = 3, 6, 3. 3
4 Lecture 11 ( 13.4): 1) The nonzero vectors u and v are orthogonal if and only if u v = 0. 2) The nonzero vectors u and v are orthogonal if and only if u v = u v. 3) If u is a nonzero vector pointing in the direction of j, and v is a nonzero vector pointing in the direction of i, then u v points in the direction of k. Lecture 12 ( 13.4): 1) If u = 1, 2, 1, v = 3, 0, 1, and w = 0, 1, 4, then u v w = 20. 2) For any three vectors u, v, and w, u v w = v u w. 3) If the unit vectors e 1, e 2, and e 3 are pairwise orthogonal (so each is orthogonal to the other two), then e 1 e2 e 3 = 1. TYPE III (for lectures 10 12): 6 th Ed. Section 13.4, # 3, 4, 7, 10, 11, 12, 14, 16, 19, 20, 22, 23, 24, 26, 31, 32, 38, th Ed. Section 12.4, numbered as above. 8 th Ed. Section 12.4, # 3, 4, 7, 10, 11, 12, 14, 16, 19, 20, 22, 23, 24, 26, 33 [31], 34 [32], 40 [38], 44 [41]. 9 th Ed. Section 11.4, # 3, 4, 7, 10, 11, 12, 18, 20, 23, 24, 27, 28, 30 [26], 37 [31], 38 [32], 44 [38], 48 [41]. Lecture 13 ( 13.5): 1) The line passing through the origin and parallel to the vector 4, 2, 6 has parametric equation x = 4t, y = 2t, z = 6t. 2) x, y, z = 1, 1, 2 + t 2, 7, 3 is a vector equation of the line passing through the point (2, 7, 3) and parallel to the vector 1, 1, 2. 3) If L 1 has parametric equations x = 1 + 3t, y = 1 + t, z = 4 t, and L 2 has parametric equations x = 7 6t, y = 2t, z = 3 + 2t, then L 1 and L 2 are parallel. Lecture 14 ( 13.5): 1) The line containing the points (3, 1, 2) and (5, 0, 1) has vector equation x, y, z = 5, 0, 1 + t 4, 2, 6. 2) Ifd L 1 is given by the equations x = 1 + 7t, y = 3 + t, z = 5 3t, and L 2 is given by the equations x = 4 t, y = 6, z = 7 + 2t, then L 1 and L 2 are skew. 3) The vector equation x, y, z = 2, 1, 4 + t 3, 0, 1, (0 t 3) describes the line segment from ( 2, 1, 4) to the point (7, 1, 1). 4
5 TYPE III (for Lectures 13,14): 6 th Ed. Section 13.5, # 2, 3, 6, 8, 9, 13, 15, 16, 20, 23, 25, 28, 30, 31, 33, 39, 42, 43, 45, th Ed. Section 12.5, numbered as above. 8 th Ed. Section 12.5, # 2, 3, 6, 8, 9, 13, 15, 16, 20, 23 [23], 25, 28, 31, 33, 45 [39], 48 [42], 49, 51 [45], th Ed. Section 11.5, # 2, 3, 6, 8, 9, 17, 19, 20, 24, 27 [23], 29, 32, 35 [31], 37 [33], 49 [39], 52 [42], 53, 55 [45], 57. Lecture 15 ( 13.6): 1) The plane given by the equation 3x 4y+z 1 = 0 is perpendicular to the vector 9, 12, 3 and contains the point (0, 0, 1). 2) If A is a plane containing the points P 1 and P 2, and P 3 is any other point in 3-space, and P 1 P 3 P 1 P 2 =0, then P 1 P 3 must be a normal vector for the plane. 3) The four points ( 2, 1, 1), (0, 2, 3), (1, 0, 1) and (5, 4, 7) all lie in the same plane. Lecture 16 ( 13.6): 1) If θ is the acute angle of intersection between the planes 2y z+4 = 0 and 3x+4y+z 3 = 0, then cosθ = 9/( 5 26). 2) The distance from the point (1, 2, 3) to the plane 2x + 4y z + 5 = 0 is 3/7. 3) The distance between the planes 3x+4y +5z +1 = 0 and 3x+4y +5z +7 = 0 is 6/(5 2). TYPE III (for lectures 15, 16): 6 th Ed. Section 13.6, # 3, 4, 7, 12, 16, 17, 20, 22, 23, 25, 26, 29, 30, 36, 37, 40, 41, th Ed. Section 12.6, numbered as above. 8 th Ed. Section 12.6, # 3, 4, 7, 12, 16, 17, 20, 22, 23, 25, 26, 29, 30, 34, 39, 42, 43, th Ed. Section 11.6, # 3, 4, 7, 12, 16, 17, 20, 26, 27, 29 [25], 30 [26], 33 [29], 34 [30], 38, 41, 44, 45, 47. Lecture 17 ( 13.7): 1) If S is the surface defined by x 2 +y 2 +z 2 = 12, then the trace of S in the plane x = 2 3 is a single point. 2) If S is the surface defined by z 2 y 2 x 2 = 1, then the trace of S in the plane z = 3 is a circle of radius ) If S is the surface defined by z 2 = x 2 + y 2, then the trace of S in the plane x = 0 is a hyperbola. 5
6 Lecture 18 ( 13.7): 1) The graph of the equation x 2 = 2z 2 + 3y 2 is an elliptic cone opening in the z direction. 2) The graph of (z 1) = (x 2) 2 /7 + (y 3) 2 /10 is an elliptic paraboloid with base at the point (2, 3, 1), and opening in the z direction. 3) If S is the surface determined by the equation x 2 /4 + y 2 /9 + z 2 /16 = 1, then any trace of S which has more than one point and lies in a plane parallel to a coordinate plane is an ellipse. TYPE III (for lectures 17, 18): 6 th Ed. Section 13.7, # 2, 7, 10, 12, 13, 18, 19, 25, 29, 30, 35, 37, 38, 39, 45, th Ed. Section 12.7, numbered as above. 8 th Ed. Section 12.7, numbered as above. 9 th Ed. Section 11.7, # 2, 7, 10, 16, 17, 22 [18], 23, 29 [25], 33, 34, 39 [35], 41 [37], 42, 43 [39], 49 [45], 53 [49]. Lecture 19 ( 13.8): (1) The point P with rectangular coordinates (0, 1, 0) has cylindrical coordinates (1, 3π/2, 0). (2) The graph of the (cylindrical coordinate) equation z 2 + r 2 = 16 is a sphere of radius 4 centered at the origin. (3) The graph of the equation z = r cosθ is a plane passing through the origin with normal vector 1, 0, 1. Lecture 20 ( 13.8): 1) The point with spherical coordinates (6, 0, 3π/4) has rectangular coordinates (3 2, 0, 3 2). 2) If the point (x 0, y 0 ) in 2-space has polar coordinates r = r 0, θ = θ 0, where r 0 0 and 0 θ 0 < 2π, then the spherical coordinates of the point (x 0, y 0, 0) are ρ = r 0, θ = θ 0, φ = π/2. 3) When converted into spherical coordinates, the equation x 2 + y 2 = 4 becomes ρ = 2. TYPE III (for lectures 19, 20): 6 th Ed. Section 13.8, # 1, 3, 5, 7, 9, 11, 17, 18, 19, 20, 26, 28, 30, 36, 38, 39, 40, 41, 42, 43, 44, 47, th Ed. Section 12.8, numbered as above. 8 th Ed. Section 12.8, numbered as above. 9 th Ed. Section 11.8, # 1, 3, 5, 7, 9, 11, 21, 22 [18], 23, 24 [20], 30, 32 [28], 34 [30], 40, 42, 43, 44 [40], 45, 46 [42], 47 [43], 48 [44], 51, 53 [49]. Lecture 21 ( 14.1): 6
7 1) Graphing (with orientation) the vector-valued function r(t) = f 1 (t)i + f 2 (t)j + f 3 (t)k, a t b, is equivalent to graphing (with orientation) the parametric equations x = f 1 (t), y = f 2 (t), z = f 3 (t), a t b. 2) The graph of the vector-valued function r(t) = 3 costi+3 sintj+5k, 0 t 2π, is a circle of radius 3 centered at (0, 0, 5) and lying in the plane z = 5. 3) If two vector-valued functions r 1 and r 2 have the same graph (with orientation), then r 1 = r 2. TYPE III: 6 th Ed. Section 14.1, # 3, 7, 11, 15, 16, 17, 18, 22, 27, 28, 32, 34, 39, 40, 41, 42, 43, 45, th Ed. Section 13.1, numbered as above. 8 th Ed. Section 13.1, numbered as above. (Note: Problem 18 in the 8th edition is slightly different from that in the earlier editions. Try the version of the problem that is stated in the solutions pdf file.) 9 th Ed. Section 12.1, # 3, 6, 8, 11, 12, 13, 14 [18], 18, 23, 24, 28, 30, 39, 40, 41, 42, 43, 45, 47. (Note: Problem 14 in the 9th edition, which is #18 in the prior editions, is slightly different from that in the prior editions. Try the version of the problem that is stated in the solutions pdf file.) Lecture 22 ( 14.2): 1) If r(t) = t 2, e t, sin 2t, then r (t) = 2t, e t, 2 cos2t. 2) r (t)dt = r(t). 3) If r (t) = 3t 2, e 2t + 1 and r(0) = i + j, then r(t) = t 3 + 1, (1/2)e 2t + t + 1/2. Lecture 23 ( 14.2): (1) The tangent vector to the graph of r(t) = (3t + 1)i + t + 2j + t 3 k at the point t = 0 is 3i + ( 2/4)j + 0k. (2) The line tangent to the curve r(t) = t, t 2, 4t at the point t = 1 is perpendicular to the plane 2x 2y 4z + 9 = 0. (3) If r(t) r (t) = 5 for all t, then r(t) r (t) + r (t) 2 = 0 for all t. TYPE III (for lectures 22, 23): 6 th Ed. Section 14.2, # 1, 5, 6, 8, 9, 11, 15, 18, 20, 26, 35, 36, 39, 42, 46, 52, 54, 57, 58, 59, th Ed. Section 13.2, # 3, 6 [18], 8 [20], 9, 13, 14, 16, 17, 19, 26, 39, 40, 43, 46, 50 [46], 54 [52], 56 [54], 57, 58, 59, th Ed. Section 13.2, # 1, 4 [18], 6 [20], 7, 10, 12, 13, 15, 22, 35, 36, 39, 42, 46, 50 [52], 52 [54], 53 [57], 54 [58], 55 [59], th Ed. Section 12.2, # 1, 4 [18], 6 [20], 7, 10, 12, 13, 15, 22, 32, 34, 37, 40, 48 [46], 50 [52], 52 [54], 53 [57], 54 [58], 55 [59], 57. 7
8 Lecture 24 ( 14.3): (1) If a curve C has a smooth parametrization, then C has infinitely many different smooth parametrizations. (2) If r 1 is a smooth vector-valued function and g is a smooth real-valued function (meaning that its derivative always exists but is nonzero), then the composition r 2 = r 1 g is a smooth vector-valued function whose derivative obeys the chain rule : r 2(τ) = g (τ) r 1(g(τ)). (3) If P is a point on C and r 1 and r 2 are smooth parametrizations of C then the tangent vector for r 1 at P is the same as the tangent vector for r 2 at P. Lecture 25 ( 14.3): (1) The arc length of a curve C depends on which smooth parametrization of C is chosen. (2) If r(s), 4 s 2, is an arc length parametrization of the curve C, then the arc length along C (moving with the orientation) from the terminal point of r( 1) to the terminal point of r(2) is 2 ( 1) = 3. (3) If r(s), a s b, is a smooth parametrization of C such that a < 0 < b and r (s) = 1 for all s in the domain of r, then r is an arc length parametrization of C. TYPE III (for lectures 24, 25): 6 th Ed. Section 14.3, # 5, 6, 7, 8, 12, 13, 17, 22, 24, 27, 28, 32, 38, 40, 42 7 th Ed. Section 13.3, numbered as above. 8 th Ed. Section 13.3, # 3, 4, 5, 6, 10, 11, 15 [17], 22, 24, 27, 28, 32, 38, 40, th Ed. Section 12.3, # 3, 4, 5, 6, 10, 11, 15 [17], 24 [22], 26, 29, 30 [28], 34 [32], 40 [38], 42 [40], 45 [42]. Lecture 26 ( 14.4): 1) If r 1 and r 2 are two smooth parametrizations of the curve C which induce the same orientation, and P is a point on C, then the tangent vectors at P for r 1 and r 2 may differ, but the unit tangent vectors (at P) must be the same. 2) If C is the graph of the smooth vector-valued function r(t), defined on the interval I, then the unit normal vector N(t) is defined for all t in I. 3) If r(s) is an arc length parametrization of C, then T(s) = r (s), N(s) = r (s)/ r (s), and B(s) = (r (s) r (s))/ r (s). TYPE III: 6 th Ed. Section 14.4, # 2, 5, 6, 7, 8, 16, 17, 20, th Ed. Section 13.4, numbered as above. 8 th Ed. Section 13.4, # 2, 7, 8, 9, 10 [8], 16, 17, 20, 21. 8
9 9 th Ed. Section 12.4, # 2, 7, 8, 9, 10 [8], 16, 17, 20. Also try the following problem, which corresponds to #21 in the prior editions (and is given in the solutions): (a) Use the formula N(t) = B(t) T(t) and Formulas (1) and (11) to show that N(t) can be expressed in terms of r(t) as N(t) = r (t) r (t) r (t) r (t) r (t) r (t). (b) Use properties of cross products to show that the formula in part (a) can be expressed as N(t) = r (t) r (t) r (t) r (t) r (t) r (t). (c) Use the result in part (b) and Exercise 45 of Section 11.4 to show that N(t) can be expressed directly in terms of r(t) as N(t) = u(t) u(t) where u(t) = r (t) 2 r (t) (r (t) r (t))r (t). Lecture 27 ( 14.5): 1) If r(t) is a parametrization of C, and the point P on C corresponds to t = 2, then the curvature κ at P is r (2). 2) If r(s) is an arc length parametrization of C, and the point P on C corresponds to s = 2, then the curvature κ at P is r (2). 3) If r 1 (t) and r 2 (τ) are any two smooth parametrizations of C, and the point P on C corresponds to t = t 0 and τ = τ 0, then assuming r 1(t 0 ) and r 2(τ 0 ) exist. r 1 (t 0) r 1 (t 0) r 1 (t 0) 3 = r 2 (τ 0) r 2 (τ 0 ) r 2 (τ 0) 3, Lecture 28 ( 14.5): 1) The line r(t) = v 0 + tv has constant curvature equal to v. 2) The curvature at any point of a circle of radius 8 is 1/8. 3) For a curve in 2-space κ = dφ ds, where φ is the angle measured counterclockwise from the positive x direction to the unit tangent vector, and s is an arc length parameter. TYPE III (for lectures 27, 28): 6 th Ed. Section 14.5, # 1, 5, 7, 8, 12, 16, 17, 20, 22, 28, 34, 39, 47, 48, 59, 61, 62, 63, th Ed. Section 13.5, numbered as above. 9
10 8 th Ed. Section 13.5, # 1, 3, 4 [34], 9, 10, 14, 18, 19 [17], 22, 24, 30, 39, 47, 48, 59, 63 [61], 64 [62], 65 [63], 66 [64]. 9 th Ed. Section 12.5, # 1, 3, 4 [34], 9, 10, 14, 18, 23 [17], 26, 27, 29, 31, 39, 45 [47], 46 [48], 55 [59], 58 [61], 59 [62], 60 [63], 61 [64]. Lecture 29 ( 14.6): 1) If a particle s position is given by r(t) = e 2t, t 3, 1 t, then at t = 1 its speed is 4e and the acceleration is 4e 2, 6, 0. 2) If a particle moves along a straight line, then the acceleration and velocity always point in the same direction. 3) For a particle moving in space at constant nonzero speed, a and N point in the same direction (assuming N exists). Lecture 30 ( 14.6): 1) If at time t an object s acceleration has norm 5 and the tangential component of its acceleration is 4, then the normal component of its acceleration is 1. 2) If a(t) = 2t, sint for all t, and v(0) = 1, 0, then v(t) = t 2 + 1, 1 cost for all t. 3) If a(t) = e 2t, 3t 2 for all t, v(1) = 1 2 e2, 1, and r(0) = 1, 2, then r(1) = 1 4 e2 + 3, 9. TYPE III (for lectures 29, 30): 6 th Ed. Section 14.6, # 1, 7, 8, 9, 12, 19, 20, 26, 29, 34, 35, 44, 48, 52, 55, 58, 62, th Ed. Section 13.6, numbered as above. 8 th Ed. Section 13.6, # 1, 7, 8, 9, 11 [19], 12, 14 [12], 26, 29, 34, 35, 44, 48, 52, 55, 58, 62, th Ed. Section 12.6, # 1, 7, 8, 9, 11 [19], 12, 14 [12], 26, 29, 34, 35, 39, 42 [48], 46, 54, 56, 60 [62], 65 [67]. Lecture 31 ( 15.1): 1) If f(x, y) = y 2 4 x 2 then the domain of f is R 2. 2) The level curve of height 5 of the function f(x, y) = 3x 2 y 2 is a hyperbola. 3) If f(x, y, z) = x 2 y 2 z 2, then the level surface of height 16 is a sphere of radius 4 centered at the origin of 3-space. TYPE III: 6 th Ed. Section 15.1, # 1, 3, 5, 7, 12, 14, 15, 18, 20, 25, 26, 30, 32, 34, 38, 42, 46, 48, 49, th Ed. Section 14.1, # 1, 3, 5, 7, 16, 18, 19, 22 [18], 24 [20], 29, 30, 34, 37, 40, 42 [34], 46 [38], 50 [42], 54 [46], 56,
11 8 th Ed. Same as 7 th Ed. 9 th Ed. Section 13.1, # 1, 3, 5, 7, 16, 18, 19, 22 [18], 24 [20], 33, 34, 38, 41, 44, 46 [34], 50 [38], 54 [42], 58 [46], 60, 61. Lecture 32 ( 15.2): 1) If D = {(x, y) y = 1}, then every point in D is a boundary point. 2) If a set D contains all of its interior points, then D is open. 3) If a set D has a finite number of elements in it, then D has no accumulation points. Lecture 33 ( 15.2): 1) If lim (x,y) (1,1) f(x, y) = 2, and C is a curve in the domain of f passing through (1, 1), then lim f(x, y) = 2. (x,y) (1,1) C 2) If lim f(x, y) = 2 for some curve C in the domain of f, then lim (x,y) (x 0,y 0) C (x,y) (x 0,y 0) f(x, y) = 2. 3) Assuming (x 0, y 0 ) is an accumulation point of dom(f), the statement lim f(x, y) = L (x,y) (x 0,y 0) is true if and only if there exist positive numbers ǫ and δ such that for all (x, y) in the domain of f, 0 < (x x 0 ) 2 + (y y 0 ) 2 < δ implies f(x, y) L < ǫ. Lecture 34 ( 15.2): 1) If f is continuous at (1, 2) and f(1, 2) = 3, then lim (x,y) (1,2) f(x, y) = 3. 2) If f is continuous at every point in its domain, then f is a continuous function. 3) If g and h are functions of one variable, g is continuous at x = 2, and h is continuous at y = 1 with h(1) 0, then f(x, y) = g(x)/h(y) must be continuous at (2, 1). TYPE III (for lectures 32 34): 6 th Ed. Section 15.2, # 1, 3, 6, 10, 15, 16, 20, 21, 26, 27, 31, 38, 39, 43, th Ed. Section 14.2, # 3, 4, 8 [20], 9, 14 [26], 15, 19 [31], 26 [38], 27, 31 [43], 32 [44], 33, 35, 38, th Ed. Same as 7 th Ed. 9 th Ed. Section 13.2, # 3, 4, 8 [20], 9, 14 [26], 15, 19 [31], 34 [38], 35, 39 [43], 40 [44], 41, 43, 46, 50. Lecture 35 ( 15.3): 11
12 1) If f is a function of two variables, then f y (1, 3) = lim h 0 f(1,3+h) f(1,3) h. 2) If f(x, y) = x 2 e 3y + y 2, then f x (x, y) = 2xe 3y + y 2. 3) If f(x, y) = e x sin y, then at every point (x, y), (f x (x, y)) 2 + (f y (x, y)) 2 = e 2x. Lecture 36 ( 15.3): 1) If f(x, y) = e 2x sin(xy), then f yy (x, y) = x 2 f(x, y). 3 f 2) To obtain, we first different f with respect to y, then differentiate that with y x x respect to x, then differentiate that with respect to x again. 3) If f(x, y, z, w) = xyzw, then f xyzw = 1. TYPE III (for lectures 35, 36): 6 th Ed. Section 15.3, # 1, 3, 4, 6, 8, 10, 11, 14, 18, 24, 29, 30, 39, 41, 42, 45, 47, 48, 67, 70, 73, 74, 76, 78, th Ed. Section 14.3, # 1, 4, 6, 7, 8 [42], 11, 14, 18, 24, 42 [78], 57 [39], 61, 64 [70], 65, 66, 69, 70 [30], 75, 77, 78 [48], 81, 83, 85, 86 [74], 88, th Ed. Section 14.3, # 1, 4, 6, 7, 8 [42], 11, 13, 16, 20, 26, 44 [78], 59 [39], 63, 66 [70], 67, 68, 71, 72 [30], 77, 78 [48], 81, 83, 85, 86 [74], 88, th Ed. Section 14.3, # 1, 4, 6, 7, 8 [42], 11, 17, 20, 24, 30, 47 [78], 63 [39], 67, 70, 73, 74, 77, 78 [30], 83, 84 [48], 87, 89, 92 [74], 94, 109. Lecture 37 ( 15.4): 1) If f x and f y exist in some open disk centered at (x 0, y 0 ), then f is differentiable at (x 0, y 0 ). 2) If f x and f y exist and are continuous at every point in some open disk centered at (x 0, y 0 ), then f is differentiable at (x 0, y 0 ) and at every other point in that open disk. 3) If f(x, y) is a polynomial in x and y, then f is differentiable everywhere. Lecture 38 ( 15.4): 1) If f is a differentiable function of x, and x is a differential function of t with x(1) = 5 and x (1) = 2, then d dt f(x(t)) t=1 = 2f (5). 2) If f is a differentiable function of x and y, and x and y are differentiable functions of t with (x(1), y(1)) = (5, 6), x (1) = 2, and y (1) = 3, then d dt f(x(t), y(t)) t=1 = 2f x (5, 6) + 3f y (5, 6). 3) Suppose f is a differentiable function of x and y, and x and y are differentiable functions of t. If f x (0, 0) = 3, f y (0, 0) = 4, x (0) = 2, y (0) = 1, (x(0), y(0)) = (0, 0), and g(t) = f(x(t), y(t)), then g (0) = 8. Lecture 39 ( 15.4): 12
13 1) If f(x, y) = 0 defines y implicitly as a differentiable function of x, then dy dx = f/ y f/ x, assuming f/ x 0. 2) If z = xy and x and y are differentiable functions of s and t, then z s = yx s + xy s. 3) If u is a differentiable function of s and t, and s and t are differentiable functions of v and w, then u w = u s s w + u t t w. TYPE III (for lectures 37 39): 6 th Ed. Section 15.4, # 1, 3, 5, 8, 10, 16, 18, 21, 25, 30, 31, 33, 37, 41, 43, 49, 52, th Ed. Section 14.5, # 1, 3, 5, 14, 16, 22 [16], 28 [18], 31, 41, 46 [30], 51, 61, 62 [43], 70 [52], 71 [53]. 8 th Ed. Section 14.5, # 1, 3, 5, 18, 20, 26 [16], 32 [18], 35, 45, 50 [30], 55, 65, 66 [43], 74 [52], 75 [53]. 9 th Ed. Section 13.5, # 1, 3, 5, 18, 20, 26 [16], 32 [18], 39, 47, 57, 58 [43], 66 [52], 67 [53]. Also try the following problem, which corresponds to #30 in the sixth edition (and is given in the solutions): Suppose that a particle moving along a metal plate in the xy-plane has velocity v = i 4j (cm/s) at the point (3, 2). Given that the temperature of the plate at points in the xy-plane is T(x, y) = y 2 lnx, x 1, in degrees Celsius, find dt/dt at the point (3, 2). Lecture 40 ( 15.5): 1) If f is differentiable at (1, 3), f x (1, 3) = 3, f y (1, 3) = 1, and f(1, 3) = 4, then the equation for the tangent plane to f at (1, 3, 4) is 3(x 1) + (y 3) (z 4) = 0. 2) There exists a differentiable function f such that the tangent plane to the graph of z = f(x, y) at the origin has normal vector 1, 2, 0. 3) At any point (x 0, y 0 ) at which f is differentiable, the total differential of f is df = f x (x 0, y 0 )dx + f y (x 0, y 0 )dy. TYPE III: 6 th Ed. Section 15.5, # 1, 2, 5, 8, 10, 12, 14, 18, 20, 23, 24, 25, 29, 30, 48, 50, th Ed. Section 14.7, # 1, 2, 5, 8, 10, 12, 14 [48], 28 [50], 29 [51]; Section 14.4, # 1, 2 [24], 26, 28, 37, th Ed. Section 14.7, same as 7 th Ed. ; Section 14.4, # 10, 12, 21, 22, 37, 38 [24]. 9 th Ed. Section 13.7, # 4, 5, 6, 9, 12, 14 [12], 16 [48], solmark32[50], 33 [51]; Section 13.4, # 10, 12, 21, 22, 41, 42 [24]. Lecture 41 ( 15.6): 1) If f is differentiable at (0, 0), f x (0, 0) = 2, f y (0, 0) = 3, and v = 2, 1, then D v f(0, 0) = 5/5. 2) If f is differentiable at (x 0, y 0 ) and v is a nonzero vector (in 2-space), then D v f(x 0, y 0 ) = D v f(x 0, y 0 ). 13
14 3) If f is differentiable at (x 0, y 0 ) and v is a nonzero vector (in 2-space), then D 3v f(x 0, y 0 ) = 3D v f(x 0, y 0 ). Lecture 42 ( 15.6): 1) If v is a nonzero vector in 2-space and f is differentiable at (x, y), then D v f(x, y) = f(x, y) v. 2) If f is differentiable at (x, y), then for every nonzero vector v, f(x, y) D v f(x, y) f(x, y). 3) If f is everywhere differentiable and level curves of f are circles centered at the origin, then at every point (x, y) (0, 0), f(x, y) is parallel to x, y. TYPE III (for lectures 41, 42): 6 th Ed. Section 15.6, # 2, 5, 6, 9, 10, 12, 13, 18, 19, 20, 22, 27, 28, 29, 32, 35, 41, 42, 43, 48, 52, 58, 59, 65, th Ed. Section 14.6, # 2, 3, 12, 13, 14, 20 [22], 29, 32, 35, 36, 39, 40, 43, 44, 45, 46 [42] 47, 50 [32], 57, 64, 68 [52], 76 [58], 77 [59], 83 [65], 84 [66]. 8 th Ed. Same as 7 th Ed. 9 th Ed. Section 13.6, # 2, 3, 12, 13, 14, 20 [22], 29, 32, 35, 36, 39, 40, 43, 44, 45, 46 [42] 47, 50 [32], 57, 68, 72 [52], 80 [58], 81 [59], 87 [65], 88 [66]. Lecture 43 ( 15.7): 1) If f is a function of x, y, z, and w, and the first partials of f exist in an open set containing (x 0, y 0, z 0, w 0 ) and are continuous at (x 0, y 0, z 0, w 0 ), then f must be continuous at (x 0, y 0, z 0, w 0 ). 2) If f(x, y, z) = xyz, x(t) = t 2, y(t) = sin t, z(t) = cost, then df dt = (2t)( cost)(sin t). 3) If f(x, y, z) = x 2 y 3 z 4, then at (2, 1 1), df = 4dx + 12dy 16dz. Lecture 44 ( 15.7): 1) If f(x, y, z, w) = xy z 2 e w, then f(x, y, z, w) = y, x, 2ze w, z 2 e w. 2) If c 1,..., c n are fixed real numbers and f is the function f(x 1,...,x n ) = c 1 x 1 + c n x n, then for every nonzero vector v, D v f(x 1,..., x n ) c c2 n. 3) If f is a differentiable function and the level surface f(x, y, z) = 5 is the paraboloid z = x 2 + y 2, then either f(0, 0, 0) = 0 or f(0, 0, 0) is parallel to the x-axis. TYPE III (for lectures 43, 44) : 6 th Ed. Section 15.7, # 1, 2, 5, 8, 9, 14, 15, 20, 23, 26, 28, 31, 33, 39, 49, 55, 56, 59, 61, 64, 71, 77, th Ed. Section 14.5, # 23, 24 [56], 47, 53 [61], 58, 69 [77]; Section 14.6, # 6, 7, 18, 51, 60, 71 [23], 85 [71]; Section 14.7, # 7, 8, 11, 16, 18 [28], 21, 23, 31 [78]. 14
15 8 th Ed. Section 14.5, # 27, 28 [56], 51, 57 [61], 62, 73 [77]; Section 14.6, # 6, 7, 18, 51, 60, 71 [23], 85 [71]; Section 14.7, # 7, 8, 11, 16, 18 [28], 21, 23, 31 [78]. 9 th Ed. Section 13.5, # 27, 28 [56], 49 [61], 54, 65 [77]; Section 13.6, # 6, 7, 18, 51, 60, 75 [23], 89 [71]; Section 13.7, # 2, 3, 4, 13, 22 [28], 25, 29, 35 [78]. Lecture 45 ( 15.8): 1) A continuous function, defined on a domain D which is both closed and bounded, must have both an absolute maximum and an absolute minimum in D. 2) If (x 0, y 0 ) is a critical point of f, then we must have f(x 0, y 0 ) = 0. 3) If (x 0, y 0 ) is a critical point of f, then f has a relative extremum at (x 0, y 0 ). Lecture 46 ( 15.8): 1) According to the Second Partials Test, if D > 0 and f xx > 0 at (x 0, y 0 ), then f has a relative maximum at (x 0, y 0 ). 2) According to the Second Partials Test, if D < 0 at (x 0, y 0 ), then f must have a saddle point at (x 0, y 0 ). 3) According to the Second Partials Test, if D = 0 at (x 0, y 0 ), then f has neither a relative maximum, nor a relative minimum, nor a saddle point at (x 0, y 0 ). TYPE III (for lectures 45, 46): 6 th Ed. Section 15.8, # 8, 10, 14, 16, 17, 25, 26, 29, 30, 34, 36, 40, 42, 44, th Ed. Section 14.8, numbered as above. 8 th Ed. Same as 7 th Ed.. 9 th Ed. Section 13.8, # 8, 10, 14, 16, 17, 29 [25], 30, 33, 34 [30], 38, 40 [36], 44, 46, 48 [44], 58 [54]. Lecture 47 ( 15.9): 1) If f(x, y) is a differentiable function, and on the smooth curve C, f C has an interior relative extremum at (x 0, y 0 ), then f(x 0, y 0 ) is orthogonal to the unit tangent vector to C at (x 0, y 0 ). 2) If on the circle x 2 + y 2 = 1 the differentiable function f(x, y) has a relative extremum at (cos θ, sin θ), then f(cosθ, sin θ) = λ 2 cosθ, 2 sin θ for some scalar λ. 3) If f(x, y) and g(x, y) are differentiable functions, g(0, 0) = 0, and f(0, 0) = λ g(0, 0) for some scalar λ, then on the constraint curve g(x, y) = 0, f must have a constrained relative extremum at (0, 0). Lecture 48 ( 15.9): 15
16 1) If f(x, y, z) is a differentiable function, A is an open subset of dom(f), and f A has an absolute maximum at (x 0, y 0, z 0 ), then d dt f(x(t), y(t), z(t)) t=t 0 = 0 where x(t), y(t), z(t) is any smooth curve in A passing through (x 0, y 0, z 0 ) when t = t 0. 2) If f(x, y, z) is a differentiable function defined on the set R, S is a smooth surface lying in R, and f S has an interior absolute minimum at (x 0, y 0, z 0 ), then f(x 0, y 0, z 0 ) is orthogonal to the tangent plane to S at (x 0, y 0, z 0 ). 3) If f(x, y, z) is a differentiable function, S is a smooth surface, and f(0, 0, 0) is orthogonal to the tangent plane to S at (0, 0, 0), then f S must have a relative extremum at (0, 0, 0). TYPE III (for lectures 47, 48): 6 th Ed. Section 15.9, # 1, 3, 4, 7, 8, 12, 14, 15, 18, 21, 27, th Ed. Section 14.9, # 1, 5, 6, 9, 10 [8], 14, 16 [14], 17, 20 [18], 21, 27, th Ed. Same as 7 th Ed. 9 th Ed. Section 13.9, # 1, 5, 6, 9, 10 [8], 18, 20 [14], 21, 24 [18], 25, 31, 34 [30]. 16
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