Teaching - Nguyen The Vinh UTC ď Exercises 2-2-1

Teaching - Nguyen The Vinh UTC ď Exercises 2-2-1 tài liệu, giáo án, bài giảng , luận văn, luận án, đồ án, bài tập lớn về...

Exercises of Mathematical analysis II In exercises - represent the domain of the function by the inequalities and make a sketch showing the domain in xy-plane √ √ z = x − y z = arcsin z = y+2 + ln y x−1 √ sin π(x2 + y ) z = ln x − ln sin y √ z = − x2 + ln(y − 4) √ x z = arcsin y √ √ z = (y + y) cos x √ z = x2 + y − − ln(9 − x2 − y ) Are the functions z= √ √ √ x sin y and z = x sin y identical? Why? 10 Are the functions z = ln xy and z = ln x + ln y identical? Why? 3x2 − y Evaluate lim lim f (x, y) and lim lim f (x, y) Does x→0 y→0 y→0 x→0 2x + y f (x, y) exist? 11 f (x, y) = lim (x,y)→(0,0) Evaluate the limit 12 lim (x,y)→(0,0) √ x2 + y x2 + y + − 1 √ 13 lim (x,y)→(0,0) x2 y + − x2 + y 14 xy (x,y)→(0,0) x2 + y 15 xy(x2 − y ) (x,y)→(0,0) x2 + y 16 x2 − y (x,y)→(0,0) x2 + y 17 sin(x3 + y ) (x,y)→(0,0) x2 + y lim lim lim lim In exercises 18 - 28 find all first-order partial derivatives of the given function √ x 2√ 18 z = x y + √ y 19 z = ln tan x y 20 z = e− y x y 21 z = sin xy − cos x √ 22 z = ln(x + x2 + y ) y 23 z = arctan √ x 24 z = xy ln(x + y) 25 w = ln(xy + ln z) 26 w = tan(x2 + y + z ) z 27 w = xy 2 28 z = (x + y ) 1− √ x2 + y √ + x2 + y x at (1; −2) 29 Evaluate the partial derivatives of z = arcsin √ x2 + y 2 30 z = x cos y − y cos x ∂z ∂z Evaluate and , if x = y = + sin x + sin y ∂x ∂y 31 w = ln(1 + x + y + z ) Evaluate ∂w ∂w ∂w + + at the point x = ∂x ∂y ∂z y=z=1 32 z = ln(x2 − y ); prove that ∂z ∂z + − =0 ∂x ∂y x + y 33 For the function z = xy +x arctan y ∂z ∂z prove that et x +y = xy +z x ∂x ∂y x 34 Find the total differential of the function z = arcsin y 35 Find the total differential of the function z = sin x y cos y x x 36 Find the total differential of the function z = ln sin y 37 Find the total differential of the function w = xyz xy 38 Find the total differential of the function z = , if x = 2, y = 1, x − y2 ∆x = 0, 01 and ∆y = 0, 03 39 Evaluate the increment ∆z and total differential dz of the function x+y 1 z= , if x = −3, y = 7, ∆x = − and ∆y = x−y 40 Evaluate the increment ∆z and total differential dz of the function x z = xy + , if x changes from −1 to −0, and y from to 2, y 41 Using total differential, compute the approximate value of 1, 963 ·2, 035 √ 82 42 Using total differential, compute the approximate value of √ 28 √ 1, 04 43 Using total differential, compute the approximate value of arcsin 2, 04 (√ ) √ 44 Using total differential, compute the approximate value of ln 0, 98 + 1, 04 − 45 Find dy , if x2 y − x4 − y = a2 dx dy , if 2y + 3x2 y + ln x = and evaluate it at x = dx √ dy x 47 Find , if y = x ln and evaluate it at the point (e2 ; e) dx y 46 Find 48 Find dy , if xy = y x and evaluate it at the point (1; 1) dx 49 Find ∂z ∂z and , if x2 − 2y + z − 4x + 2z − = ∂x ∂y ∂z ∂z 50 Find and , if z = cos xy − sin xz and evaluate these at the point ∂y ( π ∂x) ; 1; 51 Find ∂z ∂z and , if xyz = ez and evaluate these at the point (e−1 ; −1; −1) ∂x ∂y 52 Find the total differential of z, if z is determined by the equation cos2 x + cos2 y + cos2 z = dz , if z = arctan(xy + 1) and y = ln x dx √ dz x 54 Find , if z = arcsin and y = x2 + dx y 53 Find 55 Find √ dz , if z = tan(3t + 2x2 − y), x = and y = t dt t du e2x , if u = (y − z), y = sin x and z = cos x dx √ dw 57 Find , if w = x2 + u2 + v , u = sin x and v = ex dx 56 Find y dz , if z = arcsin , x = sin t and y = t2 dt x √ ∂z ∂z 59 Find and , if z = x2 + y , x = u cos v and y = v cos u ∂u ∂v 58 Find 60 Find ∂z ∂z and , if z = ln(x2 + y ), x = u cosh v and y = v sinh u ∂u ∂v 61 Find the partial derivatives if u = xy and v = x − y ∂z ∂z and of the function z = arctan uv, ∂x ∂y 62 Prove that ∂ 2z ∂2z = , if z = ex (cos y + x sin y) ∂x∂y ∂y∂x 63 Find ∂2z ∂ 2z ∂2z , if z = arcsin(xy) , and ∂x2 ∂y ∂x∂y 64 Find ∂3w , if w = exyz ∂x∂y∂z 65 Evaluate all second order derivatives of the function z = point (−1; −2) x at the y2 x 66 Evaluate all second order derivatives of the function z = arcsin √ x2 + y at the point (1; −2) 67 Evaluate ∂ 2z ∂ 2z · − ∂x2 ∂y at the point (0; −2), if z = ( ∂ 2z ∂x∂y )2 cos x2 y ∂z ∂z ∂ 2z ∂ 2z + = and 2 − 68 For the function z = ln (ex + ey ) prove that ∂x ∂y ∂x ∂y ( )2 ∂ z = ∂x∂y 69 Prove that the function z = x x2 y satisfies the equation x+y ∂ 2z ∂z ∂2z + y =2 ∂x ∂x∂y ∂x 70 Find the canonical equations of the tangent line of spatial curve x = + sin t, y = 2t − cos t, z = + t2 at t = 71 Find the canonical equations of the tangent line of screw line x = cos t, π y = sin t, z = 4t at t = 72 Find the canonical equations of the tangent line of )spatial curve x = (π √ t t − sin t, y = − cos t, z = sin at − 1; 1; 2 2 73 On the curve y = x2 , z = x3 find the points at which the tangent line is parallel to the plane x + 2y + z = −1 of 74 Find the equation of the tangent plane and the canonical(equations ) y π the normal line for the surface z = arctan at the point 2; −2; − x 75 Find the equation of the tangent plane √ and the canonical equations of the normal line for the surface z = x2 + y at (3; −4; 5) 76 Find the equation of the tangent plane and the canonical equations of y the normal line for the surface z = cos at (−1; −π; −1) x 77 Find the equation of the tangent plane and the canonical equations of the normal line for the surface x2 y + 2x + z = 16 at the point x = and y = 78 Prove that the surfaces x + 2y − ln z = −4 and x2 − xy − 8x + z = −5 have the same tangent plane at the point (2; −3; 1) √ 79 Find the gradient vector for the scalar field z = x − 3y + 3xy at the point (3; 4) points at ( which the 80 Find ( the ) ) gradient vector of the scalar field z = 16 → ln x + is − a = 1; − y √ x2 + y 81 Find the gradient vector for the scalar field w = arcsin at z the point (1; 1; 2) y 4y at the 82 Find the directional derivative of the function z = arctan − x x √ √ point (1; 3) in direction the point (2; 3) 83 Find the directional derivative of the function w = xyz at the point → A(−2; 1; 3) in the direction of − s = (4; 3; 12) 84 Find the directional derivative of the function w = x2 y − z + 2xyz at the point B(1; 1; 0) in direction forming with coordinate axes the angles 60◦ , 45◦ and 60◦ respectively 85 Find the greatest increase of the function z = ln(x2 + y ) at the point C(−3; 4) 86 Find the greatest value of the derivative of function given by the equation x2 + y − z − = at the point (3; 2; 4) 87 Find the steepest ascent of the surface z = arctan y at the point (1; 1) x 88 Find the direction of greatest increase of the function f (x, y, z) = x sin z − y cos z at the origin ( ) − → x y z 89 Find the divergence and curl of the vector field F = ; ; y z x − → 90 Find the divergence and curl of the vector field F = (ln(x2 − y ); arctan(z − y); xyz) → − 91 Find the divergence and curl of the vector field F = grad w, if w = ln(x + y − z) → − → − − → 92 Find the divergence and curl of the vector field F = rot G , if G = (x2 y; y z; x2 z) 93 Find the local extrema of the function z = 4x2 − xy + 9y + x − y and determine their type 94 Find the local extremum points of the function z = x3 y (12 − x − y), satisfying the conditions x > and y > and determine their type 95 Find the local extrema of the function z = x2 + xy + y + 1 + and x y determine their type 96 Find the local extrema of the function z = ex (x2 + y ) and determine their type 97 Find the local extrema of the function z = x3 + y − 3xy and determine their type 98 Find the greatest and the least value of the function z = x2 + 2xy − 4x + 8y in the rectangle bounded by x = 0, y = 0, x = and y = 99 Find the greatest and the least value of the function z = x2 − y in the circle x2 + y ≤ 100 Find the greatest and the least value of the function z = sin x + sin y + π π sin(x + y) in the quadrate ≤ x ≤ , ≤ y ≤ 2 101 Find the extremal values of the function z = 1 + under the condition x y x + y = 102 Find the extremal values of the function z = a cos2 x + b cos2 y under π the condition y − x = 103 Find the extremal values of the function w = x + y + z under the 1 condition + + = x y z 104 The sum of three edges of the rectangular box, passing one vertex is m Find the dimensions of this rectangular box so that the volume is the greatest 105 Find the point on the parabola y = 3x2 − closest to the pointP0 (0; 2) 106 Find the point in the plane 3x − 2z = so that the sum of squares of distances from the points A(1; 1; 1) and B(2; 3; 4) is the least ∫∫ 107 Evaluate the double integral (x2 + y )dxdy, if D is the quadrate ≤ x ≤ and ≤ y ≤ D ∫∫ 108 Evaluate the double integral dxdy , if D is the quadrate ≤ (x + y)2 D x ≤ and ≤ y ≤ √ ∫2 109 Evaluate the double integral x ∫ dx ∫1 110 Evaluate the double integral 111 Evaluate the double integral xydy x ∫x+1 dx (xy + y)dy −x ∫1 √ ∫x dx (x2 + y )dy x2 112 Sketch ∫ ∫ the domain of integration and determine the limits of integration for f (x; y)dxdy, if D is the region bounded by the line y = and D the parabola y = − x2 113 Sketch ∫ ∫ the domain of integration and determine the limits of integration f (x; y)dxdy, if D is the parallelogram bounded by the lines for D y = 0, y = a, y = x and y = x − 2a 114 Sketch ∫ ∫ the domain of integration and determine the limits of integration f (x; y)dxdy, if D is the region bounded by y = and for + x2 D y = x2 115 Sketch the√ domain of integration and change the order of integration ∫1 ∫ x for dx f (x; y)dy x3 116 Sketch the domain of integration and change the order of integration ∫1 ∫x+1 for dx f (x; y)dy √ − 1−x2 −1 117 Sketch the domain of integration and change the order of integration y2 ∫2 for ∫2 dy −2 f (x; y)dx y −2 118 Sketch the domain of integration and change the order of integration x∫2 +2 ∫1 for dx f (x; y)dy −1 x2 ∫1 119 Changing the order of integration express the sum dy ∫0 −1 ∫1 √ f (x; y)dx + y ∫y+1 dy f (x; y)dx by one double integral 120 Sketch the domain ∫ ∫ of integration, determine the limits and evaluate the double integral (x−2y)dxdy, if D is the region given by inequalities D −1 ≤ x ≤ and ≤ y ≤ x2 + 121 Sketch the domain ∫ ∫ of integration, determine the limits and evaluate the (x2 + y )dxdy, if D is bounded by the lines y = x, double integral D x + y = 2a and x = 122 Sketch the domain ∫ ∫ of integration, determine the limits and evaluate the xydxdy, if D is the least of segments bounded by double integral D the line x + y = and circle x2 + y = 2y 123 Sketch the domain ∫ ∫of integration, determine the limits and evaluate the ex+y dxdy, if D is the region bounded by y = ex , double integral D x = and y = ∫∫ 124 Convert the double integral f (x; y)dxdy to polar coordinates, if D D is the region determined by inequalities ≤ x2 + y ≤ and y ≥ ∫∫ 125 Convert the double integral f (x; y)dxdy to polar coordinates, if D D is bounded by the circles x2 + y = 4x and x2 + y = 8x and the lines y = x and y = 2x √ 2Ry−y ∫2R ∫ 126 Convert the double integral dy f (x; y)dx to polar coordi0 R nates the double integral by converting it into polar coordinates 127 Evaluate √ −x2 a a ∫ ∫ 2 dx ex +y dy 0 128 Evaluate the double integral by converting it into polar coordinates ∫∫ dxdy √ , if D is the region determined by inequalities x2 + 2 4−x −y D y ≤ 4, x ≥ 0, y ≥ 10 129 Evaluate the double integral by converting it into polar coordinates √ R ∫2 −x2 ∫R ( ) dx ln + x2 + y dy 0 130 Evaluate ∫ ∫ √ the double integral by converting it into polar coordinates R2 − x2 − y dxdy, if D is the circle x2 + y ≤ Rx D 131 Evaluate ∫ ∫ √ the double integral by converting it into polar coordinates x x2 + y dxdy, if D is bounded by the part of lemniscate (x2 + D y )2 = a2 (x2 − y ) where x ≥ ∫a 132 Evaluate the triple integral ∫x dx ∫∫∫ 133 Evaluate the triple integral ∫xy x3 y zdz dy 0 dxdydz , if V is the region (x + y + z + 1)3 V bounded by planes x = 0, y = 0, z = and x + y + z = ∫∫∫ 134 Evaluate the triple integral xyzdxdydz, if V is bounded by the V surfacesy = x2 , x = y , z = xy and z = ∫∫∫ 135 Convert the triple integral f (x; y; z)dxdydz into cylindrical coorV dinates, if V is the region bounded by the planes x = 0, y = 0, z = and cylinders x2 + y = and z = x2 + y 136 Evaluate the triple integral by converting it into cylindrical coordinates √ ∫1 ∫1−x2 dx −1 ∫1 dy √ − 1−x2 x2 +y dz √ x2 + y 137 Evaluate ∫ ∫ ∫ √the triple integral by converting it into cylindrical coordinates z x2 + y dxdydz, if V is the region determined by the inequalV ities ≤ x ≤ 2, ≤ z ≤ and ≤ y ≤ 11 √ 2x − x2 ∫∫∫ 138 Convert the triple integral f (x; y; z)dxdydz into spherical coordiV nates, if V is the region determined ≤ x2 + y + z ≤ 4, z ≥ and y ≤ by the inequalities 139 Evaluate the triple integral by converting it into spherical coordinates √ √ R2 −x2 −y R ∫R ∫2 −x2 ∫ dz √ dx dy z √ −R − R2 −x2 140 Evaluate ∫ ∫ ∫ √ the triple integral by converting it into spherical coordinates x2 + y + z dxdydz, if V is the region determined by the inV equalities ≤ y ≤ 1, ≤ x ≤ √ √ − y and ≤ z ≤ − x2 − y 141 Compute the area bounded by xy = and x + y = 142 Compute the area bounded byy = a is a positive constant 8a3 , x = 2y and x = provided x2 + 4a2 143 Compute the volume of solid bounded by the planes z = 0, y = 0, y = x and x = and paraboloid of revolution z = x2 + y 144 Compute the volume of solid bounded by the hyperbolic paraboloid (saddle surface) z = x2 − y and the planes z = and x = 145 Compute the volume of solid bounded by the surfaces z = x2 + y , z = 2(x2 + y ), y = x and y = x 146 Compute the volume of solid bounded by the sphere x2 + y + z = and paraboloid of revolution 3z = x2 + y 147 Compute the volume of solid determined by the equations y ≥ 0, y ≤ x, ≤ x2 + y ≤ and ≤ z ≤ x2 + y + 148 Compute the volume of solid determined by the equations x2 +y +z ≤ R2 and x2 + y + z ≤ 2Rz ∫ 149 Compute the line integral (x2 + y )ds where L is the line segment L from A(1; 1) to B(4; 4) 12 ∫ y ds where L is the arc of cycloid x = 150 Compute the line integral L a(t − sin t), y = a(1 − cos t) between the points O(0; 0) and C(2aπ; 0) ∫ 151 Compute the line integral (x2 + y + z)ds where L is the arc of helix L x = a cos t, y = a sin t, z = bt from t = to t = 2π ∫ 152 Compute the line integral xyzds where L is the quarter of circle √ R R R x = cos t, y = sin t, z = , which lies in the first octant 2 ∫ ydx + xdy 153 Compute the line integral where L is the segment of the x2 + y L L line y = x from (1; 1) to (2; 2) ∫ y arctan dy − dx where L is the arc of 154 Compute the line integral x L parabola y = x2 from O(0; 0) to A(1; 1) ∫ 155 Compute the line integral (x + y)dx + (x − y)dy where AB is the arc AB of ellipse x = a cos t, y = b sin t from A(a; 0) to B(0; b) ∫ 156 Compute the line integral xdy − ydx where L is the arc of astroid L 3 x = a cos t, y = a sin t from t = to t = ∫ π x dy dx + where L is the arc of cycloid y y−1 L π π x = t − sin t, y = − cos t from t = to t = ∫ xdx + ydy + zdz √ 158 Compute the line integral where AB is x2 + y + z − x − y + 2z 157 Compute the line integral AB the line segment from A(1; 1; 1) to B(4; 4; 4) 13 ∫ 159 Compute the line integral yzdx + xzdy + xydz where L is the arc of L helix x = a cos t, y = a sin t, z = bt from t = to t = 2π (1 − x3 )ydx + x(1 + y )dy to the double 160 Convert the line integral L integral over the region D where L is positively oriented, smooth, closed curve and D the region enclosed by L ex (1−cos y)dx+ex (sin y+y)dy to the double 161 Convert the line integral L integral over the region D where L is positively oriented, smooth, closed curve and D the region enclosed by L (x + y )dx + (x + y)2 dy where L is the 162 Use Green’s theorem to find L contour of triangle ABC with vertices A(1; 0), B(1; 1) and C(0; 1) with positive orientation (5x − 3y)dx + (x − 4y)dy where L is the 163 Use Green’s theorem to find L circle x2 + y = with positive orientation 2xydx + x2 dy where L is the contour of 164 Use Green’s theorem to find L square |x| + |y| = with positive orientation xy dy − x2 ydx where L is the circle 165 Use Green’s theorem to find L x2 + y = with positive orientation 166 Find the function u, if the total differential is du = x2 dx + y dy 167 Find the function u, if the total differential is du = (cos y − 2xey )dx − (x2 ey + x sin y)dy (2;1) ∫ 2xydx + x2 dy 168 Evaluate (0;0) 14 (2;3) ∫ 169 Evaluate ydx + xdy (−1;2) (2;2) ∫ 170 Evaluate xdx + ydy √ x2 + y (1;1) ∫∫ 171 Evaluate (x+y +z)dσ where S is the part of the plain x y + +z = S in the first octant ∫∫ 172 Evaluate (x2 + y )dσ where S is the surface cut from the cone S √ x2 + y by the cylinder x2 + y = ∫∫ √ 173 Evaluate R2 − x2 − y dσ where S is the upper half of the sphere z= S √ z = R − x2 − y ∫∫ √ 174 Evaluate + x2 + y dσ where S is the part of the saddle surface S z = xy cut by the cylinder x2 + y = ∫∫ xdydz + ydxdz + zdxdy where S is that part of the plane 175 Evaluate S x + y + z = which is in the first octant Choose the side where the normal forms the acute angles with coordinate axes ∫∫ 176 Evaluate x2 y zdxdy where S is the upper side of the hemisphere S √ z = R − x2 − y ∫∫ xyzdzdy where S is the lower side of the hemisphere 177 Evaluate S √ R − x2 − y ∫∫ 178 Evaluate xzdxdy + xydydz + yzdxdz+ where S is the inner side z= S 15 of the pyramid determined by the planes x = 0, y = 0, z = and x + y + z = ( )3 ( )5 179 Write the general term of the series + + + 180 Write the general term of the series − + − + 13 19 1 = − , find the nth partial sum and k(k + 1) k k+1 1 1 the sum of the series + + + + + 1·2 2·3 3·4 k(k + 1) 181 Using the identity 1 182 Find the nth partial sum and the sum of the series + + + · 6 · 9 · 12 + + 3k(3k + 3) ∞ ∑ √ ( k+2 − 183 find the nth partial sum and the sum of the series k=1 √ √ k + + k) 184 Use the Comparison Test to determine whether the series ∞ ∑ k=1 2k + 3k converges or diverges 185 Use the Comparison Test to determine whether the series ∞ ∑ k=2 (k − 1) converges or diverges 186 Use the d’Alembert Test to determine whether the series + · 2n−2 12 + + + converges or diverges 4! n! 187 Use the d’Alembert Test to determine whether the series + + 2! 3! ∞ ∑ k2 k=1 k! con- verges or diverges 188 Use the d’Alembert Test to determine whether the series ∞ ∑ k=0 converges or diverges 16 3k (3k + 1)! 189 Use the Cauchy Test to determine whether the series ∞ ∑ arcsink k=1 2k − 4k + converges or diverges 190 Use the Cauchy Test to determine whether the series ∞ ∑ lnk k=1 2k + k+1 converges or diverges 191 Use the Cauchy Test to determine whether the series ∞ ∑ ( k k=1 k+2 k+1 )−k2 converges or diverges 1 192 Use the Integral Test to determine whether the series + + + 10 + + converges or diverges 3k + 193 Use the Integral Test to determine whether the series ∞ ∑ k=2 conk(ln k)2 verges or diverges 1 194 Use the Leibnitz’s Test to determine whether the series − √ + √ − 1 √ + √ − √ + converges or diverges 195 Use the Leibnitz’s Test to determine whether the series ∞ ∑ cos kπ k=1 verges or diverges 196 Does the series 1− 1 1 n+1 + − + + (−1) + 32 52 72 (2n − 1)2 converges conditionally or absolutely? 197 Does the series 1 1 1 − · + · − + (−1)n+1 + 2 n (2)n converges conditionally or absolutely? 17 k3 con- 198 Does the series 1 1 − + − + + (−1)n + ln ln ln ln ln n converges conditionally or absolutely? In exercises 199 - 203 find the values of x for which the functional series is convergent 199 + x x2 xn + + + n + n x2 x3 x4 n+1 x + − + + (−1) + 22 32 42 n2 x x x 201 sin x + sin + sin + + 2n sin n + 200 x − 202 203 x x2 x3 xn √ + √ + √ + √ + + n+ n 1+ 2+ 3+ ∞ ∑ lnk (ex) k=0 In exercises 204 - 206 determine whether the functional series can be majorized 204 + x x2 xn + + + + ≤ x ≤ 12 22 n2 205 + x x2 x3 xn + + + + + ≤ x ≤ 1 n 206 207 sin x sin 2x sin 3x sin nx + + + + + ≤ x ≤ 2π 2 2 n2 In exercises 207 -210 find the radius of convergence and the domain of convergence ∞ ∑ k=1 208 ∞ ∑ (kx)k k=0 209 x2 k(k + 1) k! ∞ ∑ k(x − 2)k k=0 3k 18 210 ∞ ∑ 2k (x + 3)k k=0 k! In exercises 211 - 216 expand the function in powers of x and determine the domain of convergence 211 f (x) = 10 + x 212 f (x) = e−x 213 f (x) = + x2 214 f (x) = sinh x 215 f (x) = cos2 x 216 f (x) = arctan x (Remark: integrate the result of the exercise 213 in limits from to x) In exercises 217 - 221 find the Fourier series expansion of the given 2π-periodic function defined on a half-open interval { −1, −π < x < 217 f (x) = 1, ≤ x ≤ π 218 f (x) = x, if −π < x ≤ π 219 f (x) = x2 , if −π < x ≤ π 220 f (x) = sin ax, if −π < x ≤ π 221 f (x) = π−x , if < x ≤ 2π Answers No because the first function is additionally determined, if x ≤ and sin y ≤ 10 No because the first function is additionally determined, if x < and y < 11 0; -1; does not exist 12 13 14 Does not exist 15 16.√ Does not ex1 x2 x √ √ 19 ; − √ ist 17 18 2x y + √ √ 4 4y y x y y 2x x x x ; − 20 − e− y ; e− y 21 y cos xy − 2x 2x y y y sin y sin y y 19 y y y y √ √ sin ; x cos xy+ sin 22 √ ; 2 x x) + yx x x x x x + y (x + x + y x √ x y xy xy 23 − √ ; 24 y ln(x+y)+ ; x ln(x+y)+ 2 x+y x+y x(x + y ) x + y y x 2x 25 ; ; ) 26 ; 2 xy + ln z xy + ln z z(xy + ln z cos (x + y + z ) 3y 4z z z ; 27 y z xy −1 ; xy ln x · zy z−1 ; 2 2 cos (x + y + z ) cos (x + y + z ) √ √ 2 2 + y2 − x − y − x − x − y − x2 + y z xy ln x·y z ln y 28 2x· ( ; 2y· ) )2 ( √ √ 2 2 1+ x +y 1+ x +y ydx − xdy √ 29 ; 30 1; -1 31 34 dz = − x2 5 |y| y ( ) x y x y ydx + xdy 2 35 dz = x cos cos + y sin sin 36 dz = y x x x2 + y ) ( y yzdx ydx − xdy 37 dw = xyz + z ln xdy + y ln xdz 38 x x y tan y 19 19 39 ∆z = ; dz = 40 ∆z ≈ 0, 3764; 36 635 600 53 π dz = 0, 35 41 259,84 42 43 44 54 x(2x2 − y ) 0,006 45 46 − 47 48 2 y(x − 2y ) 4e 2−x 2y π e 49 ; 50 − ; − 51 ; − z+1 z+1 2+π 2+π 2 1 + ln x 52 dz = − (sin 2xdx + sin 2ydy) 53 sin 2z x2 ln2) x + 2x ln x + ( √ ) 3− − √ ( 54 55 56 2 x +4 t cos 3t + t2 − t t x + u cos x + vex t(2 − t cos t)| sin t| √ √ e2x sin x 57 58 2 x +u +v sin t sin2 t − t4 u cos2 v − v sin u cos u v cos2 u − u2 sin v cos v sinh v cosh v 59 √ ; √ 60 ; 2 2 2 2 u sinh2 v + cosh2 v u cos v + v cos u u cos v + v cos u y(2x − y) x(x − 2y) xy √ 61 ; 63 ; + x2 y (x − y)2 + x2 y (x − y)2 (1 − x2 y ) − x2 y x3 y √ √ ; 64 exyz (1 + 3xyz + 2 2 2 2 (1 − x y ) − x y (1 − x y ) − x y 4 x2 y z ) 65 0; ; − 66 − ; ; 67 25 25 25 20 √ √ x−1 y+1 z−3 x− y− √ 70 = = 71 = √ = − 2 π √ x− +1 z−π y − z − 2 √ 72 = = 73 (−1; 1; −1); 1 π ( ) z + 1 x−2 y+2 − ; ;− 74 x + y − 4z = π = = 1 27 −1 4 y+4 z−5 x−3 75 3x − 4y − 5z = 0; = = 76 z + = 0; −1 − 5 x+1 y+π z+1 x−2 = = 77 3x + 4y + 6z − 22 = 0; = 0 −1 ( ) ( ) ( ) y−1 z−2 1 = 79 2; −2 80 − ; ; ;− 4√ 4 ( ) 1 15 30 81 ; ;− 82 − 83 − 84 2 13 13 √ √ √ 85 0,4; 86 2+ ; 87 88 (0; −1; 0) ( )2 − → − → − → y z x 1 89 div F = + + ; rot F = ; ; 90 div F = y z x z(2 x2 y ) − → 1 2y 2x − + xy; rot F = xz − ; −yz; x2 − y + (z − y)2 + (z − y)2 x − y2 − → → − − → − → − → 91 div F = − ; rot F = Θ 92 div F = 0; rot F = (x + y − z) ( ) 17 (2x; 2x; 2y −2z) 93 Local minimum at − ; 94 Lo143 143 √ cal ( maximum)zmax = 6912 at (6; 4) 95 Local minimum zmin = 3 1 at √ ;√ 96 There is no local extremum at (−2; 0), local 3 3 minimum at (0; 0) 97 There is no local extremum at (0; 0), local minimum at (1; 1) 98 zmin = z(1; 0) = −3; zmax = z(1; 2) = 17 99 zmin = z(0; 2) = z(0; −2) = −4; zmax √ = z(2; 0) = z(−2; 0) = (π π ) 3 ; = 101 z(1; 1) = 100 zmin = z(0; 0) = 0; zmax = z ( ( )3 ) b π b 1 102 − arctan ; − arctan ; ; 103 104 a a 3 (√ ) ( √ ) 23 11 23 11 1 , and m 105 ; ; ; − 106 3 18 18 21 ) 63 25 21 ; 2; 107 108 ln 109 13 26 24 1−x ∫ ∫ 112 dx f (x; y)dy 35 ( −1 0 1+x2 ∫1 ∫ 114 dx 115 x √ ∫0 ∫1−y ∫2 ∫1 dy f (x; y)dx+ dy f (x; y)dx √ −1 y−1 − ∫2 √ y √ ∫3 y ∫1 f (x; y)dy −1 19 111 12 y+2a ∫ ∫a 113 dy f (x; y)dx 110 dy f (x; y)dx 116 y2 √ ∫2 117 ∫x+2 dx −2 1−y f (x; y)dy− √ − x+2 ∫ 2x f (x; y)dy dx √ − 2x ∫1 √ ∫y dy 118 √ ∫3 ∫3 ∫1 ∫y−2 f (x; y)dx+ dy f (x; y)dx− dy f (x; y)dx √ − y −1 − y−2 ∫ ∫1 119 √ x2 dx 120 −10 f (x; y)dy x−1 ∫π 123 e 124 dφ f (ρ cos φ; ρ sin φ)ρdρ R sin φ ∫2π 122 125 127 π a2 (e − 1) 8∫ cos φ dφ π f (ρ cos φ; ρ sin φ)ρdρ π 4 a arctan ∫ 2R ∫sin φ ∫ 126 121 ∫2 dφ π 20 f (ρ cos φ; ρ sin φ)ρdρ cos φ 128 ( ) R π [(1 + R2 ) ln(1 + R2 ) − R2 ] 130 π− π 129 3 √ 2 a11 131 a 132 133 ln − 134 15 110 16 96 π ∫2 ∫2 ∫ρ2 135 137 dφ dρ f (ρ cos φ; ρ sin φ; z)ρdz 136 π 138 ∫ dφ π π ∫2 f (r cos φ sin θ; r sin φ sin θ; r cos θ)r2 sin θdr dθ 22 139 8 2√ π πR R 140 ; 141 (15 − 16 ln 2) 142 16 a2 (π − 1) 143 144 27 145 146 35 √ 19 21π 5πR π 147 148 149 42 150 16 12 √ √ 256 R4 a 151 2π a2 + b2 (a2 + πb) 152 153 15 32 2 π a +b 3πab ln 154 − 155 − 156 2 16 √ √ π2 1− 157 + − ln 158 3; 159 24∫ 2 ∫ ∫∫ 160 (x3 + y )dxdy 161 ex ydxdy 162 D D 25π 163 4π 164 165 166 u(x, y) = (x + y ) + C 167 u(x, y) = x cos y − x2 ey + C 168 √ √ √ 21 π 3π 169 170 171 172 173 πR3 174 2 2πR 175 176 177 178 − 179 2) 105 ( 2k−1 k k 180 (−1)k 181 Sn = − ; S = 3k − 6k − n+1 √ √ √ 1 182 Sn = − ;S= 183 − + n + − n + 1; 9(n + 1) √ − 184 Converges 185 Diverges 186 Converges 187 Converges 188 Converges 189 Converges 190 Converges 191 Converges 192 Diverges 193 Converges 194 Converges 195 Converges 196 Converges absolutely 197 Converges absolutely 198 Converges conditionally 199 −2 < x < 200 −1 ≤ x ≤ 201 −∞ < x < ∞ 202 −1 ≤ x < 203 (e−2 ; 1) 204 Majorized 205 Not ma−1 jorized 206 Majorized 207 1; [−1; 1] 208 e ; [−e−1 ; e−1 ) x x2 x3 209 3; (−1; 5) 210 ∞; R 211 − + − + , con10 100 10 10 x2 x3 verges for −10 < x < 10 212 − x + − + , converges for −∞ < 2! 3! x < ∞ 213 − x2 + x4 − x6 + , converges for −1 < x < 214 ∞ x3 x5 ∑ (−1)n (2x)2n x+ + + ., converges for −∞ < x < ∞ 215 1+ , 3! 5! n=1 (2n)! 23 x3 x5 x7 converges for −∞ < x < ∞ 216 x − + − + , converges ∞ ∞ ∑ ∑ sin(2k + 1)x (−1)k+1 for −1 ≤ x ≤ 217 218 sin kx π k=1 2k + k k=1 ∞ ∞ ∑ π2 sin aπ ∑ (−1)k k (−1)k 219 +4 cos kx 220 sin kx 221 2 − k2 k π a k=1 k=1 ∞ ∑ sin kx k k=1 24 ... parallel to the plane x + 2y + z = −1 of 74 Find the equation of the tangent plane and the canonical(equations ) y π the normal line for the surface z = arctan at the point 2; −2; − x 75 Find the equation... equation of the tangent plane √ and the canonical equations of the normal line for the surface z = x2 + y at (3; −4; 5) 76 Find the equation of the tangent plane and the canonical equations of y the. .. Find the local extrema of the function z = ex (x2 + y ) and determine their type 97 Find the local extrema of the function z = x3 + y − 3xy and determine their type 98 Find the greatest and the