Given the vectors u, v, w and real numbers α, β, γ. Calculate vector a, which is equal to the linear combination α u + β v + γ w.

Transcription

1 Selected problems from the tetbook J. Neustupa, S. Kračmar: Sbírka příkladů z Matematiky I Problems in Mathematics I I. LINEAR ALGEBRA I.. Vectors, vector spaces Given the vectors u, v, w and real numbers α, β, γ. Calculate vector a, which is equal to the linear combination α u + β v + γ w.. u = 4,,, v = 5,,, w =,,, α =, β =, γ = Find vector, which satisfies the given equation u = u + 6v, where u =,,,, v =,,, 5 Calculate the scalar product of given vectors. Hint: Use the formula u v = u v u n v n. 5. u =,,, v = 4,, Calculate the angle between given vectors. Hint: Use the formula cos ϑ = u v/ u v.. u =,, v =, For which value of parameter α are given vectors orthogonal? Hint: Two vectors are orthogonal if their scalar product equals zero. 5. u =, α +, v =, + α Given vectors from VIE, VIE or VIE 4. a Are the vectors linearly dependent or linearly independent? b What is the dimension of the linear hull of these vectors? c Which vectors form the basis of the linear hull? Consider the fact that question b can also be formulated in this way: whose rows or columns are the given vectors? 4. u =,, v =, 4. u =,, v =, What is the rank of a matri, 44. u =, 4,, v =,, 45. u =, 5,, v =, 5, 5. =, 5, y =,, z = 5, 5 5. =,,, y =,,, z =, 9, 8 5. =,,,, y =,, 5,, z = 4, 6, 5, Given vectors from VIE, VIE or VIE 4 with parameters. a For which values of the parameters are the vectors linearly dependent or linearly independent? b What is the dimension of the linear hull of these vectors? Hint: Consider the matri whose rows are the given vectors. Determine the rank of the matri in dependence on the parameters. 68. a =,,, b =,,, c = α,, α +, d = α, α, 69. u = k,,, v =, k,, w =,, k 7. u =,, a, v =, a, a, w =,, Epress the vectors a, b as a linear combination of the vectors u, v respectively u, v, w if it is possible. Is the epression unique? Hint: Vector a has the form a = αu + βv. Write this epression in coordinates. You obtain a system of equations for the unknowns α and β. 7. a =,, 5, b = 5, 6, 7, u =,,, v =,,, w = 5,, 8

2 a Do given vectors form a basis of vector space V? b What is the dimension of the linear hull of the given vectors? Consider that question b can also be formulated as follows: rows or columns are the given vectors? 79. V = VIE, a =, 7,, b = 5,, What is the rank of a matri, whose Given a vector space V and its subset V. Determine if V is a subspace of vector space V. 85. V = VIE, V = {a, b, c; a, b, c IR, a + b = } 9. V = VIE 4, V = {u, v, w, ; u, v, w, IR, u v + w } In net problems V is a set of functions defined on interval I. The sum f + g of arbitrary two functions f and g from V is defined: f + g = f + g for I. The multiple λ f of an arbitrary function f from V by an arbitrary real number λ is defined: λ f = λ f for I. Decide if set V with the two defined operations is a vector space. 9. I =, +, V is a set of all functions of the form α sin + β cos + γ where α, β, γ IR. 94. I =, +, V is the set of all functions of the form α + β + γ where α, β, γ IR. I.. Matrices, determinants Calculate the matri A B.,,, 4,, 9. A =, B =,,,. A =, B =,, 5, 5,,,,,, 4,,,,,,. A =, B =. A =,,, B =,, 4,,,,,,,,, Calculate, 7. 4.,,,,,,, T,, Calculate the matri A B B A.,, 4,,,,. A =,,, B = 4,,. A =,,, B =,,,,,, Find, y IR that satisfy the given equation. T,, 9. = 4. y + 6, 4, + y, =, y [,,,,,, 4, 5,, ]T, Determine the rank of a given matri. If it is a square matri, decide if it is a regular matri or a singular matri.,,,,,,, 4.,, 48., 6, 9, 4, 4, 49., 9, 5,, 7, 7 4, 8,, 5,, Calculate the inverse matri if it eists.

3 6. 68.,, 6.,,,, 7.,,,,,, 66.,,,,,, 7.,,,,,,, 4, 5 cos, sin sin, cos Find matri X that satisfies the equation,,,, 74. X,, = 4,,, 75. X =,,,,, A =,,,,, B =. Find matri X, that satisfies A X = A B,,. 77. Solve the matri equation A X B = C, where A =, C =., 78. Given the matri A =, +, y, + y,. z, + z,,, B = 5,,,, For which, y, z R is matri A regular? Calculate A for =, y =, z =. Calculate the determinants 8. cos, sin sin, cos 85., 5,, 7, 4,, 4 a, a, a 9. a, a, a, a,.,,,,,, a, b,,,,, I.4. Eigenvalues and eigenvectors of square matrices Find eigenvalues and eigenvectors of the matrices ,, 6., 4 5,,,,, 4.,, 7., a a,,, 4,, 44. 4, 8, 5, 6, 8.,,,,, 5, 6 4, 6, 9, 6, 8 In net problems we assume that A is a square matri of the type, whose entries are real numbers. Decide if matri A can have given eigenvalues and eigenvectors. 45. eigenvalues:,, + i 46. eigenvalues:, + i, i, eigenvectors:, + i, i Calculate the inverse matri to matri A and find eigenvalues and eigenvectors of the inverse matri A. Compare the results with the eigenvalues and eigenvectors of matri A. 5, 6,,, 4, 5 5. A = 5. A = 5. A = 5. A =,,, 5, 5,,,

4 To given square matri A calculate the matri A and find eigenvalues and eigenvectors of the matri A. Compare the results with the eigenvalues and the eigenvectors of matri A.,,,,, 55. A = 56. A = 57. A = 58. A = 5,,,,,,, I.5. Systems of linear algebraic equations Applying Frobenius theorem decide about the solvability of the system of equations and about the number of solutions. 7. y = y = 4 5y = y + z = 9 + 5y + 4z = 5 + y + 6z = y = 5 + 4y + z = 7 + y + 5z = Applying Frobenius theorem decide about the solvability of the system of equations and about the number of solutions in dependence on parameters. 78. a + y + z = + ay + z = + y + az = 8. a y = a y = 8. y + z + u = + y z + 4u = + 7y 4z + u = λ Can the net systems be solved by means of Cramer s rule? If,,yes, find the solution applying this rule. Hint: Cramer s rule can be applied if the matri of the system is regular. 87. y + z = + y z = 7 4y z = 88. y + z = + y z = + y + z = y z = + y + z = + y + z = What is the dimension of the vector space of all solutions of the homogeneous system in dependence on parameter λ?. + y z = y + z = λ + y 4z =. 4 + y z = + y + z = λ + y λz = Solve the homogeneous systems by means of Gaussian elimination. 8. y + z = + y 5z = + y z = = + 4 = = = = = = + 4 = Solve the inhomogeneous systems by means of Gaussian elimination y + z = 4 + y z = + y + z = 8. + y + z = 5 y z = + y + 4z = = + = = = = = = =. + = 7 = 7 + = + 4 = 6

5 = 4 = + 4 = = = = + 4 = = Solve the systems of linear algebraic equations with parameters. 48. a + y + z = a y + z = + y + z = + 4y + z = 4 ay + z = y + az = 5. α + y + z = α + + αy + z = + y + αz = Specify the numbers of solutions of the systems of linear algebraic equations in dependence on parameters a and k. Solve the system for given values of the parameters. Hint: Apply Frobenius theorem. 59. a + a + y + z = a y + z = + y + z = [ a = ] 6. + y + z = 5 + y + z = k y z = [ k = 5 ] 69. Find all values of parameter λ for which the system A X = O has a non zero solution and λ, 4, 7 calculate this solution. Matri A has the form: A =, 4, 5, λ, 4 Hint: The system A X = O is therefore equal to zero. has a non zero solution if matri A is singular and its determinant 7. Applying Cramer s rule solve the system + y z = 4 4y z = 8 9z =. III. DIFFERENTIAL CALCULUS III.. Sequences of real numbers Decide if given sequences are increasing, decreasing, non increasing, non decreasing, monotone, strictly monotone, bounded below, bounded above, unbounded. Assume that n =,,,... { 575. } { + n n } { n } n + n + { + n } { } { n n + 5 } 58. n + n + Calculate the limits. 59. lim + n + n 599. lim n + 6. lim n + 5n n + 5n 4n + n 594. lim n + n n + 5 n n lim n + n 4n + n n lim n lim n + n + + n n + n n + n + 4 n + 5 n 5 + n n + n + n +

6 69. lim n + n 6. lim n + n + 5 n + n + 6. lim n nn n 6. lim n n + n n + n + 6. lim 5 + 8n n 69. lim n + n n 9n4 + III.. Functions basic notions and properties Specify domains of given functions. Hint: If the domain is not defined eplicitly then it is automatically the set of all, for which the right hand side makes sense y = 66. y = ln y = arcsin y = ln y = 668. y = ln Given the functions f a f. Write the composite functions g = f f a h = f f f =, f = sin 675. f = ln +, f = f = + 5, f = sin f = cos +, f = + Which of these functions are odd or even? 687. y = + cos 694. y = y = cos + cos Which of these functions are periodic? What is the period? 698. y = sin + cos 74. y = y = cos / Using graphs of elementary functions, sketch the graphs of these functions: 79. y = sin 78. y = arcsin 5 7. y = y = + 7. y = ln 74. y = ln 5 III.. Limit and continuity of a function Given function f and a positive number ɛ. Evaluate the limit L = lim + f and find a real number a such that f U ε L for all a, +. Hint: Use the definition of a limit f = +, ε = f = 5 + e, ε =. Calculate the limits if they eist. 79. lim lim lim tg lim + sin lim + arcsin lim lim + + / 87. lim lim + cos arctg 89. lim 84. lim 86. lim + sin 866. lim lim arctg e 869. lim e 6

7 88. lim e e sin sin Calculate the limits lim + + ln cos 887. lim 89. lim lim ln 99. lim The net three limits do not eist. Give reasons why. 9. lim 94. lim sin 96. lim + Find maimal intervals where given functions are continuous. 99. y = y = y = y = e + / 99. y = ln 94. y = e III.4. Derivative of a function, its geometrical and physical sense Calculate derivatives of given functions. Also specify for which the derivative eists. a Polynomials, rational functions, powers and roots. 96. y = y = y = y = y = y = b Trigonometric functions associated composite functions. 99. y = sin 99. y = cos 99. y = sin y = sin + +. y = tg. y = + + sin c Inverse trigonometric functions and associated composite functions. 8. y = arcsin + 9. y = arccos 6. y = arctg + d Eponential and logarithmic functions and associated composite functions. 7. y = e 8. y = e 5 +. y = e. y = e y = ln y = ln + + e Various further functions. 47. y = ln y = e + 5. y = arcsin + Calculate the derivative of a given function f and sketch the graph of function f and its derivative f. Hint: Consider the formulas = for >, = for <. Function is not does not have a derivative at the point =. 55. f = 56. f = ln 57. f = Calculate f + a f. Hint: If the right limit lim + f eists then the right derivative f + also eists and equals the limit. The same assertion also holds on the left derivative. 58. f =, = 6. f = 4, = 4 7

8 Calculate second derivatives of given functions. For which does the second derivative eist? Hint: f is equal to the derivative of function f. 66. y = y = cotg 7. y = + Write equations of the tangent line and the normal line to the graph of function f at the point [, f ]. Hint: Equation of the tangent line: y y = k, where y = f a k = f. Equation of the normal line: y y = /k. 9. f = e +, =. f = sin, = π 8. At which point does the parabola y = + 4 have a tangent line parallel with the ais? Hint: Find point where the derivative of the function + 4 equals zero. Calculate y from the equation of the parabola. 9. At which point does the parabola y = + 5 have a tangent line perpendicular to the ais of the first quadrant? Hint: The ais of the first quadrant is the straight line y =, whose slope is k =. Find point where the function +5 has the derivative k =. Calculate y from the equation of the parabola. 5. For which Df does the function f = ln + have a tangent line at the point [, f ]? Is some of the tangent lines parallel with the ais? Write the equation of the tangent line at the point [, f ] for = 5. III.5. Application of derivative, behaviour of a function Find intervals, where given function is monotone. Specify if the function is increasing or decreasing. Hint: Use the sign of the first derivative. 44. f = f = f = e 48. f = 5. f = f = ln Determine the domain Df of function f, evaluate one sided limits at the end points of intervals that form Df and find intervals of monotonicity of function f. Specify where f is increasing and where it is decreasing. Sketch the graph of function f. 6. f = e 6. f = / Decide about the eistence of the global etremes maimum and minimum on interval I. If the etremes eist, find their values and positions i.e. find points where function f takes their values. 64. f = 4 + 5, I =, 69. f = 5 4, I =, 74. f = + 6 6, I =, f = 4 4, I =, f = +, I =, 79. f = ln 6 +, I =, 6 Find local and global etremes of given functions on their domains. 4. y = ln 9. y = ln. y = 4 9. y = + 7. y = y = + 8

9 4. Given the function f = + e. Specify its domain, calculate one sided limits at the end points of the domain of f and find the local etremes. On which maimal intervals are given functions concave up and concave down? Find points of inflection. Hint: Use the sign of the second derivative. 55. y = y = y = + 6. y = ln + 6. y = ln + Find asymptotes of given functions if they eist. 64. y = arctg 67. y = y = 7. y = y = + 7. y = + arctg 75. y = + ln Eamine the behaviour of given functions. 77. y = y = 79. y = e 8. y = y = 4 9. y = + e / 9. y = 95. y = + 8. y = e 7. y = arccos + 9. y = ln 4. y = + arccotg III.6. Taylor s theorem Write Taylor s polynomial of degree n of function f at point. Write the formula for the remainder after the n th term.. f = e, n = 5, =. f = e, n = 5, =. f = e, n = 4, = 7. f = sin, n = 7, = 46. f = ln +, n = 7, = 5. f =, n = 4, = 5. f = +, n =, = 54. f =, n = 4, = Calculate approimately with the accuracy ε i.e. with an error less than or equal to ε given function values. Hint: We have to evaluate approimately the function value of f at point which is,,close to another point, where the function value f is known. Find n so large that the remainder R n+ is less than or equal to ε on some interval containing. Then calculate approimately f by means of Taylor s polynomial T n. 6. /e, ε = 6. cos 5 o, ε = 65. ln., ε = 76. Write Taylor s polynomial T of the function f = + at the point =. Using Lagrange s form of the remainder find an upper bound of f T. 9

10 IV. INDEFINITE INTEGRAL IV.. Fundamental properties of indefinite integrals, table of basic integrals Using the table of basic integrals, calculate: d 45. d 45. d 454. d 455. d 458. u du d 46., +,8 5,8 d d 464. d d 47. d 47. cos 474. cos sin d 475. sin d IV.. Integration by parts cos + cos d d Applying the integration by parts, calculate: 48. e d 48. ln d arctg d 485. cos d ; n ln d, n 494. arctg d e sin d 56. e 7 cos 5 d 5. sin d e d arcsin t dt + e d IV.. Integration by substitution Using an appropriate substitution, calculate the integrals: e 54. d 55. d e d d 5. ln 5. cos sin d 54. d d 555. e d d 58. d arcsin sin 65. d 6. d 65. ecos d 666. ln cos cos d 688. cotg d cos sin d cos d + d 4 d + d 5 e d

11 IV.4. Integration of rational functions Calculate the integrals: 8 7. d 7. 4 d 74. Using the decomposition to partial fractions, calculate the integrals: u u du 7. + u d d 748. d d d d d IV.5. Integration of trigonometric functions and their powers d d d + d + 5 d Calculate the integrals: 84. sin 7 d 85. sin d 8. sin cos 5 d 8. sin cos 5 d 88. sin d 8. sin cos d 8. sin 4 + cos 5 d 858. d sin sin d sin 865. d 87. cos cos d 87. cos sin 4 d sin + cos a + b IV.6. Integrals of the type R, S d. c + d Calculate the integrals d d d d d V. DEFINITE RIEMANN S INTEGRAL V.. Basic properties of definite integrals, Newton Leibniz formula Applying the table of basic indefinite integrals and the Newton Leibniz formula, calculate the definite integrals: a d 986. a d d π/4 e cos d 99. d. 8 d 99. π d. 4 sin d d

12 V.. Integration by parts and by substitution in the definite integral Applying integration by parts and integration by substitution, calculate the definite integrals: d. + d 7. + π/ sin t cos t dt. + e ln d 5. d 9. arcsin d 44. π e sin d. + ln d. arctg z + z dz. e d 6. 4 ln d 4. e / d 5 ln + d t 5 + 4t dt ln e d π/ arctg d sin d V.. Improper Riemann s integral Do these integrals converge? If,,yes, find their values. 5. e ln d 5. 4 d d d d ln d d 6. π/ tg d V.4. Some geometrical applications of the definite integral Calculate areas P of curvilinear trapezoids bounded by the ais and the curves with the equations: 67. y =, =, = 68. y = 4, =, = 6 Calculate areas of planar regions bounded by given curves: 69. y =, y = 7. y =, y = 74. Calculate the volume of the circular ellipsoid that arises by the revolution of the ellipse 4 + 9y 6 = a about the ais, b about the y ais. 75. Calculate the volume of the circular body that arises by the revolution of the planar region, bounded by the curves y = 8, y = a about the ais, b about the y ais. Calculate the length of the graph of a given function. Hint: Apply the formula l = b a [f ] + d. 77. y = pro, 8

13 V.5. Further problems Sketch the region bounded by given curves and evaluate its area.. y =, y =. y =, y = /. y = e, y = e, y = e 4. y = +, y = 5. Find coordinates of intersections of the graphs of given functions f and g, resp. f and h, resp. g and h: f = /8, g = 8/, h = 8/. Sketch the picture and calculate the area of the region between the graphs of these three functions. Evaluate the volume of the circular body that arises by rotation of a given curve about the ais. Sketch the region in the, y plane, bounded by the curve and the ais. 6. y = sin,, π/ 7. y =,, 8. y = e,, 9. y =,, π/4 cos. y = sin + cos,, π/. Calculate the volume of the body that arises by rotation of the planar region, bounded by the curves y =, y = / about the ais. Calculate lengths of the graphs of given functions.. y = ln 4 for, 4. y = for, 4 Calculate the mean value of function f on given interval. Hint: Apply the formula b µf = f d. b a a 4. f = sin,, π 5. f = sin,, π

14 RESULTS I.. Vectors, vector spaces. a = 7,, 8. =,,, o 5. α =, 4. LN, ; u, v 4. LZ, ; e.g. u 44. LN, ; u, v 45. LZ, ; e.g. u 5. LZ, ; e.g. 5. LZ, ; e.g., y 5. LN, ;, y, z 68. for all α; dim = for α =, dim = for α 69. k =,, ; dim = 7. a =, ; dim = 7. e.g. a = u + v, the epression is not unique, b cannot be epressed 79. no, 85. yes 9. no 9. yes 94. yes I.. Matrices, determinants, 6,, 5,, 9.., 4, 7..,, 5, 7., 9, 5,, 5,,,, 4, 7 4, 4, 4 6, 4.. 6, 4, 4. 4,, 9. = 4, y = 4. = 4, y =, 5 5, 5, 4, 4, 4,,, 4., regular 48., singular 49., singular 6. 6.,,,.5 9,,,,,, 4,,, does not eist 68., 5, 7.,,,,, 7., 9,, 4, 6, 4,,, 4,,,, 74. 4, 5,.5,.5 5, 9 8, ,.5.5, 4.5 9, 5,,.5,, y, z, y z,.5,,.5.5,,.5 8. cos a a +. a b I.4. Eigenvalues and eigenvectors of square matrices 5. λ =, X = α, λ =, X = β 4 6. λ = 7, X = α, λ =, X = β 5 7. λ = ai, X = α, λ i = ai, X = β i 8. λ =, X = α + β, α, β C, α + β = 4. λ =, X = α β, λ =, X = α γ γ, α, β C, α, β, α, β C, α, β, α, β C, α, β, α, β, γ C, α + β, γ 4

15 4. λ =, X = α, λ =, X = β 44. λ =, X = α 6, α C, α 45. no 46. no, α, β C, α, β p q, X p =, p, q C, p, q q 5. A: eigenvalues λ =, λ =,, eigenvectors X = /, / A =, eigenvalues η = /λ = /,, eigenvectors X /, / η = /λ =,, X λ 5. A: eigenvalues = 7, p 4q eigenvectors X λ =, =, X p = 5q /4, 4/4 A η =, eigenvalues = /λ = /7, eigenvectors X 5/4, /4 η = /λ = /,, X 5. A: eigenvalues, /5 A = /5, p pi λ = i 5, λ = i eigenvectors X 5, =, eigenvalues η = /λ = i/5, η = /λ = i/5,, X =, p, q C, p, q q qi, p, q C, p, q eigenvectors X, X λ 5. A: =, λ, = ± eigenvectors X, = p p, X, = ± q ± q, p, q q /,, / A = η, /, /, eigenvalues = /λ = / η /,, /, = /λ, = / ±,, eigenvectors X, X, 55. A: eigenvalues λ =, p q λ =,, eigenvectors X =, X p =, p, q C, p, q q 5, 4 A =, eigenvalues η = λ = 9, 4, 5 η = λ =,, eigenvectors X, X 56. A: eigenvalues λ =, p λ =,, eigenvectors X =, X p = q, A =, eigenvalues η = λ =, 9, 4 η = λ = 4,, eigenvectors X, X p pi, X = λ 57. A: eigenvalues = i, λ = i,, eigenvectors X = 4, A =, vl. číslo η = λ, 4 = λ = 4, eigenvectors X, X 58. A: eigenvalue λ =, eigenvectors X = p, p C, p,, A = 8, 4, 5, eigenvalue η = λ =, eigenvectors X,, p p, p, q C, p, q q qi, p, q C, p, q I.5. Systems of linear algebraic equations 7. yes 75. yes, 76. yes, 78. a,... solution, a =... solution, a =... solution 8. a... solution, a =... solution, 8. λ 5... solution, λ = 5... solution 87. no 88. yes, =, y =, z = no. for λ =, for λ. for λ =, for λ 8. =, y =, z = 9. =, =, =, 4 = 5

16 . = p, = p, = 7p, p IR 6. =, =, =, 4 = 7. = 5p + q, = q, = p, 4 = 6p, p, q IR 4. solution does not eist 8. =, y =, z =. = p, = p, = 7p, p IR 7. =, y =, z = 5, v = 4 8. = 8, = 6, =, 4 = 48. solution does not eist for a =, = a, y =, z = 4a 7 a for a 49. solution does not eist for a =, = 6p + 4, y = p +, z = p, p IR, for a = = y = z = /a for a, a 5. solution does not eist for α =, = p + 4, y = p, z = p, p IR for α = = α/ α, y =, z = / α for α, α 59. solution does not eist for a = 5, a unique solution for a 5, =, y =, z = for a = 6. solution does not eist for k 5, for k = 5 there is infinitely many solutions: = p +, y = 7p +, z = 5p, p IR 69. λ =... 6p 7p, λ =... q p q, 7. = q 45 6, y = 6, z 4 6 4p q III.. Sequences of real numbers incr. decr. non non mon. strictly lower upper bound. unbound. incr. decr. mon. bound. bound e III.. Functions basic notions and properties 657., 4, + 66., 5 66., ,, ,,, , 5/ + 5/, g = sin, h = sin 675. g = ln 5 +, h = 5 ln g = sin sin +, h = sin g = cos +, h = cos, odd 694. odd 695. even 698. yes, π 74. no 77. yes, π supremum infimum maimum minimum upper lower bound. bound. bound. 79. yes yes yes 78. π/ π/ π/ π/ yes yes yes 7. + does not eist no yes no does not eist no yes no 7. + does not eist does not eist no no no does not eist does not eist no no no 6

17 III.. Limit and continuity of a function 767. L =, e.g. a = 768. L =, e.g. a = ln π/ 865. π/ e the left limit the right limit If we choose e.g. n = π/ + πn then n +, but lim n + sin n does not eist, because sin n = n. 96. the left limit the right limit + 99.,,, + 9.,,, + 9.,,,,, + 97.,,, + 99.,,, + 94.,,, + III.4. Derivative of a function, its geometrical and physical sense ,, ,, ,, ,, 5 5, ,, / ,,, cos,, sin,, sin6 cos6,, cos + + +,, +. tg + cos, π/ + kπ, +π/ + kπ, k integer + cos. + + sin,, +, where is a solution of the equation + + sin = 8. +,, 9.,, ,, + 7. e,, + 8. e 5 +,, +. e/,, +. e 5 + e 5 +,, + 7.,, ,, ,, e e +,, + 5. sgn,,, f = sgn, 56. f = /, 57. f = sgn,, f + =, f = 6. f +4 = 4, f 4 = /,, + cos 68. sin, kπ, k integer 7. 4, 9. y =, =. y = π π, y = π/π 8. [, 4 ] 9. [, 7 4 ] 5. tangent line eists for >, parallel with the ais at the point [, f ], tangent line at the point [ 5, f 5 ] is y ln / 5 = 5/6 5 7

18 III.5. Application of derivative, behaviour of a function 44. f increasing on, and on, +, decreasing on, 45. f increasing on, and on, +, decreasing on,,,,, 46. f increasing on, and on, +, decreasing on, 48. f increasing on,, decreasing on, and on, + 5. f decreasing on, and on, +, increasing on, 56. f increasing on, and on, +, decreasing on, and on, 6. Df =,,, +, lim f =, lim f = + lim + f =, lim f =, lim + f = +, lim + f = + f increasing on,,, and on +, +, decreasing on, and on, + 6. Df =, +, lim + f =, lim + f = f increasing on, e, decreasing on e, ma I f = f = f =, min I f = f = f = ma I f = f =, min I f = f = 74. ma I f = f4 = 4, min I f = f = ma I f = f =, min I f = f = 77. ma I f = f = 4, min I f = f = 79. ma I f does not eist, min I f = f6 = ln 6 4. absolute minimum y = ln at the point = 9. absolute maimum y = /e at the point = e. local minimum y = at the point =, local maimum y = at the point =. local minimum y = at the point =, local maimum y = at the point = 7. absolute minimum y = at the point =, absolute maimum y = at the point = 8. absolute minimum y = at the point = 4. Df =, +, lim f =, lim + f = +, f has no local etremes 55. concave down on, and on,, concave up on, and on, +, points of inflection,, 56. concave down on, and on,, concave up on, and on, +, points of inflection,, 59. concave up on, + 6. concave up on,, concave down on, and on, +, points of inflection ± 6. concave up on,, concave down on, concave up on, slanted asymptote y = for and for +, vertical asymptotes =, = 68. slanted asymptote y = for and for +, vertical asymptote = 7. slanted asymptotes y = π/ for a y = + π/ for + 7. slanted asymptote y = for and for +, vertical asymptote = 74. slanted asymptote y = for and for +, vertical asymptote = 75. slanted asymptote y = for +, vertical asymptote = 77. Df =,,, +, f continuous on,,, and on, +, f even, lim f =, lim f =, lim + f = +, lim + f = +, lim ++ f =, lim + f =, 8

19 f = 4 for Df, f decreasing on, and on,, increasing on, and on, +, f has local minimum y = 4 at the point =, f = for Df, f concave down on, and on, +, concave up on,, asymptote y = for and for + and vertical asymptotes = a =. 78. Df =, +, f continuous on, +, lim f = +, lim + f =, f = for,, +, f decreasing on, and on 8 7, +, increasing on, 8 7, f has local minimum y = at the point = a local maimum y = 4 7 f = 9 4 for,, +, f concave down on, and on, +, f has no asymptotes at the point = 8 7, 79. Df =, +, f continuous on, +, lim f = lim + f =, f = e for Df, f increasing on, and decreasing on, +, f has absolute maimum y = at the point =, f = + 4 e for Df, f concave up on, / and on /, +, concave down on /, /, asymptote y = for and for + 8. Df =, +, f continuous on, +, sudá, lim f = lim + f =, f = for Df, f increasing on, and on,, decreasing on, and on, +, absolute maimum y =.5 at the points = ±, local minimum y = at the point =, the graph intersects the ais at the points ± +, f = 6 for Df, f concave down on, / and on /, +, concave up on /, /, f has no asymptotes 8. Df =, +, f continuous on, +, sudá, lim f = lim + f = +, f = 4 4 for Df, f decreasing on, and on,, increasing on, and on, +, absolute minimum y = at the points = ±, local maimum y = at the point =, the graph intersects the ais at the points ± and touches it at the point =, f = 4 for Df, f concave up on, / and on /, +, concave down on /, /, f has no asymptotes 9. Df =,, +, f continuous on, and on, +, f =, lim f =, lim f = +, lim f = +, lim f / + = e + for,, +, + f increasing on, and on, +, decreasing on, and on,, f has local maimum y = e at the point =, local minimum y = 4 e at the point = f / 5 + = e for,, +, 4 f concave down on, 5, concave up on 5, and on, +, vertical asymptote =, slanted asymptote y = + for and for + 9. Df =, +, f continuous on, +, f =, lim + f = +, 9

20 f = for, +, f increasing on, +, decreasing on,, absolute minimum y = at the point =, f + = 4 for, +, f concave up on, +, f has no asymptotes 95. Df =, +, f continuous on, +, lim f =, lim + f =, f + = + for, +, + f increasing on.5, +, decreasing on,.5, absolute minimum y = 5/ 5 at the point =.5, f = 4 + for, +, + 5/ f concave down on, + 4/8 and on + 4/8, +, concave up on + 4/8, + 4/8, asymptote y = for and y = for + 8. Df =, +, f continuous on, +, lim f = lim + f =, f = e for, +, f increasing on,, decreasing on, +, absolute maimum y = e at the point =, f = e 4 + for, +, f concave down on /, + /, concave up on, /, + /, +, asymptote y = for and for + 7. Df =, +, f continuous on, +, lim f = f = sgn + for,, +, f decreasing on,, increasing on, +, absolute minimum y = at the point =, f 4 sgn = + for,, +, f concave down on,, concave up on, +, asymptote y = π for and for + lim f = π, + 9. Df =,, f continuous on,, sudá, lim f = lim f =, + + f = 4 for,, f increasing on,, decreasing on,, absolute maimum y = ln 4 at the point =, f 8 = 4 for,, f is concave down on,, f has vertical asymptotes = a =. Df =, +, f continuous on, +, lichá, lim f =, lim f = +, f = + + for, +, f increasing on, +, f = + for, +, f concave down on,, concave up on, +, asymptote y = + π/ for a y = for + III.6. Taylor s theorem. T 5 = +! !, R 6 = eξ 6 6!

21 . T 5 = e + e! e 5! 5, R 6 = eξ 6!. T 4 = + +! +! + 4, R 5 = 4! 7. T 7 =! + 5 5! 7 7!, R 8 = sin ξ 8 8! ln 5 5! T 7 = , R 8 = 8 ξ T 4 = , R 5 = 7 56 ξ 9/ 5 5. T = , R = 8 ξ + 7/ 54. T 4 = + + 4, R 5 = 5 /ξ 6 6. e = + +! +! + 4 4! + 5 5! + 6 6! = 65 = 7 = cos 5 o =! = π =π/6 59 = ln. = ln + =. = + =.86 =. 76. T = + 9, f T 5 8 ξ 5 IV.. Table of basic integrals, fundamental properties of indefinite integrals C,, C C C C,, u u + C C 46., + 5,, 6,8 + C,, ln + C , ln, 5 + C 4 + C,, ln C 468. ln + C 474. C cotg tg 475. sin + C tg + + C, k π, k + π, k integer IV.. Integration by parts 48. e + C ln + C,, sin cos + C 484. [ + arctg ] + C 485. sin + cos + C 486. e + + C n n + ln n + + C,, arctg ln + + C 5. t arcsin t + t arcsin t t + C, t, 54. e sin e cos + C e 7 7 cos sin C, choose u = e 7 5. e C 74 IV.. Integration by substitution 54. ln + C 55. ln + e + C 56. ln sin + C, kπ, k + π, k integer C C 5. sin + C 5. C 6 5 cos5 54. ln + C,, C sin 546. ln C [ 555. C 57. e ln + ] + C,,,, +

22 579. C 4 4 ln,,,, [ ln 5 ] + C,,,, + arcsin ln + C,, 6. e cos + C +arctg +C 65. ln + + C,, ln C 666. tg lncos + tg + C, π + kπ, π + kπ, k integer 688. C e IV.4. Integration of rational functions 8 7. ln + C,,,, ln + + C u C,,,, + 7. ln + C u ln 6 ln 7. + C 6 ln 4 ln C,,,, 4, 4, ln C,,,,,, + 8 ln C,, 5 5,,,,, ln + C 75. arctg + ln + ln + C,,,, ln + + C,,,, arctg + C 76. ln + + arctg + C ln + + C,,,,,, ln + arctg + C,,,, + IV.5. Integration of trigonometric functions and their powers 84. cos cos cos5 5 cos6 6 sin + tg/ + 4 arctg5 + cos cos cos + C 85. cos + C +C 8. cos6 sin +C 88. +C sin + C 6 + C, π + kπ, π + kπ, k is an integer tg 864. ln 4 cotg + C, kπ, k + π, k is an integer 865. ln sin + cos + C, π 4 + kπ, π 4 + kπ, k is an integer 87. sin 5 + sin cos 6 cos + C C 4 4 sin 8 +C

23 IV.6. Integrals of the type R a + b, S c + d d arctg 4 + C,, C,, ln C,,,, ln C ln 8 + C,,,, + + V.. Basic properties of definite integrals, Newton Leibniz formula 985. / a 4 / / / /5. 7/6 V.. Integration by parts and by substitution in the definite integral. ln /. π. 5. π/ /6 4. 8/. π /. ln /. ln 4/ 5. π/6 6. π/ 8. / 9. ln /4 4. / 4. π/ 44. e e V.. Improper Riemann integral 5. divergent / 56. /8 57. ln π 6. divergent V.4. Some geometrical application of the definite integral 67. / 68. the sum of areas of the two parts: 76/ 69. / 7. the sum of areas of the two parts: / 74. a V = π 4/99 d = 6π, b V y = π 9/44 y dy = 4π 75. a V = 9, 6π, b V y = 4, 8π V.5. Further problems. /. / ln. 4. π / 5. 8 ln π /4 7. 9π 8. e π/ 9. π. π + π /. 8π/. 6 + ln /4. 8 /