# CALCULUS 2. 0 Repetition. tutorials 2015/ Find limits of the following sequences or prove that they are divergent.

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1 CALCULUS tutorials 5/6 Repetition. Find limits of the following sequences or prove that they are divergent. a n = n( ) n, a n = n 3 7 n 5 n +, a n = ( n n 4n + 7 ), a n = n3 5n + 3 4n 7 3n, 3 ( ) 3n 6n + 5 a n =, 6n + (f) a n = n ( ln(n + n + 3) ln(n + ) ),. Compute the limits or prove that they do not exist. lim x tan x, lim x x x + arctan x x x, lim x arctan x x, tan π x lim x x x ( lim ln x ln(x + x + 3) ), x ( ) (f) lim x cos, x + x (g) lim x ln(4x), x + x 4 x (h) lim, x x (i) lim xsin x. x +.3 Find (if possible) values of parameters a, b R for which the following function is continuous: a + 3 x for x < f(x) = b for x =. ( + x ) x for x >.4 Find the equation of the line tangent to the graph of f at the point with x, if f(x) = 3 cot x and x = π 3 f(x) = x arctan x and x =.

2 .5 Investigate asymptotes, monotonicity and convexity of the function f and sketch its graph. f(x) = x x +, x 3 f(x) = ln x x..6 Use monotonicity of an appropriate function to verify how many solutions does the following equation have: ( x ) 4 arctan = x +..7 Find the smallest perimeter (sum of side lengths) of a rectangle of the given area S >..8 Find the maximal possible volume of a cylinder inscribed in a ball with the given diameter d >..9 Use the second order (i.e. degree) Taylor polynomial to approximate the value 4.4 (as a decimal). Estimate the error of the approximation using the form of the remainder.. Use the third order Taylor polynomial to approximate the value ln(.3) (as a decimal). Estimate the error of the approximation using the form of the remainder.. Verify if the Lagrange Mean Value Theorem can be applied to the functions f(x) = x + ln x 3 for x [, e] and g(x) = 3 x x for x [, ]. If yes, find the point which is said to exist by the Lagrange Theorem.. Is it possible to apply the Rolle s Theorem to the function f(x) = arccos x for x [, ]?.3 Find the following indefinite integrals: dx x 3 ln x, (f) (g) (x ) e x/ dx, e x cos(x) dx, arcsin(x) dx, dx x + 6 x, x 4 + x 3 + 5x + 4x + dx. x 4 + 3x + cos x sin x(5 cos x) dx,

3 (h) (i) (j) (k) sin 3 x 5 cos x dx, sin 4 x dx, + sin x dx 4x + 5 dx, 3 x x (l) x + 6x + dx..4 Find the directional derivative f (P ) of the function f(x, y) = sinh(xy) + x ln y y v at point P = (, e) in the direction of the unit vector v = ( 4, 3 5 5)..5 Use differential to approximate the quantities (.9).3 and ln(.8) + sinh(.) +..6 Find all extreme points & values and saddle points of the following functions f(x, y) = x xy + y x + y + 3, f(x, y) = x 3 + y 6xy 48x, f(x, y, z) = x + y + 9z + xy 3x + 8z 7, f(x, y, z) = x y + yz 4x z +. Ordinary Differential Equations First order ODE s. Solve the following separable equations and initial value problems: y = y cos x y = xy x y and y(3) = 5 xy = y and y( 4) = 6 (x )y + xy = y = x+y (f) x y = cos(y) + and y() = π 4.. Transform the following equations to separable equations (applying appropriate substitutions) and find their general or particular solutions. y = cos(y x) xy = x + y xy y = x tan y x y = y x + e y x (x y )dx + xy dy = 3

4 (f) x + y xyy = and y( e) = e..3 Find general solutions of the nonhomogeneous linear first-order equations and the particular solution of the linear initial value problem. xy y = x 4 y x x + y = + x xy + (x + )y = 3x e x y = x(y x cos x) x y = xy and y( ) = (f) y y tan x = and y() =. cos x.4 Solve the following Bernoulli equations. y = 4y x + x y xy + y = xy ln x. Linear Homogeneous Differential Equations with Constant Coefficients.5 Find a general solution: y + y y = y + y + 5y = y + 6y + 9y = y 4y + 5y y = y (4) + 4y + 4y = (f) y (5) + y y = (g) y 3y + y 3y = (h) y (5) 3y (4) + 3y y =..6 Find the linear homogeneous equation with constant coefficients knowing the basis of the linear space of its solutions. y = 3, y = 3e x, y 3 = e x y =, y = 5x, y 3 = x, y 4 = e x..7 Find the particular solution of the following homogeneous linear initial value problems: y 4y + 3y =, y() = 6, y () = y y =, y() = e, y () = e, y () = e. 4

5 Nonhomogeneous Linear Equations with Constant Coefficients.8 Find general solutions of the following nonhomogeneous linear equations using the Method of Undetermined Coefficients (trial functions): y 3y + y = x 3 y + y y = e x y 7y + 6y = sin x y + y = sin x + cos x y y = 6x + e x (f) y y = x + e x ( + e x ) (g) y + y + y + y = xe x..9 Find general solutions of the following nonhomogeneous linear equations using the Method of Variation of Parameters: y + y = tan x y y + y = e x ln x y + y = e x + y + y = cos x y + y = tan x.. Find particular solutions of the following nonhomogeneous linear initial value problems: y y y = x +, y() =, y () = y 4y = sin(x), y() =, y () = y y = e x (x + x 3), y() =, y () = y 4 + 4y = cos(x), y() = 3, y () =. Definite integral. Calculate the following definite integrals: ( x x + ) dx, + x π 4 e sin 4 x cos x dx, x 3 + x 8 dx, x + 3 ln x dx, arcsin x dx, 5

6 (f) (g) (h) (i) π 4 e x 4 x dx, x cos x dx, ( x x ) sin(nπx) dx, n N, x n ln x dx, n N.. Find the formula of the function F (x) = x Applications of the definite integral.3 Find area of the region bounded by curves: y = x x and y = x, y = and y = x + x, y = x 4 and y = x, y = ln x and y = ln x y = sin x, y = cos x, x = and x = π..4 Find lengths of the given curves. y = ln x for x [ 3, 8 ] y = ln( x ) for x [, ] x(t) = t, y(t) = t t3 3 for t [, ] t e t dt. x(t) = e t cos t, y(t) = e t sin t, z(t) = e t for t [, ]..5 Compute mass of an arc of the curve y = ln x for x [ 3, 8 ] if the density equals ρ(x, y) = x..6 Find mass of an arc of the spiral x(t) = t cos t, y(t) = t sin t, z(t) = 3 (t) 3, t [, ], ( ) if its density equals ρ(x, y, z) = 3 x + y +..7 Find mass of an arc of the logarithmic spiral x(t) = e t cos t, y(t) = e t sin t, z(t) = e t, t [, ], if its density ρ(x, y, z) is inversely proportional to the distance of the point (x, y, z) from the origin (,, )..8 Calculate volume of the solid obtained by revolving around the Ox axis the region between the curve y = x and the Ox axis for x [, 3] 6

7 the region between the curves y = x and y = x. Sketch pictures of the solids..9 Calculate surface of the solid obtained by revolving arout the Ox axis part of the curve y = x for x [, 3] part of the sinusoid y = sin x for x [, π] part of the curve y = x for x [ 4, ].. Investigate monotonicity and convexity of the function f(x) = Improper integrals x ln t dt. t 3. Test convergence of the following improper integrals. Compute these which are convergent. (f) (g) (h) (i) e x dx e x dx 3 x dx arctan x + x dx (x + )e x dx arcsin x x dx x x 3 dx x dx ln x dx. 3 Infinite series 3. Show by definition that the following series converge. Find their sums. n + 3 n 6 n n. n! 7

8 3. Do the following series satisfy the necessary condition for convergence? n + 3n n n 5 n ( ) n n 3 n + 5 4n 3 5 n + n 3. Series with nonnegative terms 3.3 Use the Comparison Test to verify if the following series are convergent or divergent: (f) (g) (h) n n 7 n + n e n n + n + n ( ) n + n 3 + n= sin π n n n sin n n 3 ln n tan n n. 3.4 Apply the Limit Comparison Test to check convergence of the series. 5 n n n n ( + n n ln ) n ( + n ). 8

9 3.5 Determine if the following series converge using the Ratio (d Alembert) Test. n + 3 n! (n)! n!(n + )! arctan n n n n n! n n. 3.6 Determine if the following series converge using the Root (Cauchy) Test. ( ) n n + en + 3 ( ) n n 3n ( arctan n) n ( ) n n 5 n (n + ) n 3 n n n. 3.7 Is any of two test, the Ratio Test or the Root Test, conclusive for the following series? Are they convergent? n + n n + 3 sin n. 3.8 Do the series satisfy assumptions of the Integral Test? If yes, verify their convergence. n=3 n=3 n ln n n e n3 ln n n 9

10 Alternating series 3.9 Are the following series: absolutely convergent, conditionally convergent or divergent? (f) ( ) n n + cos(nπ) 3 n + 3 ( ) n+ sin n n ( ) n tan n cos(πn) arctan n n! ( n) n. 4 Function sequences and series Function sequences 4. Find (pointwise) limits of the following function sequences: f n (x) = x n for x [, ] f n (x) = n sin x for x [, π] n { for x [ ), f n (x) = n for x [, ]. n Function series 4. Find regions of convergence and sums of the following series: x n x n ( ) n x n e nx.

11 4.3 Find regions of convergence of the following series (sin x) n ( ) n n x n + x + ( ) n 4 (arctan x) n. π Power series 4.4 Find centers and radii of convergence of the following power series: ( ) n n (x 3) n n + sinh n x n ( (arctan n) n x + π ) n n n n! (x e)n. 4.5 Find regions of convergence of the following power series: (f) (g) (h) n x n (x ) n n x n n x n n n x 3n n ( x) n n! (n!) (n)! (x + 5)n 9 n (x + ) n

12 (i) (j) (k) (6 3x) n 3 n + n sin n (x 3)n ( 7 n + 7 n )x n. Taylor series 4.6 Find the Maclaurin series of the functions 4.7 Find the Taylor series of f about x, if f(x) = x and x = f(x) = sin x and x = π f(x) = x and x = f(x) = e x and x = f(x) = ln x and x = Knowing that for all x R we have f(x) = sin x, g(x) = cos x, h(x) = ln( + x). e x = find the Maclaurin series of sinh x, cosh(x) and x 7 e x Find the Maclaurin series of the following functions without calculating their derivatives. f(x) = x 3 sin(x) f(x) = sin x f(x) = sinh x f(x) = ln(4 + 9x ) f(x) = x7 x 3 (f) f(x) = x 7 cos(x ). 4. Use an appropriate Taylor series of f to find values of the given derivatives. f(x) = 4+x, f (5) () =? f(x) = e x3, f (33) () =? f(x) = cosh(x ), f (6) () =? f(x) = cos(x 3), f () (3) =? 4. Find sums of the series: π π3 3 3! + π5 5 5! π7 7 7! +. x n n!

13 Fourier series 4. Find the Fourier series of the given function f. Draw sum of its Fourier series and check if it is equal to f on the given set. {, x [ π, ] f(x) = x, x (, π] f(x) = 3x for x [, ] f(x) = x + for x [, ] f(x) = x x for x [ π, π], x ( 3, ) f(x) =, x { 3,, 3}, x (, 3), x (, ) (f) f(x) =, x {,, }, x (, ) (g) f(x) = sin x for x [ π, π] 4.3 Use the Fourier expansion of the function f(x) = π x for x [ π, π] to find ( ) n+ sum of the series. n 4.4 Consider a π-periodic function such that { x f(x) =, x (, π) π, x = Use its Fourier expansion to find sums of the series. n and (n ) 4.5 Find the half-range sine and cosine expansions of the following functions after extending them properly in an odd and even way. Draw the extensions. f(x) = π for x (, π) f(x) = sin x for x (, π) f(x) = 3 x for x (, ) f(x) = x x for x (, ) {, x (, ) f(x) = 3, x (, ). 3

14 5 Multiple integrals Double Integral 5. Calculate the following double integrals. Draw the set D. (x y + ) dx dy, where D = {(x, y) R : x, x y x} (f) D D D D D D xy dx dy, where D = {(x, y) R : y 3, x y} xy dx dy, where D is bounded by the curves y = 9x and y = x + x + y dx dy, where D = { (x, y) R : x + y 4, x y 3x } xy dx dy, where D = {(x, y) R : x + y x, y } y dx dy, where D = {(x, y) R : 4x + y 4, y }. 5. Find area of the region bounded by the given curves. x + y = x, x + y = y and the point (, ) belongs to the region (x + y ) = a xy, where a >. 5.3 Find the moment of inertia with respect to the origin (of the Cartesian system of coordinates) of a homogeneous region bounded by the cardioid r(ϕ) = a( + cos ϕ) for ϕ [, π], where a >. { 5.4 Find the mass of the ring D = (x, y) R : r } x + y R, where < r < R, and d denotes the eu- if the density is given by the function ρ(x, y) = clidean distance on the plane. d ((x,y),(,)) 5.5 Find the center of mass of an isosceles right triangle if the density at every point is proportional to its distance from the hypotenuse. 5.6 Find the mass of that part of the paraboloid z = x + y which lies between planes z = and z = 4, if the density is given by ρ(x, y) = xyz. 5.7 Find the center of mass of the homogeneous -simplex described by x + y + z = for x, y, z. Triple Integral 5.8 Find volumes of solids bounded by the given surfaces. y = x, y =, x + y + z = 4, z = z = 4 x y, z = + x + y 4

15 x + y + z = R, x + y = Rx, where R >. 5.9 Calculate the following triple integrals. Draw the set V. dx dy dz, where V is bounded by the surfaces: x + y + z =, + x + y V x =, y =, z = x + y + z dx dy dz over the ball V = {(x, y, z) R 3 : x + y + z 4} V y dx dy dz over the solid V bounded by the cone y = x + z and V the plane y = h, where h >. x +y dx dy dz, where V = {(x, y, z) R 3 : 4 x + y + z 9, z > }. V 5. Find the mass of the solid bounded by spheres: x + y + z = R and x + y + z = Rz, where R >, if the density at any point is proportional to its distance from the Oxy plane. 5. Find the center of mass of a homogeneous half-globe of radius R >. 5. Find the center of mass of a homogeneous solid bounded by the cone x + y = 3z and the paraboloid x + y = 6z. 5.3 Find the moment of inertia with respect to the axis of symmetry of a cone of base radius R > and height H >. 5.4 Find the mass of a cylinder of base radius R > and height H > if the density at any point is equal to its distance from the axis of symmetry. 5.5 Find the moment of inertia with respect to the Oz axis of the solid V bounded by the surfaces z = x + y and x +y +z =, for z, if the density at the point (x, y, z) equals ρ(x, y, z) = z x + y. 5.6 Find the area of that part of the cone z = x + y which lies inside the cylinder x + y = x. 6 Line integral and the Green s Theorem 6. Compute the line integrals: (x+y)dl, where is the boundary of the triangle with the vertices A = (, ), B = (, ), C = (, ) (x + y )dl, where (t) = a(cos t + t sin t)i + a(sin t t cos t)j for t [, π] and some a > 5

16 x y dl, where is the part of the circle centered at the origin and of radius r > lying in the first quadrant y dl, where an arc of the cycloid x = a(t sin t), y = a( cos t) for t [, π], where a >, (x + y )dl, where is the circle: x + y = ax, a >. 6. Compute the mass of the first coil of the helix (t) = a cos ti + a sin tj + btk, where a, b >. Density at every point is equal to the square of the distance from the origin. 6.3 Compute the following directed line integrals of the given vector fields: x dx y dy, where is a part of parabola 4y = x from the point (, ) (f) to (, ) y dx+x dy, where is the part of the circle x +y = x lying in the first quadrant and oriented clockwise ( x + xy ) dx + ( y xy ) dy, where is the interval AB for A = (, ), B = (, ) (x + y)dx + y dy, where (t) = a(t sin t)i + a( cos t)j for t [, π], where a > y dx + x dy, where is the upper half of the ellipse boundary described by: x = a cos t, y = b sin t, oriented counter-clockwise (y + )dx + y dy + (x + z)dz, where the broken line ABC for A = (,, ), B = (3,, ), C = (3,, ). 6.4 Use the Green s Theorem to compute the integrals ( ) x y dx + x ( + y ) dy, where is the circle centered at the origin and of radius R >, positively oriented ( x + y ) dx + (x + y) dy, where is the boundary of the triangle with the vertices A = (, ), B = (, ), C = (, 3), positively oriented. Draw the curves and mark their orientations. 6

17 6.5 Use the Green s Theorem to compute the integral ( y 4 + e x + x ) dx + ( x y arctan y + y ) dy, where = D is oriented counterclockwise and the region D is bounded by the curves y = x and y = x. Draw the curve and mark its orientation. 6.6 Verify the Green s Theorem for the vector field F(x, y) = i + xyj and positively oriented closed curve consisting of arc of parabola y = x and segment of the line y = x. Draw the curve and mark its orientation. 6.7 Verify the Green s Theorem for the vector field F(x, y) = xi xy j and negatively oriented boundary of the region D = {(x, y) : x } y y. Draw the curve and mark its orientation. Potential field 6.8 Show that F(x, y) = ye x i + (y e x ) j is a potential field, find the potential and compute the work needed to move a unique mass (along an arbitrary curve) from the point A = (, ) to B = (, ). 6.9 Show that F(x, y, z) = xyzi + (x z + z ) j + (x y + yz) k is a potential field, find the potential and compute the work needed to move a unique mass (along an arbitrary curve) from the point A = (,, ) to B = (,, ). choice by Agnieszka Badeńska Literature:. G. B. Thomas, M. D. Weir, J. R. Hass, "Thomas Calculus", Pearson Addison Wesley. R. A. Adams, C. Essex, "Calculus. A complete course", Pearson Addison Wesley 3. J. Marsden, A. Weinstein, "Calculus ", Springer (available on the Springer s website) 4. G. Teschl, "Ordinary Differential Equations and Dynamical Systems", Graduate Studies in Mathematics, American Mathematical Society. 7

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