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


 Lisa Strickland
 2 years ago
 Views:
Transcription
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 firstorder 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 πperiodic 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 halfrange 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 halfglobe 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 counterclockwise (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
Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum
Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic
More informationMATH 2300 review problems for Exam 3 ANSWERS
MATH 300 review problems for Exam 3 ANSWERS. Check whether the following series converge or diverge. In each case, justify your answer by either computing the sum or by by showing which convergence test
More informationSolutions for Review Problems
olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector
More informationPractice Final Math 122 Spring 12 Instructor: Jeff Lang
Practice Final Math Spring Instructor: Jeff Lang. Find the limit of the sequence a n = ln (n 5) ln (3n + 8). A) ln ( ) 3 B) ln C) ln ( ) 3 D) does not exist. Find the limit of the sequence a n = (ln n)6
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More informationDefinition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: f (x) =
Vertical Asymptotes Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: lim f (x) = x a lim f (x) = lim x a lim f (x) = x a
More informationThis makes sense. t 2 1 + 1/t 2 dt = 1. t t 2 + 1dt = 2 du = 1 3 u3/2 u=5
1. (Line integrals Using parametrization. Two types and the flux integral) Formulas: ds = x (t) dt, d x = x (t)dt and d x = T ds since T = x (t)/ x (t). Another one is Nds = T ds ẑ = (dx, dy) ẑ = (dy,
More informationSection 12.6: Directional Derivatives and the Gradient Vector
Section 26: Directional Derivatives and the Gradient Vector Recall that if f is a differentiable function of x and y and z = f(x, y), then the partial derivatives f x (x, y) and f y (x, y) give the rate
More informationSolutions to Vector Calculus Practice Problems
olutions to Vector alculus Practice Problems 1. Let be the region in determined by the inequalities x + y 4 and y x. Evaluate the following integral. sinx + y ) da Answer: The region looks like y y x x
More informationSolutions to Practice Problems for Test 4
olutions to Practice Problems for Test 4 1. Let be the line segmentfrom the point (, 1, 1) to the point (,, 3). Evaluate the line integral y ds. Answer: First, we parametrize the line segment from (, 1,
More informationMATH 132: CALCULUS II SYLLABUS
MATH 32: CALCULUS II SYLLABUS Prerequisites: Successful completion of Math 3 (or its equivalent elsewhere). Math 27 is normally not a sufficient prerequisite for Math 32. Required Text: Calculus: Early
More informationMath 209 Solutions to Assignment 7. x + 2y. 1 x + 2y i + 2. f x = cos(y/z)), f y = x z sin(y/z), f z = xy z 2 sin(y/z).
Math 29 Solutions to Assignment 7. Find the gradient vector field of the following functions: a fx, y lnx + 2y; b fx, y, z x cosy/z. Solution. a f x x + 2y, f 2 y x + 2y. Thus, the gradient vector field
More informationL 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has
The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:
More informationSAT Subject Test Practice Test II: Math Level II Time 60 minutes, 50 Questions
SAT Subject Test Practice Test II: Math Level II Time 60 minutes, 50 Questions All questions in the Math Level 1 and Math Level Tests are multiplechoice questions in which you are asked to choose the
More informationMATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4.
MATH 55 SOLUTIONS TO PRACTICE FINAL EXAM x 2 4.Compute x 2 x 2 5x + 6. When x 2, So x 2 4 x 2 5x + 6 = (x 2)(x + 2) (x 2)(x 3) = x + 2 x 3. x 2 4 x 2 x 2 5x + 6 = 2 + 2 2 3 = 4. x 2 9 2. Compute x + sin
More informationSolutions  Homework sections 17.717.9
olutions  Homework sections 7.77.9 7.7 6. valuate xy d, where is the triangle with vertices (,, ), (,, ), and (,, ). The three points  and therefore the triangle between them  are on the plane x +
More informationMark Howell Gonzaga High School, Washington, D.C.
Be Prepared for the Calculus Exam Mark Howell Gonzaga High School, Washington, D.C. Martha Montgomery Fremont City Schools, Fremont, Ohio Practice exam contributors: Benita Albert Oak Ridge High School,
More information3. Double Integrals 3A. Double Integrals in Rectangular Coordinates
3. Double Integrals 3A. Double Integrals in ectangular Coordinates 3A1 Evaluate each of the following iterated integrals: c) 2 1 1 1 x 2 (6x 2 +2y)dydx b) x 2x 2 ydydx d) π/2 π 1 u (usint+tcosu)dtdu u2
More informationAP Calculus BC. All students enrolling in AP Calculus BC should have successfully completed AP Calculus AB.
AP Calculus BC Course Description: Advanced Placement Calculus BC is primarily concerned with developing the students understanding of the concepts of calculus and providing experiences with its methods
More informationTOPIC 3: CONTINUITY OF FUNCTIONS
TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let
More informationPRACTICE FINAL. Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 10cm.
PRACTICE FINAL Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 1cm. Solution. Let x be the distance between the center of the circle
More informationLearning Objectives for Math 165
Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given
More informationPROBLEM SET. Practice Problems for Exam #1. Math 1352, Fall 2004. Oct. 1, 2004 ANSWERS
PROBLEM SET Practice Problems for Exam # Math 352, Fall 24 Oct., 24 ANSWERS i Problem. vlet R be the region bounded by the curves x = y 2 and y = x. A. Find the volume of the solid generated by revolving
More informationSome Notes on Taylor Polynomials and Taylor Series
Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited
More information7.3 Volumes Calculus
7. VOLUMES Just like in the last section where we found the area of one arbitrary rectangular strip and used an integral to add up the areas of an infinite number of infinitely thin rectangles, we are
More informationMATHEMATICS (CLASSES XI XII)
MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)
More informationMULTIPLE INTEGRALS. h 2 (y) are continuous functions on [c, d] and let f(x, y) be a function defined on R. Then
MULTIPLE INTEGALS 1. ouble Integrals Let be a simple region defined by a x b and g 1 (x) y g 2 (x), where g 1 (x) and g 2 (x) are continuous functions on [a, b] and let f(x, y) be a function defined on.
More information15.1. Vector Analysis. Vector Fields. Objectives. Vector Fields. Vector Fields. Vector Fields. ! Understand the concept of a vector field.
15 Vector Analysis 15.1 Vector Fields Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. Objectives! Understand the concept of a vector field.! Determine
More informationTOPIC 4: DERIVATIVES
TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the
More informationCentroid: The point of intersection of the three medians of a triangle. Centroid
Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:
More information1. Firstorder Ordinary Differential Equations
Advanced Engineering Mathematics 1. Firstorder ODEs 1 1. Firstorder Ordinary Differential Equations 1.1 Basic concept and ideas 1.2 Geometrical meaning of direction fields 1.3 Separable differential
More information1.7 Cylindrical and Spherical Coordinates
56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a twodimensional coordinate system in which the
More informationTaylor and Maclaurin Series
Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions
More informationDerive 5: The Easiest... Just Got Better!
Liverpool John Moores University, 115 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de Technologie Supérieure, Canada Email; mbeaudin@seg.etsmtl.ca 1. Introduction Engineering
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationTechniques of Integration
CHPTER 7 Techniques of Integration 7.. Substitution Integration, unlike differentiation, is more of an artform than a collection of algorithms. Many problems in applied mathematics involve the integration
More informationRAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I  ORDINARY DIFFERENTIAL EQUATIONS PART A
RAJALAKSHMI ENGINEERING COLLEGE MA 26 UNIT I  ORDINARY DIFFERENTIAL EQUATIONS. Solve (D 2 + D 2)y = 0. 2. Solve (D 2 + 6D + 9)y = 0. PART A 3. Solve (D 4 + 4)x = 0 where D = d dt 4. Find Particular Integral:
More informationReview Solutions MAT V1102. 1. (a) If u = 4 x, then du = dx. Hence, substitution implies 1. dx = du = 2 u + C = 2 4 x + C.
Review Solutions MAT V. (a) If u 4 x, then du dx. Hence, substitution implies dx du u + C 4 x + C. 4 x u (b) If u e t + e t, then du (e t e t )dt. Thus, by substitution, we have e t e t dt e t + e t u
More informationDear Accelerated PreCalculus Student:
Dear Accelerated PreCalculus Student: I am very excited that you have decided to take this course in the upcoming school year! This is a fastpaced, collegepreparatory mathematics course that will also
More information1 3 4 = 8i + 20j 13k. x + w. y + w
) Find the point of intersection of the lines x = t +, y = 3t + 4, z = 4t + 5, and x = 6s + 3, y = 5s +, z = 4s + 9, and then find the plane containing these two lines. Solution. Solve the system of equations
More informationGRE Prep: Precalculus
GRE Prep: Precalculus Franklin H.J. Kenter 1 Introduction These are the notes for the Precalculus section for the GRE Prep session held at UCSD in August 2011. These notes are in no way intended to teach
More informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More informationApr 23, 2015. Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23, 2015 sketching 1 / and 19pa
Calculus with Algebra and Trigonometry II Lecture 23 Final Review: Curve sketching and parametric equations Apr 23, 2015 Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23,
More informationCalculus with Analytic Geometry I Exam 10 Take Home part
Calculus with Analytic Geometry I Exam 10 Take Home part Textbook, Section 47, Exercises #22, 30, 32, 38, 48, 56, 70, 76 1 # 22) Find, correct to two decimal places, the coordinates of the point on the
More informationSOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve
SOLUTIONS Problem. Find the critical points of the function f(x, y = 2x 3 3x 2 y 2x 2 3y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Partial derivatives
More information4.2. LINE INTEGRALS 1. 2 2 ; z = t. ; y = sin
4.2. LINE INTEGRALS 1 4.2 Line Integrals MATH 294 FALL 1982 FINAL # 7 294FA82FQ7.tex 4.2.1 Consider the curve given parametrically by x = cos t t ; y = sin 2 2 ; z = t a) Determine the work done by the
More informationSAT Subject Math Level 2 Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses
More informationTwo vectors are equal if they have the same length and direction. They do not
Vectors define vectors Some physical quantities, such as temperature, length, and mass, can be specified by a single number called a scalar. Other physical quantities, such as force and velocity, must
More informationSolutions to old Exam 1 problems
Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections
More informationMathematical Procedures
CHAPTER 6 Mathematical Procedures 168 CHAPTER 6 Mathematical Procedures The multidisciplinary approach to medicine has incorporated a wide variety of mathematical procedures from the fields of physics,
More informationLimit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)
SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) f
More informationCalculus. Contents. Paul Sutcliffe. Office: CM212a.
Calculus Paul Sutcliffe Office: CM212a. www.maths.dur.ac.uk/~dma0pms/calc/calc.html Books One and several variables calculus, Salas, Hille & Etgen. Calculus, Spivak. Mathematical methods in the physical
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationChapter 17. Review. 1. Vector Fields (Section 17.1)
hapter 17 Review 1. Vector Fields (Section 17.1) There isn t much I can say in this section. Most of the material has to do with sketching vector fields. Please provide some explanation to support your
More informationThe Fourth International DERIVETI92/89 Conference Liverpool, U.K., 1215 July 2000. Derive 5: The Easiest... Just Got Better!
The Fourth International DERIVETI9/89 Conference Liverpool, U.K., 5 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de technologie supérieure 00, rue NotreDame Ouest Montréal
More informationx 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1
Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs
More informationGeometry A Solutions. Written by Ante Qu
Geometry A Solutions Written by Ante Qu 1. [3] Three circles, with radii of 1, 1, and, are externally tangent to each other. The minimum possible area of a quadrilateral that contains and is tangent to
More informationLine and surface integrals: Solutions
hapter 5 Line and surface integrals: olutions Example 5.1 Find the work done by the force F(x, y) x 2 i xyj in moving a particle along the curve which runs from (1, ) to (, 1) along the unit circle and
More information4. An isosceles triangle has two sides of length 10 and one of length 12. What is its area?
1 1 2 + 1 3 + 1 5 = 2 The sum of three numbers is 17 The first is 2 times the second The third is 5 more than the second What is the value of the largest of the three numbers? 3 A chemist has 100 cc of
More informationEstimated Pre Calculus Pacing Timeline
Estimated Pre Calculus Pacing Timeline 20102011 School Year The timeframes listed on this calendar are estimates based on a fiftyminute class period. You may need to adjust some of them from time to
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationPractice Problems for Midterm 2
Practice Problems for Midterm () For each of the following, find and sketch the domain, find the range (unless otherwise indicated), and evaluate the function at the given point P : (a) f(x, y) = + 4 y,
More informationVector Calculus Solutions to Sample Final Examination #1
Vector alculus s to Sample Final Examination #1 1. Let f(x, y) e xy sin(x + y). (a) In what direction, starting at (,π/), is f changing the fastest? (b) In what directions starting at (,π/) is f changing
More informationwww.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates
Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c
More information1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives
TRIGONOMETRY Chapter Trigonometry Objectives After studying this chapter you should be able to handle with confidence a wide range of trigonometric identities; be able to express linear combinations of
More informationIn mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data.
MATHEMATICS: THE LEVEL DESCRIPTIONS In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data. Attainment target
More informationBX in ( u, v) basis in two ways. On the one hand, AN = u+
1. Let f(x) = 1 x +1. Find f (6) () (the value of the sixth derivative of the function f(x) at zero). Answer: 7. We expand the given function into a Taylor series at the point x = : f(x) = 1 x + x 4 x
More informationStudent Performance Q&A:
Student Performance Q&A: AP Calculus AB and Calculus BC FreeResponse Questions The following comments on the freeresponse questions for AP Calculus AB and Calculus BC were written by the Chief Reader,
More informationDifferentiation and Integration
This material is a supplement to Appendix G of Stewart. You should read the appendix, except the last section on complex exponentials, before this material. Differentiation and Integration Suppose we have
More informationAlgebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123
Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from
More information2008 AP Calculus AB Multiple Choice Exam
008 AP Multiple Choice Eam Name 008 AP Calculus AB Multiple Choice Eam Section No Calculator Active AP Calculus 008 Multiple Choice 008 AP Calculus AB Multiple Choice Eam Section Calculator Active AP Calculus
More information(a) We have x = 3 + 2t, y = 2 t, z = 6 so solving for t we get the symmetric equations. x 3 2. = 2 y, z = 6. t 2 2t + 1 = 0,
Name: Solutions to Practice Final. Consider the line r(t) = 3 + t, t, 6. (a) Find symmetric equations for this line. (b) Find the point where the first line r(t) intersects the surface z = x + y. (a) We
More informationFundamental Theorems of Vector Calculus
Fundamental Theorems of Vector Calculus We have studied the techniques for evaluating integrals over curves and surfaces. In the case of integrating over an interval on the real line, we were able to use
More informationAPPLIED MATHEMATICS ADVANCED LEVEL
APPLIED MATHEMATICS ADVANCED LEVEL INTRODUCTION This syllabus serves to examine candidates knowledge and skills in introductory mathematical and statistical methods, and their applications. For applications
More informationFunctions and Equations
Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c
More informationTaylor Polynomials and Taylor Series Math 126
Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will
More informationSolutions to Exercises, Section 5.1
Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle
More informationTrigonometric Functions: The Unit Circle
Trigonometric Functions: The Unit Circle This chapter deals with the subject of trigonometry, which likely had its origins in the study of distances and angles by the ancient Greeks. The word trigonometry
More informationSection 10.7 Parametric Equations
299 Section 10.7 Parametric Equations Objective 1: Defining and Graphing Parametric Equations. Recall when we defined the x (rcos(θ), rsin(θ)) and ycoordinates on a circle of radius r as a function of
More information4B. Line Integrals in the Plane
4. Line Integrals in the Plane 4A. Plane Vector Fields 4A1 Describe geometrically how the vector fields determined by each of the following vector functions looks. Tell for each what the largest region
More informationHOMEWORK 4 SOLUTIONS. All questions are from Vector Calculus, by Marsden and Tromba
HOMEWORK SOLUTIONS All questions are from Vector Calculus, by Marsden and Tromba Question :..6 Let w = f(x, y) be a function of two variables, and let x = u + v, y = u v. Show that Solution. By the chain
More informationFind the length of the arc on a circle of radius r intercepted by a central angle θ. Round to two decimal places.
SECTION.1 Simplify. 1. 7π π. 5π 6 + π Find the measure of the angle in degrees between the hour hand and the minute hand of a clock at the time shown. Measure the angle in the clockwise direction.. 1:0.
More informationIntroduction. The Aims & Objectives of the Mathematical Portion of the IBA Entry Test
Introduction The career world is competitive. The competition and the opportunities in the career world become a serious problem for students if they do not do well in Mathematics, because then they are
More informationAB2.5: Surfaces and Surface Integrals. Divergence Theorem of Gauss
AB2.5: urfaces and urface Integrals. Divergence heorem of Gauss epresentations of surfaces or epresentation of a surface as projections on the xy and xzplanes, etc. are For example, z = f(x, y), x =
More informationCalculus 1: Sample Questions, Final Exam, Solutions
Calculus : Sample Questions, Final Exam, Solutions. Short answer. Put your answer in the blank. NO PARTIAL CREDIT! (a) (b) (c) (d) (e) e 3 e Evaluate dx. Your answer should be in the x form of an integer.
More informationSection 2.6 Cylindrical and Spherical Coordinates A) Review on the Polar Coordinates
Section.6 Cylindrical and Spherical Coordinates A) Review on the Polar Coordinates The polar coordinate system consists of the origin O,the rotating ray or half line from O with unit tick. A point P in
More informationDifferentiation of vectors
Chapter 4 Differentiation of vectors 4.1 Vectorvalued functions In the previous chapters we have considered real functions of several (usually two) variables f : D R, where D is a subset of R n, where
More information4 More Applications of Definite Integrals: Volumes, arclength and other matters
4 More Applications of Definite Integrals: Volumes, arclength and other matters Volumes of surfaces of revolution 4. Find the volume of a cone whose height h is equal to its base radius r, by using the
More informationx(x + 5) x 2 25 (x + 5)(x 5) = x 6(x 4) x ( x 4) + 3
CORE 4 Summary Notes Rational Expressions Factorise all expressions where possible Cancel any factors common to the numerator and denominator x + 5x x(x + 5) x 5 (x + 5)(x 5) x x 5 To add or subtract 
More informationIntroduction to Calculus
Introduction to Calculus Contents 1 Introduction to Calculus 3 11 Introduction 3 111 Origin of Calculus 3 112 The Two Branches of Calculus 4 12 Secant and Tangent Lines 5 13 Limits 10 14 The Derivative
More informationSequences and Series
Sequences and Series Consider the following sum: 2 + 4 + 8 + 6 + + 2 i + The dots at the end indicate that the sum goes on forever. Does this make sense? Can we assign a numerical value to an infinite
More informationModule 1 : A Crash Course in Vectors Lecture 2 : Coordinate Systems
Module 1 : A Crash Course in Vectors Lecture 2 : Coordinate Systems Objectives In this lecture you will learn the following Define different coordinate systems like spherical polar and cylindrical coordinates
More informationReview for Calculus Rational Functions, Logarithms & Exponentials
Definition and Domain of Rational Functions A rational function is defined as the quotient of two polynomial functions. F(x) = P(x) / Q(x) The domain of F is the set of all real numbers except those for
More informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
More informationMaximum / Minimum Problems
171 CHAPTER 6 Maximum / Minimum Problems Methods for solving practical maximum or minimum problems will be examined by examples. Example Question: The material for the square base of a rectangular box
More information( 1)2 + 2 2 + 2 2 = 9 = 3 We would like to make the length 6. The only vectors in the same direction as v are those
1.(6pts) Which of the following vectors has the same direction as v 1,, but has length 6? (a), 4, 4 (b),, (c) 4,, 4 (d), 4, 4 (e) 0, 6, 0 The length of v is given by ( 1) + + 9 3 We would like to make
More information13. Write the decimal approximation of 9,000,001 9,000,000, rounded to three significant
æ If 3 + 4 = x, then x = 2 gold bar is a rectangular solid measuring 2 3 4 It is melted down, and three equal cubes are constructed from this gold What is the length of a side of each cube? 3 What is the
More informationInverse Circular Function and Trigonometric Equation
Inverse Circular Function and Trigonometric Equation 1 2 Caution The 1 in f 1 is not an exponent. 3 Inverse Sine Function 4 Inverse Cosine Function 5 Inverse Tangent Function 6 Domain and Range of Inverse
More informationcorrectchoice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:
Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that
More information