Notes of Calculus II (MTH 133) 2013 Summer. Hongli Gao

Transcription

1 Notes of Calculus II (MTH 133) 2013 Summer Hongli Gao June 16, 2013

2 Chapter 6 Some Applications of the Integral 6.2 Volume by parallel cross-sections; Disks And Washers Definition 1. A cross-section of a solid S is the plane region formed by intersecting S with a plane. How to find the volumes integrating cross-sections? 1. The general case V = b a A(x)dx. (Difficulty: find a simple typical cross-section.) 2. Regions of revolution V = V = b a b a π[r(x)] 2 dx. (about x-axis or y = c) π[r(y)] 2 dy. (about y-axis or x = c) 3. Certain regions with holes V = V = b a b a π ( [R(x)] 2 [r(x)] 2) dx. (about x-axis or y = c) π ( [R(y)] 2 [r(y)] 2) dy. (about y-axis or x = c) Example 1 A pyramid 3 m high has a square base that is 3 m on a side. The cross-section of the pyramid perpendicular to the altitude x m down from the vertex is a square x m on a side. Find the volume of the pyramid. 1

3 Example 2 A curved wedge is cut from a circular cylinder of radius 3 by two planes. One plane is perpendicular to the axis of the cylinder. The second plane crosses the first plane at a 45 angel at the center of the cylinder. Find the volume of the wedge. Example 3 The region between the curve y = x,0 x 4, and the x-axis is revolved about the x-axis to generate a solid. Find its volume. Example 4 Find the volume of the solid generated by revolving the region bounded by y = x and the lines y = 1, x = 4 about the line y = 1. 2

4 Example 5 Find the volume of the solid generated by revolving the region between the y axis and the curve x = 2/y, 1 y 4, about the y axis. Example 6 Find the volume of the solid generated by revolving the region between the parabola x = y 2 +1 and the line x = 3 about the line x = 3. Example 7 The region bounded by the curve y = x 2 +1 and the line y = x+3 is revolved about the x axis to generate a solid. Find the volume of the solid. 3

5 Example 8 The region bounded by the parabola y = x 2 and the line y = 2x in the first quadrant is revolved about the y axis to generate a solid. Find the volume of the solid. 4

6 6.5 The Notion of Work The work done by a variable force F(x) in the direction of motion along the x axis from x = a to x = b is W = b a F(x)dx, Particularly, if the force is a constant, then W = Fd, where d is the distance. CASE 1 Spring Problem F = kx (Hooker s Law) x is the stretched or compressed length, k is the force constant, measured in force units per unit length. So the work is W = d where d is the stretched or compressed distance. CASE 2 Rope Problem 0 W = kxdx = 1 2 kd2, c 0 kxdx where k is weight density, measured in force units per unit length, c is the length of the rope. CASE 3 Tank Problem W = h1 0 ka(y)(h y)dy where k is weight density, measured in force units per unit volume, A(y) is the horizontal cross-section area, h 1 is the initial height of the liquid in the tank, h is the lifting distance. Example 1 Find the work required to compress a spring from its natural length of 1 ft to a length of 0.75 ft if the force constant is k = 16 lb/ft. 5

7 Example 2 A spring has a natural length of 1 m. A force of 24 N holds the spring stretched to a total length of 1.8 m. (a) Find the force constant k. (b) How much work will it take to stretch the spring 2 m beyond its natural length? (c) How far will a 45-N force stretch the spring? Example 3 A 5-lb bucket is lifted from the ground into the air by pulling in 20 ft of rope at a constant speed. the rope weighs 0.08 lb/ft. How much work was spent lifting the bucket and rope? Example 4Averticalrightcircularcylindricaltankhasheighth = 15fthighanddiameter d = 8 ft. It is full of kerosene weighing 51.2 lb per cubic ft. How much work does it to pump all of the kerosene from the tank to an outlet which is at the level of the top of the tank? 6

8 Example 5 To design the interior surface of a huge stainless-steel tank, you revolve the curve y = 3x 2, 0 x 3, about the y axis. The container, with dimensions in meters, is to be filled with seawater, which weighs 9000 N/m 3. How much work does it take to empty the tank by pumping the water to the tank s top? 7

9 Chapter 7 The Transcendental Functions 7.1 One-to One Functions; Inverses Definition 1 A function f(x) is one-to-oneon a domain D if f(x 1 ) f(x 2 ) whenever x 1 x 2 in D. How to test if a function is one-to-one from graph? Horizontal Test: A function y = f(x) is one-to-one if and only if its graph intersects each horizontal line at most once. Example 1 (a) (b) (c) (d) (e) (f) Definition 2 Suppose that f is a one-to-one function on a domain D with range R. The inverse function f 1 is defined by f 1 (b) = a if f(a) = b. The domain of f 1 is R and the range of f 1 is D. 8

10 Finding Inverses Example 2 (From a graph) y x Example 3 (From a formula) Find the inverse of y = 1 x+1, expressed as a function of x. 2 Example 4 (From a formula) Given the function f(x) = 2(x 3) 2 +4, for x 3, find f 1 (x). 9

11 The Derivative Rule for Inverses, In Leibniz notation, If f(a) = b, (f 1 ) (x) = 1 f (f 1 (x)) dx dy = 1 dy/dx (f 1 ) (b) = 1 f (a) = 1 f (f 1 (b)) Example 4 Let f(x) = x 3 2. Find the value of df 1 /dx at x = 6 = f(2) without finding a formula for f 1 (x). Example 5 Suppose that the differentiable function y = g(x) is invertible and that the graph of g passes through the origin with slope 13. Find the slope of the graph of g 1 at the origin. Find the slope of the graph of g 1 at the origin. 10

12 7.2 & 7.3 The Logarithm Function Definition 1 The natural logarithm is the function ln(x) = x 1 dt t, x (0, ) In particular, ln(1) = 0, ln(e) = 1 The derivative: ln (x) = 1 x ; and ln (u) = u u (u is a function) Integrals involving logarithms: dx x = ln( x ) +c; and f (x) f(x) = ln( f(x) ) +c (f(x) is a function) Properties: (I) ln(ab) = ln(a) + ln(b), (product rule); (II) ln( a ) = ln(a) ln(b), (quotient rule); b (III) ln( 1 ) = ln(a), (reciprocal rule); a (IV) ln(a b ) = bln(a), (power rule) Some integrals tanxdx = ln cosx +c = ln secx +C cotxdx = ln sinx +c secxdx = ln secx+tanx +c cscxdx = ln cscx cotx +c 11

13 Example 1 Find the derivative of (1) y(x) = ln(3x) and z(x) = ln(2x 2 +cos(x)) Example 2 [ (x+1) 2] Compute the derivative of y(x) = ln 3(x+2) Example 3 (1) tan(x)dx (2) cot(x)dx 12

14 Example 4 (1) y(t) = 3sin(t) dt, (2) y(x) = (2+cos(t)) x 2 lnx dx, (2) y(x) = 1 4x3 x dx. Example 5 Find the derivative of y(x) = x3 (x+2) 2 cos 3 (x) by logarithmic differentiation. Example 6 d dx log 2(3x 2 +1) 13

15 7.4 & 7.5 The Exponential Function Part I. The exponential function e x Definition 1 For every x R we denote e x = ln 1 (x) = exp(x) Example 1 Find the solution of e 3x+1 = 2 Definition 2 The exponential function, exp : R (0, ) is the inverse of the natural logarithm,that is, exp(x) = y x = ln(y) In particular, exp(0) = 1, exp(1) = e The derivative: (e x ) = e x ; and Integral: (e u ) = e u u (u is a function) e ax dx = eax a +c; Properties: (I) e a+b = e a e b ; (II) e a = 1 e a; (III) e a b = ea e b; (IV) ( e a)q = e qa. (V) lne x = x, e lnx = x 14

16 Example 2 Find the derivative of y(x) = e sin(3x2 )ln(x 2 +1) Example 3 Find y for y(x) = e 3x2 +5 Example 4 Evaluate the integrals: e 4/x e 3x (1) dx (2) dx (3) 6x2 1+e3x 2ln3 0 xe x2 /2 dx 15

17 Example 5 Find I = π/4 0 e 3sin(2x) cos(2x)dx Example 6 Find the solution to the initial value problem y (x) = 18e 3x, y(0) = 1, y (0) = 2 Example 7 Simplify the expression ( e x ln(2) e ) 3 16

18 Part II. Other bases: a x Properties: (I) a c+b = a c a b ; (II) a b = 1 a b; (III) a c b = ac a b; (IV) ( a b)q = a qb. The derivative: (a x ) = ln(a)a x ; and (a u ) = ln(a)a u u (u is a function) Integral: a x dx = ax ln(a) +c; Example 1 Find the following integrals: (1) 4 x dx ( 1) sin(x)cos(x)dx (2) I = 7 (3) x5 x2 dx 17

19 Example 2 Differentiate the following functions: (1) y = x 7 (2) y = 7 x (3)y(x) = 3 cos(3x) (4) y(x) = (3sinx) x (5) y = (x+14) x 18

20 7.6 Solving differential equations & Exponential Growth and Decay Part I: Separable Equations Definition 1 Suppose we have a general form of differential equation dy dx = f(x,y), where f is a function of both x and y. A solution of the differential equation is a differentiable function y = y(x) such that d y(x) = f(x,y(x)). dx Remark: Differential equations have infinitely many solutions. Definition 2 A differential equation is separable if f(x,y) can be expressed as a product or quotient of a function of x and a function of y, i.e. dy dx = g(x)h(y) or dy dx = g(x) h(y). How to solve separable equations? Step 1 Rewrite the equation. (Separate terms with x and y, respectively.) Step 2 Integral both sides. Step 3 Determine the constant c if the initial value is given. Example 1 Verify that the functions y(x) = ce 2x 3, for every c R, are solutions to the 2 differential equation y = 2y +3 Example 2 Determine each equation is separable or not. (1) dy dx = cos(x) y 2 (2) dy dx = ex (1+y) (3) dy dx = x+y (4) 3(x+1)y dy dx = 2(1+y2 ) 19

21 Example 3 Find the function y = y(x) which satisfies the initial value problem dy dx = sinx 4y y(0) = 2 Example 4 Solve for y = y(x) for 9x 12y x 2 +1 dy = 0, y(0) = 5. dx Example 5 Solve for y = y(x) for 10x 2 + dy dx = 5x2 y, y(0) = 8. 20

22 Part II: Application We will discuss three cases of application problems: Population growth (can also be applied to interest problem), radioactive decay, Newton s Law of Cooling. Population Growth The problem can be modeled by the differential equation dy dt = ky, y(0) = y 0 where y, a function of time t, gives the number of population at time t. We can solve this initial problem y(t) = y 0 e kt Radioactive Decay The problem can be described by the differential equation dy dt = ky, y(0) = y 0 where y, a function of time t, gives the amount of some radioactive substance at time t. We can solve this initial problem y(t) = y 0 e kt The most common problem of this case is called half-life problem. Newton s Law of Cooling The temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. The temperature T(t) at time t is given by T(t) T s = (T 0 T s )e kt where T 0 is the initial temperature, T s is the temperature of the surrounding medium. The key point of the application questions: Find the constant k!!!!!!!! 21

23 Example 1 You have just placed d dollars in a bank account that pays 5 percent(annual) interest, compounded continuously. (a) How much do you have in the account after 8 years? (b) How long (in years) will it take your money to double? (c) How long (in years) will it take your money to triple? Example 2 Suppose the half-life of carbon-14 is 5700 years. The charcoal from a tree killed in the volcanic eruption that formed Crater Lake in Oregon Contained 37.4 percent of the carbon-14 found in living matter. About how old is Crater Lake? Example 3 Suppose that the temperature of a cup of soup obeys Newton s law of cooling. If the soup has a temperature of 175 F when freshly served, and 6 minutes later has cooled to 165 F in a room at 64 F, how much longer must it take the soup to reach a temperature of 115 F? 22

24 7.7 The Inverse Trigonometric Functions Domain restrictions that make the trigonometric functions one-to-one and graphs of the six basic inverse trigonometric functions. 23

25 Definition y = sin 1 x is the number in [ π/2,π/2] for whichsiny = x y = cos 1 x is the number in [0,π] for whichcosy = x y = tan 1 x is the number in ( π/2,π/2) for whichtany = x y = cot 1 x is the number in (0,π) for whichcoty = x And we also have ( 1 ) ( 1 ) sec 1 x = cos 1 and csc 1 x = sin 1 x x From the definition, we create the following table of common values for arcsin, arccos and arctan The derivative of inverse trigonometric functions d 1 dx sin 1 (x) =, d 1 1 x 2 dx cos 1 (x) = 1 x 2 d dx tan 1 (x) = 1 d 1+x 2, dx cot 1 (x) = 1 1+x 2 d 1 d dx sec 1 (x) = 1 x x 2 1, dx csc 1 (x) = x x 2 1 The Integrals dx ( x ) a2 x = 2 sin 1 +c a dx a 2 +x = 1 ( x ) 2 a tan 1 +c a dx x x 2 a = 1 ( x ) 2 a sec 1 +c a 24

26 Example 1 Evaluate the following. sin 1 ( 1 2 ) = 3 ) sin ( 1 = 2 tan 1 ( 1) = tan 1 ( 3 ) = sec 1 ( 2 3 ) = sec 1 ( 2) = ( tan (sin 1 3 )) = 2 Example 2 Given that x = sec 1 ( 5 5), find sin(x), cos(x), tan(x), cot(x), csc(x). 25

27 Example 3 Compute the derivative of (1) y(x) = sec 1 (3x+7) (2) y(x) = tan 1 (4lnx) (3) y(x) = tan(arcsin(x)) Example 4 Evaluate the following integrals. 5 (1) dx 1 9(x 1) 2 26

28 (2) e t(1+ln 2 t) dt (3) t t 2 dt 27

29 7.8 & 7.9 The Hyperbolic functions Definition 1 The hyperbolic sine and hyperbolic cosine functions are defined by the equations And we also have sinhx = ex e x tanhx = sinhx coshx = ex e x e x +e x 2 and coshx = ex +e x and cothx = coshx sinhx = ex +e x e x e x sechx = 1 coshx = 2 e x +e x and cschx = 1 sinhx = 2 e x e x 2 Identity for hyperbolic functions (the most common one) cosh 2 x sinh 2 x = 1 Derivatives of hyperbolic functions d d (sinhx) = coshx dx d dx (tanhx) = 1 cosh 2 x d sinhx (sechx) = dx cosh 2 x (coshx) = sinhx dx d dx (cothx) = 1 sinh 2 x d (cschx) = coshx dx sinh 2 x Integral formulas for hyperbolic functions sinhudu = coshu+c coshudu = sinhu+c sech 2 udu = tanhu+c csch 2 udu = cothu+c sech(u)tanhudu = sech(u)+c csch(u)cothudu = csch(u)+c 28

30 Example 1 Simplify as much as possible. cosh(ln 9) Example 2 If sinh(x) = 3, then tanh(x) =? Example 3 Find the derivative of f(x) = sinh 3 (1/x) ( )) Example 4 Find the derivative of f(x) = sinh cosh (x 4 Example 5 Evaluate tanh 7xdx Example 6 Evaluate π/8 π/8 8cosh(tan2t)sec 2 2tdt 29

31 Chapter Integration Techniques Example 1 Techniques of Integration x 2 2x+26 dx Example 2 x 2 64+x 2dx 30

32 Example 3 6t 2 4 3t+10 dt Example 4 1 7t 20t 2 +5 dt 31

33 8.2 Integration by parts Integration by parts formula f(x)g (x)dx = f(x)g(x) f (x)g(x)dx It is common to write in differential form. udv = uv vdu Integration by parts formula for definite integrals b a f(x)g (x)dx = f(x)g(x) b a b a f (x)g(x)dx Example 1 xcos3xdx 32

34 Example 2 3x ln(3x)dx Example 3 x 3 e 7x dx 33

35 Example 4 π 2 0 x 2 sin(2x)dx 34

36 Example 5 e 3y cos6ydy 35

37 8.3 Trigonometric Integrals We begin with integrals of the form: where m and n are nonnegative integers. sin m xcos n xdx Case 1 If m is odd, we write m as 2k +1 and use the identity sin 2 x = 1 cos 2 x to obtain sin m x = sin 2k+1 x = (sin 2 x) k sinx = (1 cos 2 x) k sinx. Then we combine the single sinx with dx in the integral and set sinxdx equal to d(cosx). Case 2 If m is even and n is odd in sin m xcos n xdx, we write n as 2k + 1 and use the identity cos 2 x = 1 sin 2 x to obtain cos n x = cos 2k+1 x = (cos 2 x) k cosx = (1 sin 2 x) k cosx. Then we combine the single cosx with dx in the integral and set cosxdx equal to d(sinx). Case 3 If both m and n are even in sin m xcos n xdx, we substitute sin 2 x = 1 cos2x, cos 2 x = 1+cos2x 2 2 to reduce the integrand to one in lower powers of cos2x. Example 1 π/2 0 7sin 5 (7x)dx 36

38 Example 2 π/3 0 sin 3 (9x)cos 9 (9x)dx Example 3 π/6 0 sin 2 (3x)cos 2 (3x)dx 37

39 Eliminating square roots Example 4 π/8 0 1 cos4x dx 2 Example 5 π/2 π/3 5sin 2 x 1 cosx dx 38

40 Integrals of Powers of tanx and secx Example 6 π/16 π/16 2tan 4 (4x)dx Example 7 5sec 3 (3x)dx 39

41 8.4 Trigonometric Substitutions Case 1 Integrals involving a 2 +x 2, we let x = atanθ, then a 2 +x 2 = a 2 +a 2 tan 2 θ = a 2 (1+tan 2 θ) = a 2 sec 2 θ. Case 2 Integrals involving a 2 x 2, we let x = asinθ, then a 2 x 2 = a 2 a 2 sin 2 θ = a 2 (1 sin 2 θ) = a 2 cos 2 θ. Case 3 Integrals involving x 2 a 2, we let x = asecθ, then x 2 a 2 = a 2 sec 2 θ a 2 = a 2 (sec 2 θ 1) = a 2 tan 2 θ Example 1 81 x2 dx, x 9, 40

42 Example 2 dx 16x2 81, x > 9 4, Example 3 x 3 dx x

43 Example x 2 (64 x 2 ) 3/2dx Example 5 7dx (4x 2 +1) 2 42

44 8.5 Integration of Rational Function by Partial Fractions f(x)/g(x) I. the degree of f(x) < g(x) Example 1 5x 3 x 2 2x 3 dx Example 2 1 x 3 x 2 6x dx 43

45 For the partial fractions f(x)/g(x), if x r is a linear factor of g(x). Suppose that (x r) m is the highest power of x r that divides g(x). Then to this factor, assign the sum of the m partial fractions: Example 3 A 1 (x r) + A 2 (x r) A m 2 (x r) m 1 (x 2 1) dx 2 Example 4 6x+6 (x 2 +1)(x 1) 3 dx 44

46 II. the degree of f(x) the degree of g(x) Example 5 2x 3 4x 2 +1 dx x 2 2x Example 6 x x 2 +6x+8 dx 45

47 Chapter 11 Sequences; Indeterminate forms; Improper integrals 11.2 & 11.3 Sequences Definition 1 An infinite sequences of numbers is an ordered set of real numbers. A sequence is called convergent iff it has a limit, otherwise it is called divergent. Remark: A sequence is denoted as For example, {a 1,a 2,...,a n,...} or {a n } n=1 or a n = f(n) { n }, a n = n { 1 n+1 n=1 n+1, 2, 2 3,..., n } n+1,... {( 1) n n} n=3, a n = ( 1) n n, { 3, 4, 5,...} { {cos(nπ/6)} n=0, a n = cos(nπ/6), 1, } 3, 1,0, How to find the sequence limit? (1) Properties of sequence limits If the sequence {a n } A and {b n } B, then (a) lim{a n +b n } = A+B; n (b) lim{a n b n } = A B; n (c) lim{ka n } = ka; n (d) lim{a n b n } = AB; n { an } (b) If B 0, thenlim = A n b n B. (2) The Sandwich Theorem for sequences: If the sequence {a n }, {b n } and {c n } satisfy and if both a n L and c n L, then b n L. a n b n c n, for n > N, (3) The Continuous Function Theorem for sequences: If a sequence {a n } L and a continuous function f is defined both at L and every a n, then the sequence { f(a n ) } f(l). (4) L Hôpital s Rule 46

48 Example 1 Write a formula for the nth term of this sequence: {6, 6,6, 6,6, 6,...} Example 2 Write a formula for the nth term of this sequence: { } 1, 1, 1, 1, 1, Example 3 Find the limit of the sequence diverge? { a n = 3 3n4 3n 4 +7n 3 }. Does this sequence converge or Example 4 Find the limit of the sequence diverge? { a n = 7n2 2n+1 4n 3 }. Does this sequence converge or Example 5 Find the limit of the sequence {a n = 4+( 1) n 2}. Does this sequence converge or diverge? Example 6 Find the limit of the sequence converge or diverge? { a n = ( n )( n )}. Does this sequence Example 7 Find the limit of the sequence diverge? { a n = 3sin(2n) 2n }. Does this sequence converge or Example 8 Find the limit of the sequence diverge? { ( 5 ) 1 } 3n a n =. Does this sequence converge or 2n 47

49 11.4, 11.5 & 11.6 Limits using L Hôpital s Rule L Hôpital s Rule Suppose that f(a) = g(a) = 0, that f and g are differentiable on an open interval I containing a, and that g (x) 0 on I if x a. Then f(x) lim x a g(x) = lim f (x) x a g (x), assuming that the limit on the right side of this equation exists. Remark 1: To find f(x) lim x a g(x) by L Hôpital s Rule, continue to differentiate f and g,so long as we still get the form 0/0 at x = a. But as soon as one or the other of these derivative is different from zero at x = a we stop differentiating. L Hôpital s Rule does not apply when either the numerator or denominator has a finite nonzero limit. Remark 2: L Hôpital s Rule applies to the limits of the form 0/0, /, 0,, 1, 0 0, 0. Example 1 Let α be any positive number. Evaluate the limit: lim function grows fastest? Which function grows slowest, as x? x lnx and lim xα x e x x α Which 48

50 sinx Example 2 Evaluate the limit: lim. Does the function sinx grow faster, at the same x 0 x rate, or slower than the function x as x? Example 3 Evaluate the limit: lim x 3 x x 5. 3x+1 Example 4 Evaluate the limit: lim. x 2x+1 6 x Example 5 Evaluate the limit: lim x ln 2 x. 49

51 Example 6 Evaluate the limit: lim(5x) 1 4x. x ( Example 7 Evaluate the limit: lim x. x x) 50

52 11.7 Improper Integrals Definition 1 Integrals with infinite limits of integration are Improper Integrals of Type I 1. (Upper limit) If f(x) is continuous on [a, ), then a b f(x)dx = lim f(x)dx. b a 2. (Lower limit) If f(x) is continuous on (,b], then b b f(x)dx = lim f(x)dx. a a 3. (Both limits) If f(x) is continuous on (, ), then f(x)dx = c f(x)dx+ c f(x)dx. where c is any real number. In each case, if the limit is finite we that the improper integral converges and that the limit is the values of the improper integral. If the limit fails to exist, the improper integral diverges. Definition 2 Integrals of functions that become infinite at a point within the interval of integration are Improper Integrals of Type II 1. (Upper endpoint) If f(x) is continuous on [a,b) and discontinuous at b, then b a c f(x)dx = lim f(x)dx. c b a 2. (Lower endpoint) If f(x) is continuous on (a,b] and discontinuous at a, then b a f(x)dx = lim c a + b c f(x)dx. 3. (Interior point) If f(x) is discontinuous at c, where a < c < b, and continuous on [a,c) (c,b], then where c is any real number. b a f(x)dx = c a f(x)dx+ b c f(x)dx. In each case, if the limit is finite we that the improper integral converges and that the limit is the values of the improper integral. If the limit fails to exist, the improper integral diverges. 51

53 Direct Comparison Test Let f and g be continuous on [a, ) with 0 f(x) g(x) for all x a. Then a f(x)dx converges if g(x)dx diverges if a a a g(x)dx converges. f(x)dx diverges. Limit Comparison Test If the positive functions f and g are continuous on [a, ) and if f(x) lim a g(x) = L, 0 < L <. then both converge or both diverge. a f(x)dx and a g(x)dx Example 1 Evaluate the integral: 0 xe 3x dx Example 2 Evaluate the integral: 0 e 4x dx 52

54 Example 3 Evaluate the integral: 4 0 x+1 16 x 2 dx Example 4 Evaluate the integral: 6 6 xln x dx 53

55 Example 5 Determine whether 0 3 dx converges or diverges? 4x+e2x Example 6 Determine whether 1 3 dx converges or diverges? x

56 Example 7 Determine whether 2 6x dx converges or diverges? 9x4 +x3 55

57 Chapter Infinite Series Infinite Series Definition 1 An infinite series is a sum of infinite terms. a 1 +a 2 +a 3...+a n +... = Remark: The series a 1 +a 2 +a 3...+a n +... can be denoted as a n, a k, an n=1 k=1 Definition 2 Given an infinite series n=1 a n, the sequence of partial sums of the series is the sequence {s n } given by s n = k=1 a k, that is, s 1 = a 1 s 2 = a 1 +a 2 s 3 = a 1 +a 2 +a 3. The series converges to L iff sequence of partial sums {s n } converges to L, and in this case we write a n = L. The series diverges iff the sequence of partial sums {s n } diverges. n=1 Geometric series Definition 3 A geometric series is a series of the form ar n = a+ar +ar 2 +ar n=0 where a and r are real numbers. Theorem 1 If the geometric series n=1 ar n has ratio r < 1, then converges and n=0 n=0 If r > 1, then ar n = a 1 r diverges. n=0 n-term test for a divergent series Theorem 2 If the series ar n = a 1 r. a n converges, then a n 0. n=1 Remark: If lim a n 0, then n n=1 a n diverges. 56 a n

58 Example 1 Evaluate the infinite series Does it converge or diverge? 1 (2)(3) + 1 (3)(4) + 1 (4)(5) + 1 (5)(6) +... Example 2 Evaluate the infinite series Does it converge or diverge? Example 3 Evaluate the infinite series ( 1) n n=0 5 n. Does it converge or diverge? Example 4 Evaluate the infinite series n=0 ( 8 2 n 1 3 n ). Does it converge or diverge? 57

59 Example 5 Evaluate the infinite series n=0 3 n+1. Does it converge or diverge? 8n Example 6 Evaluate the geometric series ( 5) n x n and give the interval of convergence. n=0 Example 7 Evaluate the geometric series ( 1) n (x+5) n andgivetheinterval ofconvergence. n=0 58

60 Example 8 Express this number as the ratio of two integers:1.492 = Example 9 Use the nth term test to determine the infinite series diverges. n=0 (1 1 8n )n converges or Example 10 Use the nth term test to determine the infinite series or diverges. 2 cos(2nπ) converges n=0 59

61 12.3-Part I The Integral Test Theorem 1 If f : [1, ) R is a continuous, positive, decreasing function, and a n = f(n), then the following holds: n=1 a n converges 1 f(x)dx converges. Example 1 Show whether the series 6 n=1 n 8n converges or not. Example 2 Show whether the series n=1 lnn 4n converges or not. 60

62 Example 3 Show whether the series n=1 4 n n+8 converges or not. Example 4 Show whether the series n=2 8 converges or not. n(lnn) 5 Example 5 Show whether the series n=1 2 n 2 +1 converges or not. 61

63 12.3-Part II Comparison Test Direct comparison test for series If the sequence satisfy 0 a n b n for all n N, then (a) b n converges a n converges; (b) n=1 a n diverges n=1 n=1 b n diverges. n=1 Limit comparison test for series Assume that a n > 0 and b n > 0 for n N. a n (a) If lim = L > 0, then a n and n b n a n (b) If lim = 0 and n b n a n (c) If lim = and n b n n=1 b n both converge or both diverge. n=1 b n converges, then n=1 b n diverges, then n=1 a n converges. n=1 a n diverges. n=1 Example 1 Determine whether the series n=1 cos 2 (n) n n 3 converges or not. 62

64 Example 2 Determine whether the series n=1 5n 8 n 4 +3 n 6n 10 n 3 +5 converges or not. Example 3 Determine whether the series (2+4n) 2 converges or not. n=1 Example 4 Determine whether the series n=1 1 converges or not. 6n 1/2 +n1/4 63

65 Example 5 Determine whether the series n=1 ln(n+5) n converges or not. Example 6 Determine whether the series n=2 4 converges or not. (lnn) 2 Example 7 Determine whether the series n=2 ( 1 ) tan converges or not. 9n 64

66 12.4 The Ratio and Root Tests The Ratio Test Let {a n } be a positive sequence with lim n a n+1 a n = ρ exists. (a) If ρ < 1, the series a n converges. (b) If ρ > 1, the series a n diverges. (c) If ρ = 1, the test is inconclusive. The Root Test Let {a n } be a positive sequence with lim n n a n = ρ exists. (a) If ρ < 1, the series a n converges. (b) If ρ > 1, the series a n diverges. (c) If ρ = 1, the test is inconclusive. Example 1 Determine whether the series n=1 n 3 converges or not. 2n Example 2 Determine whether the series 2n!e 5n converges or not. n=1 65

67 Example 3 Determine whether the series n=1 ln5n n 6 converges or not. Example 4 Determine whether the series n=1 (n+4)(n+5) n! converges or not. Example 5 Determine whether the series n=1 n2 n (n+5)! 9 n n! converges or not. 66

68 12.5 Alternating Series, Absolute and Conditional Convergence Definition 1 An infinite series a n is an alternating series iff holds either a n = ( 1) n a n or a n = ( 1) n+1 a n. The Alternating Series Test (Leibniz s Test) The series ( 1) n+1 a n = a 1 a 2 +a 3 a n=1 converges if all three of the following conditions are satisfied: (1) The a n s are all positive; (2) The positive a n s are nonincreasing: a n a n+1 ; (3) a n 0. Definition 2 (1) A series a n is absolutely convergent iff the series a n converges; (2) A series converges conditionally iff it converges but does not converges absolutely. Example 1 Determine whether the series ( 1) n+1 2n 2n 5 +5 n=1 converges absolutely. 67

69 Example 2 Determine whether the series n=1 ( 1) n (8n)! 8 n n!n converges absolutely. Example 3 Determine whether the series n=1 ( 1) n (n!)2 3 n (2n+1)! converges absolutely. Example 4 Determine whether the series ( 1) n 7 n 1 n=1 7 n+1 n converges absolutely. 68

70 12.6 & 12.7 Taylor polynomials and Taylor series Taylor Polynomial Definition 1 The Taylor polynomial of order n centered at a D of an n-differentiable function f : D R R is given by P n (x) = Particularly, if a = 0, n P n (x) = n k=0 f (k) (a) (x a) k k! = f(a)+f (a)(x a)+ f (a) 2! k=0 Taylor Series f (k) (0) k! x k = f(0)+f (0)x+ f (0) 2! (x a) f(n) (a) (x a) n n! x f(n) (0) x n n! Definition 2 The Taylor series centered at a D of an infinitely differentiable function f : D R R is given by f(x) = n=0 f (n) (a) n! Particularly, if a = 0, f(x) = (x a) n = f(a)+f (a)(x a)+ f (a) 2! n=0 f (n) (0) n! x n = f(0)+f (0)x+ f (0) 2! (x a) f(n) (a) (x a) n +... n! x f(n) (0) x n +... n! Taylor Theorem If f has n+1 derivatives on the open interval I that contains a, then for each x I, with f(x) = f(a)+f (a)(x a)+ f (a) 2! R n (x) = 1 f (n+1) (t)(x t) n dt n! a Lagrange formula for the remainder x (x a) f(n) (a) (x a) n +R n (x) n! R n (x) = f(n+1) (c) (n+1)! (x a)n+1 where c is some number between a and x. And ( ) x a n+1 R n (x) max t is between a and x f(n+1) (t) (n+1)! 69

71 Taylor Theorem for a = 0 If f has n+1 derivatives on the open interval I that contains 0, then for each x I, with f(x) = f(0)+f (0)x+ f (0) 2! R n (x) = 1 n! x Lagrange formula for the remainder where c is some number between a and x. And 0 x f(n) (0) x n +R n (x) n! f (n+1) (t)(x t) n dt R n (x) = f(n+1) (c) (n+1)! xn+1 ( ) x n+1 R n (x) max t is between 0 and x f(n+1) (t) (n+1)! some Taylor series of basic functions at a = 0 1. e x x n x2 = = 1+x+ n! 2! + x3 3! +... n=0 2. cosx = ( 1) n x2n x2 = 1 (2n)! 2! + x4 4! x6 6! +... n=0 3. sinx = ( 1) n x 2n+1 (2n+1)! n= x = n=0 5. ln(1+x) = = x x3 3! + x5 5! x7 7! +... x n = 1+x+x 2 +x 3 +x k=1 ( 1) (k+1) k x k = x x2 2 + x

72 Example 1 Find the nth order Taylor polynomial P n (x) in powers of x for the function e x, and, using the Lagrange formula for the remainder, write the remainder R n (x) as a function of x and c. Example 2 Find the Taylor polynomials in powers of x of orders 0, 1, 2 and 3 generated by the function f(x) = 3, and, using the Lagrange formula for the remainder, write the remainder x+1 R 3 (x) as a function of x and c. 71

73 Example 3Findthe3rdorderTaylorpolynomialP 3 (x)inpowersofxforthefunctioncosh(6x), and, using the Lagrange formula for the remainder, write the remainder R 3 (x) as a function of x and c. Example 4 Find the Taylor polynomials of order 3 generated by the function f(x) = 1 x in powers of x 6 and, using the Lagrange formula for the remainder, write the remainder R 3 (x) as a function of x and c. 72

74 Example 5 Find the Taylor polynomials of order 3 generated by the function f(x) = x in powers of x 3 and, using the Lagrange formula for the remainder, write the remainder R 3 (x) as a function of x and c. ( x Example 6FindtheTaylorpolynomialsoforder1generatedbythefunctionf(x) = arctan 7) inpowers ofx 7 and, using the Lagrangeformula forthe remainder, write the remainder R 1 (x) as a function of x and c. 73

75 Example 7 Estimate the maximum error made in approximating e x by the polynomial 1+x+ 1 2 x2 over the interval x [ 0.2,0.2]. Example 8 Estimate the maximum error made in approximating sinh(x) by the polynomial x+ 1 3! x3 over the interval x [ 0.1,0.1]. 74

76 12.8 & 12.9 Power Series, Binomial Series Definition 1 An power series centered at x 0 is the function y : D R R y(x) = c n (x x 0 ) n, c n R. n=0 Definition 2 The power series y(x) = c n (x x 0 ) n has radius of convergence ρ 0 iff n=0 the following conditions holds: (a) The series converges absolutely for x x 0 < ρ; (b) The series diverges for x x 0 > ρ. How to find the radius (or interval) of convergence? Ratio Test! Theorem If The Differentiability Theorem f(x) = then f is differentiable on ( c,c) and a k x k k=0 for all x in ( c,c). f (x) = k=0 d ( ak x k) for all x in ( c,c). dx Theorem If then Term-by-Term Integration f(x) = f(x)dx = k=0 a k x k k=0 converges on ( c,c). a k k +1 xk+1 +C converges on ( c,c). 75

77 The Binomial Series For 1 < x < 1, where we define and ( m k ) = ( m 1 (1+x) m = 1+ ) = m, ( m 2 ( m k k=1 ) ) x k, = m(m 1) 2! m(m 1)(m 2) (m k +1) k! for k 3. Example 1 Determine the interval of convergence of ( 1) n (2x+9) n. n=1 Example 2 Determine the interval of convergence of n=1 x 5n+1 n!. 76

78 Example 3 Determine the interval of convergence of n=1 nx n 3 n (n 2 +3). ( 3x+4 ) Example 4Findthefirst5nonzerotermsoftheTaylorseriesatx = 0forf(x) = x 2 cos. Example 5 Find the first 5 nonzero terms of the Taylor series at x = 0 for f(x) = sinx 2x+ x 3 /3!. 77

79 Example 6 Expand arctanx in terms of x. Example 7 Find the first 4 nonzero terms of the power series representation for the function F(x) = x 0 3t 2 costdt Example 8 Find the first 4 terms of the binomial series for the function f(x) = 1+x 78

80 Example 9 Find the first 4 terms of the binomial series for the function f(x) = (1+ x 4 ) 4 Example 10 Find the first 4 terms of the binomial series for the function f(x) = (1+x 2 )

81 Chapter Polar Coordinates To define polar coordinates, we first fix an origin O (called the pole) and an initial ray from O. Then each point P can be located by assigning to it a polar coordinate pair (r,θ) in which r gives the directed distance from O to P and θ gives the directed angle from the initial ray to ray OP. Polar equations and graphs Equation r = a θ = θ 0 Graph Circle of radius a centered ato. Line through O making an angle θ 0 with the initial ray. Transformation rules Polar-Cartesian x = rcosθ, y = rsinθ,,r 2 = x 2 +y 2, tanθ = y x. 80

82 Example 1 Decide if the points given in polar coordinates are the same. ( ) ( ) (a) 15, 26π and 15, π 3 3 ( ) ( ) (b) 9, π and 9, π 3 3 Example 2 For each Polar coordinates (r, θ), find the equivalent Cartesian coordinates (x, y) below. ( ) ) ) (a) (b) ( 6, 43 π (c) (8, 83 π 4, 11π 6 81

83 Example 3 Graph the following inequalities. (a) θ = π 4 (b) 1 r 2 (c) r 0, 5π 4 θ 7π 4 (d) 2 r 1, 3π 4 θ 7π 4 Example 4 Consider the region graphed below. The two arcs shown are circular, and the region is between the two arcs and between the y axis and the line graphed, which is y = 1 3 x. Give inequalities for r and θ which describe the region below in polar coordinates. (3, (2, 2 3 ) 3 3 ) 82

84 Example 5 Consider the region graphed below. The arc shown is circular, and the region extends indefinitely in the y direction. Give inequalities for r and θ which describe the region below in polar coordinates. 1 3 Example 6 Find the equation of the curve in Cartesian coordinates for rcosθ = 9 83

85 Example 7 Find the equation of the curve in Cartesian coordinates for r = 6 sinθ 4cosθ Example 8 Find the equation of the curve in Cartesian coordinates for r 2 sin2θ = 7 Example 9 Find the equation of the curve in Cartesian coordinates for r 2 = 6rcosθ 84

86 10.3 Graphing in Polar Coordinates How to graph the polar equation r = f(θ)? Method 1: make a table of (r, θ) values, plot the corresponding points, and connecting them in order of increasing θ. Method 2: step 1: graph r = f(θ) in Cartesian rθ plane. step 2: then use the Cartesian graph as a table and guide to sketch the polar coordinate graph. Use the symmetry to reduce the work needed to graph. x axis symmetry: (r,θ) and (r, θ) or ( r,π θ) belong to the graph. Origin symmetry: (r,θ) and ( r,θ) or (r,π +θ) belong to the graph. y axis symmetry: (r,θ) and ( r, θ) or (r,π θ) belong to the graph. Example 1 Graph the polar equation in xy plane. r = 2+cosθ. Example 2 Graph the polar equation in xy plane. r = 2+4cosθ. 85

87 Example 3 Graph the polar equation in xy plane. ( θ r = 7sin. 2) Example 4 Graph the polar equation in xy plane. r = 5sin(2θ). 86

88 10.4 Areas in Polar Coordinates Area of the Fan-Shaped Region Between the Origin and the Curve A = β α 1 2 r2 dθ. Area of the Region 0 r 1 (θ) r r 2 (θ), α θ β A = = β α β α β 1 2 r2 2 dθ ( r2 2 r α ) dθ. 1 2 r2 1 dθ Example 1 Find the point(s) of intersection of the pair of curves r = 2+2cosθ and r = 2 2cosθ Give your answer in Cartesian coordinates. 87

89 Example 2 Find the point(s) of intersection of the pair of curves r = 9sinθ and r = 9sin2θ Give your answer in Cartesian coordinates. Example 3 Find the area inside the cardioid r = 3+3cosθ Example 4 Find the area inside one leaf of the four-leaved rose r = 3cos2θ 88

90 Example 5 Find the area shared by the circle r = 6 and the cardioid r = 6(1 cosθ) Example 6 Find the area inside the lemniscate r 2 = 150cos2θ and outside the circle r = 5 3. Example 7Findtheareainsideofthecircler = 6cosθandoutsidethecardioidr = 6(1 cosθ). 89

91 10.5 Parametrizations of Plane Curves Definition 1 If x and y are given as functions x = f(t), y = g(t) over an interval I of t values, then the set of points (x,y) = (f(t),g(t)) defined by these functions is a parametric curve. The equations are parametric equations for the curve. Example 1 Describe the curve x(t) = 5cos2t, y = 5sin2t, t [ ] 0, π. 2 Example 2 Describe the curve x(t) = 4sin4t, y = 5cos4t, t [ ] 0, π. 4 ( ) Example 3 Describe the curve x(t) = 5sec5t, y = 5tan5t, t π, π

92 Example 4 Find the parametrization of the line segment joining ( 5, 3) to (5, 4). Express the functions as x = a function of t and y = a function of t, where t [0,1]. Example 5 Find the parametrization of the curve y = x from ( 3, 19) to (5,133). Express the functions as x = a function of t and y = a function of t, where t [ 3,5]. Example 6 Find the parametrization of the bottom part of the curve x2 16 +y2 = 1 which starts 4 at (4,0) to ( 4,0). Express the functions as x = a function of t and y = a function of t, where t [0,π]. 91

93 10.6 & 10.7 Tangents to Curves Given Parametrically Arc Length Parametric Formula for dy/dx If all three derivatives exist and dx/dt 0, b dy dx = dy/dt dx/dt. Arc Length Case 1: The length of y = f(x), a x b a a L = 1+[f (x)] 2 dx = 1+ ( dy) 2dx dx The length of x = g(x), c y d a L = c d 1+[g (y)] 2 dy = b c d 1+ ( dx) 2 dy dy Case 2: The length of x = f(t) and y = g(t), a t b b [f L = (t) ] 2 [ + g (t) ] b (dx ) 2 2 ( dy ) 2dt dt or equivalently L = + dt dt Case 3: The length of r = ρ(θ), α θ β L = β α [ρ(θ) ] 2 + [ ρ (θ) ]2 dθ a 92

94 Example 1 Find the length of the curve y = x3/2 1, 0 x 1. Example 2 Find the length of the curve x = 3y , 1 y 2. 36y3 93

95 Example 3 Find the equation of the tangent line to the parametric curve x = 6sin3t, y = 6cos3t, at the point on the curve associated with t = π 12. Example 4 Find the length L of the curve given by x = 1 6 (4t + 36)3/2 and y = 8t + t2 2 for 0 t 6. 94

96 Example 5 Find the equation in x and y for the line tangent to the polar curve r = 4 4sinθ at the value θ = 0. 95