Section 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables


 Wesley Simon
 2 years ago
 Views:
Transcription
1 The Clculus of Functions of Severl Vribles Section 2.3 Motion Along Curve Velocity ccelertion Consider prticle moving in spce so tht its position t time t is given by x(t. We think of x(t s moving long curve C prmetrized by function f, where f : R R n. Hence we hve x(t f(t, or, more simply, x f(t. For us, n will lwys be 2 or 3, but there re physicl situtions in which it is resonble to hve lrger vlues of n, most of wht we do in this section will pply to those cses eqully well. This is lso good time to introduce the Leibniz nottion for derivtive, thus writing dx Df(t. (2.3. At given time t 0, the vector x(t 0 + h x(t 0 represents the mgnitude direction of the chnge of position of the prticle long C from time t 0 to time t 0 + h, s shown in Figure Dividing by h, we obtin vector, x(t 0 + h x(t 0, (2.3.2 h with the sme direction, but with length pproximting the verge speed of the prticle over the time intervl from t 0 to t 0 + h. Assuming differentibility tking the limit s h pproches 0, we hve the following definition. x( t 0 x ( t + h 0 t 0 x( t 0 + h  x( Figure 2.3. Motion long curve C Copyright c by Dn Sloughter 200
2 2 Motion Along Curve Section 2.3 Definition Suppose x(t is the position of prticle t time t moving long curve C in R n. We cll the velocity of the prticle t time t we cll the speed of the prticle t time t. Moreover, we cll v(t d x(t (2.3.3 s(t v(t (2.3. (t d v(t (2.3.5 the ccelertion of the prticle t time t. Exmple is Consider prticle moving long n ellipse so tht its position t ny time t x (2 cos(t, sin(t. Then its velocity is its speed is its ccelertion is For exmple, t t s we hve v ( 2 sin(t, cos(t, sin 2 (t + cos 2 (t 3 sin 2 (t +, ( 2 cos(t, sin(t. x v s ( 2, 2, ( 2,, 2 5 2, ( 2,. 2 See Figure Notice tht, in this exmples, x for ll vlues of t.
3 Section 2.3 Motion Along Curve 3 v 0.5 x Figure Position, velocity, ccelertion vectors for motion on n ellipse Curvture Suppose x is the position, v is the velocity, s is the speed, is the ccelertion, t time t, of prticle moving long curve C. Let T (t be the unit tngent vector N(t be the principl unit norml vector t x. Now T (t dx dx v v v s, (2.3.6 so v s T (t. (2.3.7 Thus Since we hve dv d ds st (t ds N(t T (t + sdt (t (2.3.8 DT (t DT (t, (2.3.9 T (t + s DT (t N(t. (2.3.0 Note tht (2.3.0 expresses the ccelertion of prticle s the sum of sclr multiples of the unit tngent vector the principl unit norml vector. Tht is, T T (t + N N(t, (2.3. where T ds (2.3.2 N s DT (t. (2.3.3
4 Motion Along Curve Section 2.3 However, since T (t N(t re orthogonl unit vectors, we lso hve T (t ( T T (t + N N(t T (t T (T (t T (t + N (T (t N(t (2.3. T N(t ( T T (t + N N(t N(t T (T (t N(t + N (N(t N(t N. (2.3.5 Hence T is the coordinte of in the direction of T (t N is the coordinte of in the direction of N(t. Thus (2.3.0 writes the ccelertion s sum of its component in the direction of the unit tngent vector its component in the direction of the principl unit norml vector. In prticulr, this shows tht the ccelertion lies in the plne determined by T (t N(t. Moreover, T is the rte of chnge of speed, while N is the product of the speed s DT (t, the mgnitude of the rte of chnge of the unit tngent vector. Since T (t for ll t, DT (t reflects only the rte t which the direction of T (t is chnging; in other words, DT (t is mesurement of how fst the direction of the prticle moving long the curve C is chnging t time t. If we divide this by the speed of the prticle, we obtin strd mesurement of the rte of chnge of direction of C itself. Definition Given curve C with smooth prmetriztion x f(t, we cll κ DT (t s(t (2.3.6 the curvture of C t f(t. Using (2.3.6, we cn rewrite (2.3.0 s ds T (t + s2 κn(t. (2.3.7 Hence the coordinte of ccelertion in the direction of the tngent vector is the rte of chnge of the speed the coordinte of ccelertion in the direction of the principl norml vector is the squre of the speed times the curvture. Thus the greter the speed or the tighter the curve, the lrger the size of the norml component of ccelertion; the greter the rte t which speed is incresing, the greter the tngentil component of ccelertion. This is why drivers re dvised to slow down while pproching curve, then to ccelerte while driving through the curve. Exmple is given by Suppose prticle moves long line in R n so tht its position t ny time t x tw + p,
5 Section 2.3 Motion Along Curve 5 where w 0 p re vectors in R n. Then the prticle hs velocity v dx w speed s w, so the unit tngent vector is T (t v s Hence T (t is constnt vector, so DT (t 0 w w. κ DT (t s 0 for ll t. In other words, line hs zero curvture, s we should expect since the tngent vector never chnges direction. Exmple Consider prticle moving long circle C in R 2 of rdius r > 0 center (, b, with its position t time given by x (r cos(t +, r sin(t + b. Then its velocity, speed, ccelertion re s v ( r sin(t, r cos(t, r 2 sin ( t + r 2 cos 2 (t r ( r cos(t, r sin(t, respectively. Hence the unit tngent vector is T (t v s ( sin(t, cos(t. Thus DT (t ( cos(t, sin(t DT (t cos 2 (t + sin 2 (t. Hence the curvture of C is, for ll t, κ DT (t s r.
6 6 Motion Along Curve Section 2.3 Thus circle hs constnt curvture, nmely, the reciprocl of the rdius of the circle. In prticulr, the lrger the rdius of circle, the smller the curvture. Also, note tht so, from (2.3.0, we hve ds d r 0, rn(t, which we cn verify directly. Tht is, the ccelertion hs norml component, but no tngentil component. Exmple time t is Now consider prticle moving long n ellipse E so tht its position t ny x (2 cos(t, sin(t. Then, s we sw bove, the velocity speed of the prticle re v ( 2 sin(t, cos(t s 3 sin( 2 +, respectively. For purposes of differentition, it will be helpful to rewrite s s cos(2t s ( cos(2t Then the unit tngent vector is 2 T (t ( 2 sin(t, cos(t. 5 3 cos(2t Thus 2 DT (t 5 3 cos(2t So, for exmple, t t, we hve 3 2 sin(2t ( 2 cos(t, sin(t ( 2 sin(t, cos(t. (5 3 cos(2t 3 2 x v s ( 2, 2, ( 2,, 2 5 2,
7 Section 2.3 Motion Along Curve 7 ( T ( DT 5 ( 2,, 5 (, 8, 5 ( DT ( 2, Hence the curvture of E t is 2 κ , where the finl numericl vlue hs been rounded to four deciml plces. Although the generl expression for κ is complicted, it is esily computed plotted using computer lgebr system, s shown in Figure Compring this with the plot of this ellipse in Figure 2.3.2, we cn see why the curvture is gretest round (2, 0 ( 2, 0, corresponding to t 0, t, t 2, smllest t (0, (0,, corresponding to t 2 t 3 2. Finlly, s we sw bove, the ccelertion of the prticle is so Now if we write ( 2 cos(t, sin(t, then we my either compute, using (2.3.7, T ds or, using (2.3. (2.3.5, T ( 2,. 2 T T (t + N N(t, 2 (5 3 cos(2t 2 (3 sin(2t N s 2 k ( T , 5 0 ( 2, ( 2, N ( N ( 2, 2 5 (, 8 0.
8 8 Motion Along Curve Section Figure Curvture of n ellipse Hence, in either cse, 3 ( T + ( N. 0 0 Arc length Suppose prticle moves long curve C in R n so tht its position t time t is given by x f(t let D be the distnce trveled by the prticle from time t to t b. We will suppose tht s(t v(t is continuous on [, b]. To pproximte D, we divide [, b] into n subintervls, ech of length t b n, lbel the endpoints of the subintervls t 0, t,..., t n b. If t is smll, then the distnce the prticle trvels during the jth subintervl, j, 2,..., n, should be, pproximtely, s t, n pproximtion which improves s t decreses. Hence, for sufficiently smll t (equivlently, sufficiently lrge n, n s(t j t (2.3.8 j will provide n pproximtion s close to D s desired. Tht is, we should define D lim n j n s(t j t. (2.3.9 But (2.3.8 is Riemnn sum (in prticulr, lefth rule sum which pproximtes the definite integrl b s(t. (2.3.20
9 Section 2.3 Motion Along Curve 9 Hence the limit in (2.3.9 is the vlue of the definite integrl (2.3.20, so we hve the following definition. Definition Suppose prticle moves long curve C in R n so tht its position t time t is given by x f(t. Suppose the velocity v(t is continuous on the intervl [, b]. Then we define the distnce trveled by the prticle from time t to time t b to be b v(t. (2.3.2 Note tht the distnce trveled is the length of the curve C if the prticle trverses C exctly once. In tht cse, we cll (2.3.2 the length of C. In generl, for ny t such tht the intervl [, t] is in the domin of f, we my clculte σ(t which we cll the rc length function for C. Exmple t Consider the helix H prmetrized by f(t (cos(t, sin(t, t. v(u du, ( If we let L denote the length of one complete loop of the helix, then prticle trveling long H ccording to x f(t will trverse this distnce s t goes from 0 to 2. Since we hve Hence Exmple by v(t v(t ( sin(t, cos(t,, L sin 2 (t + cos 2 (t Suppose prticle moves long curve C so tht its position t time t is given x (( + 2 cos(t cos(t, ( + 2 cos(t sin(t. Then C is the curve in Figure 2.3., which is clled limçon. The prticle will trverse this curve once s t goes from 0 to 2. Now so v ( ( + 2 cos(t sin(t 2 sin(t cos(t, ( + 2 cos(t cos(t 2 sin 2 (t, v 2 v v ( + 2 cos(t 2 sin 2 (t + ( + 2 cos(t sin 2 (t cos(t + sin 2 (t cos 2 (t + ( + 2 cos(t 2 cos 2 (t ( + 2 cos(t sin 2 (t cos(t + sin (t ( + 2 cos(t 2 (sin 2 (t + cos 2 (t + sin 2 (t cos 2 (t + sin (t ( + 2 cos(t 2 + sin 2 (t cos 2 (t + sin (t,
10 0 Motion Along Curve Section Figure 2.3. A limçon Hence the length of C is 2 0 ( + 2 cos(t 2 + sin 2 (t cos 2 (t + sin (t 3.369, where the integrtion ws performed with computer the finl result rounded to four deciml plces. Note tht integrting from 0 to would find the distnce the prticle trvels in going round C twice, nmely, 0 ( + 2 cos(t 2 + sin 2 (t cos 2 (t + sin (t Problems. For ech of the following, suppose prticle is moving long curve so tht its position t time t is given by x f(t. Find the velocity ccelertion of the prticle. ( f(t (t 2 + 3, sin(t (b f(t (t 2 e 2t, t 3 e 2t, 3t (c f(t (cos(3t 2, sin(3t 2 (d f(t (t cos(t 2, t sin(t 2, 3t cos(t 2 2. Find the curvture of the following curves t the given point. ( f(t (t, t 2, t (b f(t (3 cos(t, sin(t, t
11 Section 2.3 Motion Along Curve (c f(t (cos(t, sin(t, t, t 3 (d f(t (cos(t, sin(t, e t, t 0 3. Plot the curvture for ech of the following curves over the given intervl I. ( f(t (t, t 2, I [ 2, 2] (b f(t (cos(t, 3 sin(t, I [0, 2] (c g(t (( + 2 cos(t cos(t, ( + 2 cos(t sin(t, I [0, 2] (d h(t (2 cos(t, sin(t, 2t, I [0, 2] (e f(t ( cos(t + sin(t, sin(t + sin(t, I [0, 2]. For ech of the following, suppose prticle is moving long curve so tht its position t time t is given by x f(t. Find the coordintes of ccelertion in the direction of the unit tngent vector in the direction of the principl unit norml vector t the specified point. Write the ccelertion s sum of sclr multiples of the unit tngent vector the principl unit norml vector. ( f(t (sin(t, cos(t, t 3 (b f(t (cos(t, 3 sin(t, t (c f(t (t, t 2, t (d f(t (sin(t, cos(t, t, t 3 5. Suppose prticle moves long curve C in R 3 so tht its position t time t is given by x f(t. Let v, s, denote the velocity, speed, ccelertion of the prticle, respectively, let κ be the curvture of C. ( Using the fcts v st (t ds T (t + s2 κn(t, show tht v s 3 κ(t (t N(t. (b Use the result of prt ( to show tht κ v v Let H be the helix in R 3 prmetrized by f(t (cos(t, sin(t, t. Use the result from Problem 5 to compute the curvture κ of H for ny time t. 7. Let C be the ellipticl helix in R 3 prmetrized by f(t ( cos(t, 2 sin(t, t. Use the result from Problem 5 to compute the curvture κ of C t t. 8. Let C be the curve in R 2 which is the grph of the function ϕ : R R. Use the result from Problem 5 to show tht the curvture of C t the point (t, ϕ(t is κ ϕ (t. ( + (ϕ (t 2 3 2
12 2 Motion Along Curve Section Let P be the grph of f(t t 2. Use the result from Problem 8 to find the curvture of P t (, (2,. 0. Let C be the grph of f(t t 3. Use the result from Problem 8 to find the curvture of C t (, (2, 8.. Let C be the grph of g(t sin(t. Use the result from Problem 8 to find the curvture of C t ( ( 2,, For ech of the following, suppose prticle is moving long curve so tht its position t time t is given by x f(t. Find the distnce trveled by the prticle over the given time intervl. ( f(t (sin(t, 3 cos(t, I [0, 2] (b f(t (cos(t, sin(t, 2t, I [0, ] (c f(t (t, t 2, I [0, 2] (d f(t (t cos(t, t sin(t, I [0, 2] (e f(t (cos(2t, sin(2t, 3t 2, t, I [0, ] (f f(t (e t cos(t, e t sin(t, I [ 2, 2] (g f(t ( cos(t + sin(t, sin(t + sin(t, I [0, 2] 3. Verify tht the circumference of circle of rdius r is 2r.. The curve prmetrized by f(t (sin(2t cos(t, sin(2t sin(t hs four petls. Find the length of one of these petls. 5. The curve C prmetrized by h(t (cos 3 (t, sin 3 (t is clled hypocycloid (see Figure in Section 2.3. Find the length of C. 6. Suppose ϕ : R R is continuously differentible let C be the prt of the grph of ϕ over the intervl [, b]. Show tht the length of C is b + (ϕ (t Use the result from Problem 6 to find the length of one rch of the grph of f(t sin(t. 8. Let h : R R n prmetrize curve C. We sy C is prmetrized by rc length if Dh(t for ll t. ( Let σ be the rc length function for C using the prmetriztion f let σ be its inverse function. Show tht the function g : R R n defined by g(u f(σ (u prmetrizes C by rc length.
13 Section 2.3 Motion Along Curve 3 (b Let C be the circulr helix in R 3 with prmetriztion f(t (cos(t, sin(t, t. Find function g : R R n which prmetrizes C by rc length. 9. Suppose f : R R n is continuous on the closed intervl [, b] hs coordinte functions f, f 2,..., f n. We define the definite integrl of f over the intervl [, b] to be ( b b b b f(t f (t, f 2 (t,..., f n (t. Show tht if prticle moves so its velocity t time t is v(t, then, ssuming v is continuous function on n intervl [, b], the position of the prticle for ny time t in [, b] is given by x(t t v(sds + x(. 20. Suppose prticle moves long curve in R 3 so tht its velocity t ny time t is v(t (cos(2t, sin(2t, 3t. If the prticle is t (0,, 0 when t 0, use Problem 9 to determine its position for ny other time t. 2. Suppose prticle moves long curve in R 3 so tht its ccelertion t ny time t is (t (cos(t, sin(t, 0. If the prticle is t (, 2, 0 with velocity (0,, t time t 0, use Problem 9 to determine its position for ny other time t. 22. Suppose projectile is fired from the ground t n ngle α with n initil speed v 0, s shown in Figure Let x(t, v(t, (t be the position, velocity, ccelertion, respectively, of the projectile t time t. v(0 α Figure The pth of projectile
14 Motion Along Curve Section 2.3 ( Explin why x(0 (0, 0, v(0 (v 0 cos(α, v 0 sin(α, (t (0, g for ll t, where g 9.8 meters per second per second is the ccelertion due to grvity. (b Use Problem 9 to find v(t. (c Use Problem 9 to find x(t. (d Show tht the curve prmetrized by x(t is prbol. Tht is, let x(t (x, y show tht y x 2 + bx + c for some constnts, b, c. (e Show tht the rnge of the projectile, tht is, the horizontl distnce trveled, is R v 0 sin(2α g conclude tht the rnge is mximized when α. (f When does the projectile hit the ground? (g Wht is the mximum height reched by the projectile? When does it rech this height? 23. Suppose, 2,..., m re unit vectors in R n, m n, which re mutully orthogonl (tht is, i j when i j. If x is vector in R n with show tht x i x i, i, 2,..., m. x x + x x m m,
5.2 The Definite Integral
5.2 THE DEFINITE INTEGRAL 5.2 The Definite Integrl In the previous section, we sw how to pproximte totl chnge given the rte of chnge. In this section we see how to mke the pproximtion more ccurte. Suppose
More informationSection 4.3. By the Mean Value Theorem, for every i = 1, 2, 3,..., n, there exists a point c i in the interval [x i 1, x i ] such that
Difference Equtions to Differentil Equtions Section 4.3 The Fundmentl Theorem of Clculus We re now redy to mke the longpromised connection between differentition nd integrtion, between res nd tngent lines.
More informationArc Length. P i 1 P i (1) L = lim. i=1
Arc Length Suppose tht curve C is defined by the eqution y = f(x), where f is continuous nd x b. We obtin polygonl pproximtion to C by dividing the intervl [, b] into n subintervls with endpoints x, x,...,x
More information5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous relvlued
More informationReview guide for the final exam in Math 233
Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered
More informationExample A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
More informationLesson 10. Parametric Curves
Return to List of Lessons Lesson 10. Prmetric Curves (A) Prmetric Curves If curve fils the Verticl Line Test, it cn t be expressed by function. In this cse you will encounter problem if you try to find
More information1 Numerical Solution to Quadratic Equations
cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll
More informationSolutions to Section 1
Solutions to Section Exercise. Show tht nd. This follows from the fct tht mx{, } nd mx{, } Exercise. Show tht = { if 0 if < 0 Tht is, the bsolute vlue function is piecewise defined function. Grph this
More informationNet Change and Displacement
mth 11, pplictions motion: velocity nd net chnge 1 Net Chnge nd Displcement We hve seen tht the definite integrl f (x) dx mesures the net re under the curve y f (x) on the intervl [, b] Any prt of the
More informationAREA OF A SURFACE OF REVOLUTION
AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.
More informationThe Calculus of Variations: An Introduction. By Kolo Sunday Goshi
The Clculus of Vritions: An Introduction By Kolo Sundy Goshi Some Greek Mythology Queen Dido of Tyre Fled Tyre fter the deth of her husbnd Arrived t wht is present dy Liby Irbs (King of Liby) offer Tell
More informationMath 22B Solutions Homework 1 Spring 2008
Mth 22B Solutions Homework 1 Spring 2008 Section 1.1 22. A sphericl rindrop evportes t rte proportionl to its surfce re. Write differentil eqution for the volume of the rindrop s function of time. Solution
More informationScalar Line Integrals
Mth 3B Discussion Session Week 5 Notes April 6 nd 8, 06 This week we re going to define new type of integrl. For the first time, we ll be integrting long something other thn Eucliden spce R n, nd we ll
More informationAnswer, Key Homework 8 David McIntyre 1
Answer, Key Homework 8 Dvid McIntyre 1 This printout should hve 17 questions, check tht it is complete. Multiplechoice questions my continue on the net column or pge: find ll choices before mking your
More informationHarvard College. Math 21a: Multivariable Calculus Formula and Theorem Review
Hrvrd College Mth 21: Multivrible Clculus Formul nd Theorem Review Tommy McWillim, 13 tmcwillim@college.hrvrd.edu December 15, 2009 1 Contents Tble of Contents 4 9 Vectors nd the Geometry of Spce 5 9.1
More informationReview Problems for the Final of Math 121, Fall 2014
Review Problems for the Finl of Mth, Fll The following is collection of vrious types of smple problems covering sections.,.5, nd.7 6.6 of the text which constitute only prt of the common Mth Finl. Since
More informationPROBLEMS 13  APPLICATIONS OF DERIVATIVES Page 1
PROBLEMS  APPLICATIONS OF DERIVATIVES Pge ( ) Wter seeps out of conicl filter t the constnt rte of 5 cc / sec. When the height of wter level in the cone is 5 cm, find the rte t which the height decreses.
More informationCURVES ANDRÉ NEVES. that is, the curve α has finite length. v = p q p q. a i.e., the curve of smallest length connecting p to q is a straight line.
CURVES ANDRÉ NEVES 1. Problems (1) (Ex 1 of 1.3 of Do Crmo) Show tht the tngent line to the curve α(t) (3t, 3t 2, 2t 3 ) mkes constnt ngle with the line z x, y. (2) (Ex 6 of 1.3 of Do Crmo) Let α(t) (e
More informationUniform convergence and its consequences
Uniform convergence nd its consequences The following issue is centrl in mthemtics: On some domin D, we hve sequence of functions {f n }. This mens tht we relly hve n uncountble set of ordinry sequences,
More informationCurve Sketching. 96 Chapter 5 Curve Sketching
96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of
More information5.6 Substitution Method
5.6 Substitution Method Recll the Chin Rule: (f(g(x))) = f (g(x))g (x) Wht hppens if we wnt to find f (g(x))g (x) dx? The Substitution Method: If F (x) = f(x), then f(u(x))u (x) dx = F (u(x)) + C. Steps:
More informationDerivatives and Rates of Change
Section 2.1 Derivtives nd Rtes of Cnge 2010 Kiryl Tsiscnk Derivtives nd Rtes of Cnge Te Tngent Problem EXAMPLE: Grp te prbol y = x 2 nd te tngent line t te point P(1,1). Solution: We ve: DEFINITION: Te
More informationr 2 F ds W = r 1 qe ds = q
Chpter 4 The Electric Potentil 4.1 The Importnt Stuff 4.1.1 Electricl Potentil Energy A chrge q moving in constnt electric field E experiences force F = qe from tht field. Also, s we know from our study
More informationApplications to Physics and Engineering
Section 7.5 Applictions to Physics nd Engineering Applictions to Physics nd Engineering Work The term work is used in everydy lnguge to men the totl mount of effort required to perform tsk. In physics
More information11. Fourier series. sin mx cos nx dx = 0 for any m, n, sin 2 mx dx = π.
. Fourier series Summry of the bsic ides The following is quick summry of the introductory tretment of Fourier series in MATH. We consider function f with period π, tht is, stisfying f(x + π) = f(x) for
More information1 Line Integrals of Scalar Functions
MA 242  Fll 2010 Worksheet VIII 13.2 nd 13.3 1 Line Integrls of Sclr Functions There re (in some sense) four types of line integrls of sclr functions. The line integrls w.r.t. x, y nd z cn be plced under
More information1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator
AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.
More informationAntiderivatives/Indefinite Integrals of Basic Functions
Antiderivtives/Indefinite Integrls of Bsic Functions Power Rule: x n+ x n n + + C, dx = ln x + C, if n if n = In prticulr, this mens tht dx = ln x + C x nd x 0 dx = dx = dx = x + C Integrl of Constnt:
More informationThe Chain Rule. rf dx. t t lim " (x) dt " (0) dx. df dt = df. dt dt. f (r) = rf v (1) df dx
The Chin Rule The Chin Rule In this section, we generlize the chin rule to functions of more thn one vrible. In prticulr, we will show tht the product in the singlevrible chin rule extends to n inner
More information4.11 Inner Product Spaces
314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces
More information4: RIEMANN SUMS, RIEMANN INTEGRALS, FUNDAMENTAL THEOREM OF CALCULUS
4: RIEMA SUMS, RIEMA ITEGRALS, FUDAMETAL THEOREM OF CALCULUS STEVE HEILMA Contents 1. Review 1 2. Riemnn Sums 2 3. Riemnn Integrl 3 4. Fundmentl Theorem of Clculus 7 5. Appendix: ottion 10 1. Review Theorem
More informationMATLAB: Mfiles; Numerical Integration Last revised : March, 2003
MATLAB: Mfiles; Numericl Integrtion Lst revised : Mrch, 00 Introduction to Mfiles In this tutoril we lern the bsics of working with Mfiles in MATLAB, so clled becuse they must use.m for their filenme
More informationArea Between Curves: We know that a definite integral
Are Between Curves: We know tht definite integrl fx) dx cn be used to find the signed re of the region bounded by the function f nd the x xis between nd b. Often we wnt to find the bsolute re of region
More information2.2. Volumes. A(x i ) x. Once again we recognize a Riemann sum at the right. In the limit as n we get the so called Cavalieri s principle: V =
2.2. VOLUMES 52 2.2. Volumes 2.2.1. Volumes by Slices. First we study how to find the volume of some solids by the method of cross sections (or slices ). The ide is to divide the solid into slices perpendiculr
More informationSo there are two points of intersection, one being x = 0, y = 0 2 = 0 and the other being x = 2, y = 2 2 = 4. y = x 2 (2,4)
Ares The motivtion for our definition of integrl ws the problem of finding the re between some curve nd the is for running between two specified vlues. We pproimted the region b union of thin rectngles
More informationDouble Integrals over General Regions
Double Integrls over Generl egions. Let be the region in the plne bounded b the lines, x, nd x. Evlute the double integrl x dx d. Solution. We cn either slice the region verticll or horizontll. ( x x Slicing
More informationNotes #5. We then define the upper and lower sums for the partition P to be, respectively, U(P,f)= M k x k. k=1. L(P,f)= m k x k.
Notes #5. The Riemnn Integrl Drboux pproch Suppose we hve bounded function f on closed intervl [, b]. We will prtition this intervl into subintervls (not necessrily of the sme length) nd crete mximl nd
More informationIntegration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
More informationThe Definite Integral
Chpter 4 The Definite Integrl 4. Determining distnce trveled from velocity Motivting Questions In this section, we strive to understnd the ides generted by the following importnt questions: If we know
More informationto the area of the region bounded by the graph of the function y = f(x), the xaxis y = 0 and two vertical lines x = a and x = b.
5.9 Are in rectngulr coordintes If f() on the intervl [; ], then the definite integrl f()d equls to the re of the region ounded the grph of the function = f(), the is = nd two verticl lines = nd =. =
More informationName: Lab Partner: Section:
Chpter 4 Newton s 2 nd Lw Nme: Lb Prtner: Section: 4.1 Purpose In this experiment, Newton s 2 nd lw will be investigted. 4.2 Introduction How does n object chnge its motion when force is pplied? A force
More informationGraphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
More informationIntegration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
More information6 Energy Methods And The Energy of Waves MATH 22C
6 Energy Methods And The Energy of Wves MATH 22C. Conservtion of Energy We discuss the principle of conservtion of energy for ODE s, derive the energy ssocited with the hrmonic oscilltor, nd then use this
More informationUsing Definite Integrals
Chpter 6 Using Definite Integrls 6. Using Definite Integrls to Find Are nd Length Motivting Questions In this section, we strive to understnd the ides generted by the following importnt questions: How
More informationm, where m = m 1 + m m n.
Lecture 7 : Moments nd Centers of Mss If we hve msses m, m 2,..., m n t points x, x 2,..., x n long the xxis, the moment of the system round the origin is M 0 = m x + m 2 x 2 + + m n x n. The center of
More informationWritten Homework 6 Solutions
Written Homework 6 Solutions Section.10 0. Explin in terms of liner pproximtions or differentils why the pproximtion is resonble: 1.01) 6 1.06 Solution: First strt by finding the liner pproximtion of f
More information1. 1 m/s m/s m/s. 5. None of these m/s m/s m/s m/s correct m/s
Crete ssignment, 99552, Homework 5, Sep 15 t 10:11 m 1 This printout should he 30 questions. Multiplechoice questions my continue on the next column or pge find ll choices before nswering. The due time
More informationJackson 2.23 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jckson.3 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: A hollow cube hs conducting wlls defined by six plnes x =, y =, z =, nd x =, y =, z =. The wlls z =
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More informationBrief review of prerequisites for ECON4140/4145
1 ECON4140/4145, August 2010 K.S., A.S. Brief review of prerequisites for ECON4140/4145 References: EMEA: K. Sdsæter nd P. Hmmond: Essentil Mthemtics for Economic Anlsis, 3rd ed., FT Prentice Hll, 2008.
More informationMODULE 3. 0, y = 0 for all y
Topics: Inner products MOULE 3 The inner product of two vectors: The inner product of two vectors x, y V, denoted by x, y is (in generl) complex vlued function which hs the following four properties: i)
More informationFor a solid S for which the cross sections vary, we can approximate the volume using a Riemann sum. A(x i ) x. i=1.
Volumes by Disks nd Wshers Volume of cylinder A cylinder is solid where ll cross sections re the sme. The volume of cylinder is A h where A is the re of cross section nd h is the height of the cylinder.
More informationVolumes of solids of revolution
Volumes of solids of revolution We sometimes need to clculte the volume of solid which cn be obtined by rotting curve bout the xxis. There is strightforwrd technique which enbles this to be done, using
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More informationQuadratic Functions. Analyze and describe the characteristics of quadratic functions
Section.3  Properties of rphs of Qudrtic Functions Specific Curriculum Outcomes covered C3 Anlyze nd describe the chrcteristics of qudrtic functions C3 Solve problems involving qudrtic equtions F Anlyze
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationWorksheet 24: Optimization
Worksheet 4: Optimiztion Russell Buehler b.r@berkeley.edu 1. Let P 100I I +I+4. For wht vlues of I is P mximum? P 100I I + I + 4 Tking the derivtive, www.xkcd.com P (I + I + 4)(100) 100I(I + 1) (I + I
More informationA.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324
A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................
More information14.2. The Mean Value and the RootMeanSquare Value. Introduction. Prerequisites. Learning Outcomes
he Men Vlue nd the RootMenSqure Vlue 4. Introduction Currents nd voltges often vry with time nd engineers my wish to know the men vlue of such current or voltge over some prticulr time intervl. he men
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More information6.2 Volumes of Revolution: The Disk Method
mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine soclled volumes of
More informationVectors 2. 1. Recap of vectors
Vectors 2. Recp of vectors Vectors re directed line segments  they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms
More informationBasic Math Review. Numbers. Important Properties. Absolute Value PROPERTIES OF ADDITION NATURAL NUMBERS {1, 2, 3, 4, 5, }
ƒ Bsic Mth Review Numers NATURAL NUMBERS {1,, 3, 4, 5, } WHOLE NUMBERS {0, 1,, 3, 4, } INTEGERS {, 3,, 1, 0, 1,, } The Numer Line 5 4 3 1 0 1 3 4 5 Negtive integers Positive integers RATIONAL NUMBERS All
More informationThe notation r(t) conforms to the notation used in the textbook. In most applications, the curves will be either
Lecture 27 Line integrls: Integrtion long curves in R n (Relevnt section from Stewrt, lculus, Erly Trnscendentls, Sixth Edition: 16.2, pp. 134141) In this section, we shll be integrting sclrvlued functions
More information9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
More informationThe Quadratic Formula and the Discriminant
99 The Qudrtic Formul nd the Discriminnt Objectives Solve qudrtic equtions by using the Qudrtic Formul. Determine the number of solutions of qudrtic eqution by using the discriminnt. Vocbulry discriminnt
More informationRadius of the Earth  Radii Used in Geodesy James R. Clynch February 2006
dius of the Erth  dii Used in Geodesy Jmes. Clynch Februry 006 I. Erth dii Uses There is only one rdius of sphere. The erth is pproximtely sphere nd therefore, for some cses, this pproximtion is dequte.
More informationAe2 Mathematics : Fourier Series
Ae Mthemtics : Fourier Series J. D. Gibbon (Professor J. D Gibbon, Dept of Mthemtics j.d.gibbon@ic.c.uk http://www.imperil.c.uk/ jdg These notes re not identicl wordforword with my lectures which will
More informationChapter G  Problems
Chpter G  Problems Blinn College  Physics 2426  Terry Honn Problem G.1 A plne flies horizonlly t speed of 280 mês in position where the erth's mgnetic field hs mgnitude 6.0µ105 T nd is directed t n
More informationLecture 15  Curve Fitting Techniques
Lecture 15  Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting  motivtion For root finding, we used given function to identify where it crossed zero where does fx
More informationSlow roll inflation. 1 What is inflation? 2 Equations of motions for a homogeneous scalar field in an FRW metric
Slow roll infltion Pscl udrevnge pscl@vudrevnge.com October 6, 00 Wht is infltion? Infltion is period of ccelerted expnsion of the universe. Historiclly, it ws invented to solve severl problems: Homogeneity:
More informationCONIC SECTIONS. Chapter 11
CONIC SECTIONS Chpter 11 11.1 Overview 11.1.1 Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig. 11.1). Fig. 11.1 Suppose we
More informationLecture 3 Basic Probability and Statistics
Lecture 3 Bsic Probbility nd Sttistics The im of this lecture is to provide n extremely speedy introduction to the probbility nd sttistics which will be needed for the rest of this lecture course. The
More informationConics and Polar Coordinates
CHAPTER 11 Conics nd Polr Coordintes Ü11.1. Qudrtic Reltions We will see tht curve defined b qudrtic reltion between the vribles x is one of these three curves: ) prbol, b) ellipse, c) hperbol. There re
More informationExperiment 6: Friction
Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht
More informationAn OffCenter Coaxial Cable
1 Problem An OffCenter Coxil Cble Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Nov. 21, 1999 A coxil trnsmission line hs inner conductor of rdius nd outer conductor
More informationLecture 5. Inner Product
Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right
More information1+(dy/dx) 2 dx. We get dy dx = 3x1/2 = 3 x, = 9x. Hence 1 +
Mth.9 Em Solutions NAME: #.) / #.) / #.) /5 #.) / #5.) / #6.) /5 #7.) / Totl: / Instructions: There re 5 pges nd totl of points on the em. You must show ll necessr work to get credit. You m not use our
More informationAnswer, Key Homework 4 David McIntyre Mar 25,
Answer, Key Homework 4 Dvid McIntyre 45123 Mr 25, 2004 1 his printout should hve 18 questions. Multiplechoice questions my continue on the next column or pe find ll choices before mkin your selection.
More informationVolumes of Revolution by Slicing
Volumes of Revolution by Slicing 72425 Strt with n re plnr region which you cn imgine s piece of crdbord. The crdbord is ttched by one edge to stick (the xis of revolution). As you spin the stick, the
More informationSection 54 Trigonometric Functions
5 Trigonometric Functions Section 5 Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form
More informationNumerical integration
Chpter 4 Numericl integrtion Contents 4.1 Definite integrls.............................. 4. Closed NewtonCotes formule..................... 4 4. Open NewtonCotes formule...................... 8 4.4
More information15.6. The mean value and the rootmeansquare value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style
The men vlue nd the rootmensqure vlue of function 5.6 Introduction Currents nd voltges often vry with time nd engineers my wish to know the verge vlue of such current or voltge over some prticulr time
More informationAAPT UNITED STATES PHYSICS TEAM AIP 2010
2010 F = m Exm 1 AAPT UNITED STATES PHYSICS TEAM AIP 2010 Enti non multiplicnd sunt preter necessittem 2010 F = m Contest 25 QUESTIONS  75 MINUTES INSTRUCTIONS DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD
More information9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes
The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is soclled becuse when the sclr product of two vectors
More informationLectures 8 and 9 1 Rectangular waveguides
1 Lectures 8 nd 9 1 Rectngulr wveguides y b x z Consider rectngulr wveguide with 0 < x b. There re two types of wves in hollow wveguide with only one conductor; Trnsverse electric wves
More informationExponential and Logarithmic Functions
Nme Chpter Eponentil nd Logrithmic Functions Section. Eponentil Functions nd Their Grphs Objective: In this lesson ou lerned how to recognize, evlute, nd grph eponentil functions. Importnt Vocbulr Define
More informationContinuous probability distributions
Chpter 8 Continuous probbility distributions 8.1 Introduction In Chpter 7, we explored the concepts of probbility in discrete setting, where outcomes of n experiment cn tke on only one of finite set of
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationOn the Meaning of Regression Coefficients for Categorical and Continuous Variables: Model I and Model II; Effect Coding and Dummy Coding
Dt_nlysisclm On the Mening of Regression for tegoricl nd ontinuous Vribles: I nd II; Effect oding nd Dummy oding R Grdner Deprtment of Psychology This describes the simple cse where there is one ctegoricl
More informationLecture 3.1 Scalars and Vectors, Kinematics in Two and Three Dimensions
1. Sclrs n Vectors Lecture 3.1 Sclrs n Vectors, Kinemtics in Two n Three Dimensions Phsics is quntittive science, where everthing cn be escribe in mthemticl terms. As soon s the sstem of units hs been
More informationA5682: Introduction to Cosmology Course Notes. 4. Cosmic Dynamics: The Friedmann Equation. = GM s R 2 s(t).
4. Cosmic Dynmics: The Friedmnn Eqution Reding: Chpter 4 Newtonin Derivtion of the Friedmnn Eqution Consider n isolted sphere of rdius R s nd mss M s, in uniform, isotropic expnsion (Hubble flow). The
More informationAMTH247 Lecture 16. Numerical Integration I
AMTH47 Lecture 16 Numericl Integrtion I 3 My 006 Reding: Heth 8.1 8., 8.3.1, 8.3., 8.3.3 Contents 1 Numericl Integrtion 1.1 MonteCrlo Integrtion....................... 1. Attinble Accurcy.........................
More information1 SuperBrief Calculus I Review.
CALCULUS MATH 66 FALL 203 (COHEN) LECTURE NOTES For the purposes of this clss, we will regrd clculus s the study of limits nd limit processes. Without yet formlly reclling the definition of limit, let
More informationTheory of Forces. Forces and Motion
his eek extbook  Red Chpter 4, 5 Competent roblem Solver  Chpter 4 relb Computer Quiz ht s on the next Quiz? Check out smple quiz on web by hurs. ht you missed on first quiz Kinemtics  Everything
More informationEQUATIONS OF LINES AND PLANES
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in pointdirection nd twopoint
More informationLINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
More informationwww.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values)
www.mthsbo.org.uk CORE SUMMARY NOTES Functions A function is rule which genertes ectl ONE OUTPUT for EVERY INPUT. To be defined full the function hs RULE tells ou how to clculte the output from the input
More information