Sum: lim( f + g) = lim f + limg. Difference: lim( f g) = lim f limg. Product: lim( f g) = (lim f )(limg)

MATH 5: Clculus, SET8 SUMMARIES [Belmonte, 207] 2 Limits nd Derivtives 2.2 The Limit of Function Limit of sclr function: We write lim x f (x) = L nd sy the limit of f (x) s x pproches equls L if nd only if we cn mke vlues of f (x) rbitrrily close to L by choosing x to be sufficiently close to (but not equl to ). We write f (x) L s x nd sy f (x) pproches L s x pproches. Left-hnd limit: lim f (x) = L; sme ide s bove but with x x < ; i.e., x pproches through vlues less thn. Right-hnd limit: lim f (x) = L; sme ide s bove but x + with x > ; i.e., x pproches through vlues greter thn. Theorem: lim f (x) = L if nd only if lim f (x) = L nd x x lim f (x) = L; i.e., the two-sided limit exists if nd only if x + the one-sided limits exist nd re equl. (This lso pplies to infinite limits, which we now describe.) Positive infinite limit: lim x f (x) = if nd only if vlues of f (x) cn be mde rbitrrily positively lrge by choosing x to be sufficiently close to (but not equl to ). We lso write f (x) s x nd sy f (x) tends to positive infinity s x pproches. Note tht is not rel number. The limit symbolism is just shorthnd description of behvior. Negtive infinite limit: lim x f (x) = if nd only if vlues of f (x) cn be mde rbitrrily negtively lrge by choosing x to be sufficiently close to (but not equl to ). We lso write f (x) s x nd sy f (x) tends to negtive infinity s x pproches. Agin, is not rel number. The limit symbolism is shorthnd description of behvior. We lso hve one-sided infinite limits: lim f (x) =, x f (x) =, lim f (x) =, lim f (x) =. + + lim x x x The line x = is verticl symptote provided one of the six forementioned infinite limits is mnifested. Limit of vector function: Limits of r(t) s t pproches re formulted in mnner nlogous to the preceding. Tht sid, from prcticl perspective, the limit the vector is the vector of the limits. For exmple, let r(t) = [x(t), y(t)] nd suppose x(t) [ b nd y(t) ] b 2 s t. Then we hve lim r(t) = lim x(t),limy(t) = [b,b 2 ] = b. t t t 2.3 Clculting Limits Using the Limit Lws Let nd c be rel constnts with f (x) nd g(x) functions of x. Suppose both lim f nd limg exist s x, x, or x +. Sum: lim( f + g) = lim f + limg Difference: lim( f g) = lim f limg Product: lim( f g) = (lim f )(limg) Quotient: lim( f /g) = (lim f )/(limg), provided limg 0 Constnt: lim(c f ) = c(lim f ) Power: lim( f n ) = (lim f ) n, where n is positive integer Root: lim ( n f ) = n lim f, where n is positive integer (If n is even, then we further ssume lim f 0.) Two specific limits: limc = c, limx = Direct substitution: lim f = f () for in the domin of polynomil or rtionl function. Monotonicity: Let f g in open intervl contining (except perhps t itself). If lim f = L nd limg = M, then L M. Squeeze Theorem: Let f g h in open intervl tht contins (except perhps t itself). If lim f = L = limh, then limg = L. Vector limits: The sum, difference, product, nd constnt lws bove hold for vector functions f nd g of t. The product opertion involved is the dot product. 2.5 Continuity A function f is continuous t if lim x f (x) = f (). It is continuous on n intervl if it is continuous t every point in the intervl. Geometriclly, the function s grph hs no breks or holes in it. You cn drw it without lifting your pencil from the pper. A function f is discontinuous t if it is not continuous there. Be fmilir with jump, removble, nd infinite discontinuities. A function f is continuous from the left t if we hve lim f (x) = f (). It is continuous from the right t if x lim f (x) = f (). x + Continuity nd the lgebr of functions: Let f nd g be continuous t nd let c be constnt. Then the functions f + g, f g, c f, f g, nd f /g re lso continuous t. [For f /g, we require g() 0.] The following types of functions re continuous on their domins: polynomil, rtionl, root, trigonometric, inverse trigonometric, exponentil, logrithmic. Continuity of composite function: If g is continuous t nd f is continuous t g(), then the composition ( h = f ) g is continuous t ; so lim f (g(x)) = f (g()) = f lim g(x). x x

Intermedite Vlue Theorem: If f is continuous on [,b], f () f (b), nd N is between f () nd f (b), then f (c) = N for some c in (,b). Continuity of vector functions: A vector function r is continuous t if lim t r(t) = r(). The continuity of r(t) = [x(t),y(t)] t is equivlent to the continuity of its constituent functions x nd y t. 2.6 Limits t Infinity; Horizontl Asymptotes Limit t infinity: Let f be function defined on (, ). We write lim f (x) = L or f L s x nd sy tht the limit x of f s x pproches is L. This signifies tht the vlues of f cn be mde s close to L s we plese by tking sufficiently lrge positive vlues of x. Similrly, lim x f (x) = L provided vlues of f cn be mde s close to L s we like by tking sufficiently lrge negtive vlues of x. In ech of the preceding cses, the line y = L is clled horizontl symptote of the curve y = f (x). Most limit lws in Section 2.3 pply to limits t infinity. The symbolism lim f (x) = mens tht vlues of f become x rbitrrily lrge once x becomes sufficiently positively lrge. The following hve nlogous connottions: lim f (x) =, x lim f (x) =, lim f (x) =. x x 2.7 Derivtives nd Rtes of Chnge These ides lso pply to vector functions. Sy r(t) gives the position of prticle s function of time. Then verge rte of chnge of r with respect to t is verge velocity nd instntneous rte of chnge is instntneous velocity. 2.8 The Derivtive s Function We sy tht function f is differentible t if f () exists. The set {(x, f (x))} of ll ordered pirs such tht f is differentible t x constitutes the derivtive function f. Other nottions for the derivtive of y = f (x) re f (x) = y = dy dx = d f dx = d dx ( f (x)) = D f (x) = D x f (x) where D nd d re termed differentil opertors. dx Theorem: If f is differentible t, then f is continuous t. Geometriclly, if f is differentible t, its grph is smooth there (but not verticl). If f is not differentible t b, its grph my exhibit kink, cusp, or discontinuity of some sort. If f is differentible function, its derivtive function f my lso hve derivtive function, denoted by f or d2 f dx 2 nd clled the second derivtive. Similrly, f my hve derivtive function, denoted by f or d3 f nd clled the third derivtive, nd so on. For n 4, dx3 the n th derivtive is typiclly denoted f (n) or dn f dx n. Using vector functions, let r be position. Then v = r is velocity, = v = r is ccelertion, nd = r is jerk. The derivtive of function f t, denoted f (), is defined s f f ( + h) f () () = lim, provided this limit exists. h 0 h Equivlently, f f (t) f () () = lim, by setting t = + h. t t Let y depend on x. Then y is function of x nd we write y = f (x). As x chnges from x to x 2, the chnge in x is x = x 2 x. The chnge in y is y = f (x 2 ) f (x ). Here re rtes of chnge (r.o.c.) of y with respect to x. verge r.o.c = y x = f (x 2) f (x ) x 2 x instntneous r.o.c = y lim x 0 x = lim f (x 2 ) f (x ) x 2 x x 2 x Note the instntneous rte of chnge is the derivtive f (x ). Consider the grph of y = f (x) nd denote y k = f (x k ). The verge rte of chnge bove is the slope of secnt line t (x,y ) through nerby point (x 2,y 2 ). The instntneous rte of chnge is the slope of the tngent line to the curve y = f (x) t the point (x, f (x )); i.e., the derivtive f (x ). 2

MATH 5: Clculus, SET8 SUMMARIES [Belmonte, 207] 3 Differentition Rules Let c nd r be rel constnts, x rel vrible, nd f nd g rel-vlued differentible functions of x. Let e x = exp(x) be the nturl exponentil function. Its bse e 2.7828 is the number e h such tht lim = 0. Below d h 0 h dx nd signify differentition. 3. Derivtives of Polynomils nd Exponentil Functions Derivtive of Constnt Function: d dx (c) = 0 Power Rule: d dx (x r ) = rx r (r 0; if r = 0, see bove.) Constnt Multiple Rule: (c f ) = c f Sum Rule: ( f + g) = f + g Difference Rule: ( f g) = f g Use preceding rules to differentite polynomil functions. Derivtive of exp(x): d dx (e x ) = e x. 3.2 Product nd Quotient Rules Product Rule: ( f g) = f g + f g Quotient Rule: ( ) f = g f f g g g 2 3.3 Derivtives of Trigonometric Functions Limits: lim θ 0 sinθ θ =, lim θ 0 cosθ θ f (x) sin x tn x sec x f (x) cosx sec 2 x secxtnx g(x) cosx cotx cscx g (x) sinx csc 2 x cscxcotx 3.4 The Chin Rule = 0. Derivtives: In the following, the requisite derivtives re ssumed to exist. Chin Rule: Let h = f g; i.e., h(x) = f (g(x)). Then we hve h (x) = f (g(x))g (x). Equivlently, let y = f (u) nd u = g(x). Then dy dx = dy du du dx. Generl Power Rule: Let r be nonzero rel number. Setting y = u r bove, we hve d dx (ur ) = ru r du dx. Derivtive of generl exponentil function: d dx (bx ) = b x lnb, where b > 0 with b is the bse. 3.5 Implicit Differentition Let F (x,y) = G(x,y) implicitly define y s function of x. Here re three wys to compute dy/dx without explicitly solving for y in terms of x (which my be impossible). Clc method: Differentite the eqution with respect to x, then solve for dy/dx. (This is tedious nd error-prone.) Clc 3 method: Let H(x,y) = F (x,y) G(x,y). Then dy dx = H/ x, which involves prtil derivtives. To H/ y compute H H, regrd y s constnt. To compute x y, regrd x s constnt. (This is fster becuse you hve formul.) MATLAB: A locl function implements the Clc 3 wy! function dy_dx = impdiff(eq,x,y) H = lhs(eq) - rhs(eq); dy_dx = simplify(-diff(h,x) / diff(h,y)); end Derivtives of inverse trigonometric functions f (x) sin x tn x sec x f (x) x 2 + x 2 x x 2 g(x) cos x cot x csc x g (x) x 2 + x 2 x x 2 3.6 Derivtives of Logrithmic Functions Consider the log b function with bse b > 0 where b. For b = e this is the nturl logrithm function log e = ln. Let u = g(x) be differentible function. Derivtives f (x) log b x lnx ln x lnu lng(x) f (x) xlnb x x du u dx g (x) g(x) Logrithmic Differentition: This technique mkes it esier to differentite function tht involves products, powers, nd/or quotients.. Tke nturl logrithms of both sides of n eqution y = f (x). Use lws of logrithms to simplify. 2. Differentite with respect to x. 3. Solve for y = dy/dx. ( Representtions: e = lim ( + x) /x = lim + n. x 0 n n)

3.7 Rtes of Chnge in the Nturl nd Socil Sciences The derivtive s rte of chnge hs pplicbility in engineering, physics, chemistry, geology, biology, economics, sociology, etc. 3.8 Exponentil Growth nd Decy The lw of nturl growth/decy sttes tht the time rte of chnge of quntity y is proportionl to y itself. The constnt of proportionlity k is positive for growth nd negtive for decy. Problems re expressed s differentil eqution together with n initil condition. dy dt = ky, y(0) = y 0 The solution is y(t) = y 0 e kt. We work from this solution. 3.9 Relted Rtes A given quntity Q(t) my be relted to one or more other quntities. Here t represents time. An eqution expressing the reltionship my be explicit or implicit. We differentite it with respect to t, then solve for dq/dt. This revels how the rte of chnge of Q is relted to the rtes of chnge of the other quntities. Drwing digrm helps to formulte reltionship between the quntities involved. 3.0 Liner Approximtion nd Differentils Let f be differentible t. The tngent line to the grph of f t is given by the liner function L(x) = f () + f ()(x ), clled the lineriztion of f t. Ner, the tngent line lies very close to the curve represented by f. The pproximtion f (x) L(x) = f () + f ()(x ) is clled the liner pproximtion or lterntively the tngent line pproximtion of f t. The differentil dx is n independent vrible which cn tke on ny rel vlue. The differentil dy, defined by dy = f (x) dx, is vrible tht depends on x nd dx. The geometricl significnce of dy is tht ner point on the grph of y = f (x), the ctul chnge y in the function long the curve is pproximtely equl to the chnge dy long the tngent line f (x + dx) f (x) = y dy = f (x) dx, which implies f (x + dx) f (x) + f (x) dx. 2

MATH 5: Clculus, SET8 SUMMARIES [Belmonte, 207] 4 Applictions of Differentition 4. Mximum nd Minimum Vlues Let c be in D, the domin of function f. Ner c mens tht c is in some open intervl contined in D. We chrcterize the vlue f (c) s follows. (The symbol mens for ll. ) bsolute mximum vlue of f on D if f (c) f (x) x D. bsolute minimum vlue of f on D if f (c) f (x) x D. locl mximum vlue of f if f (c) f (x) for x ner c. locl minimum vlue of f if f (c) f (x) for x ner c. An extremum is mximum or minimum; plurl: extrem. Extreme Vlue Theorem: If f is continuous on [,b], then f hs bsolute mximum & bsolute minimum vlues on [,b]. Fermt s Theorem: Suppose f hs locl extremum t c. If f is differentible t c, then f (c) = 0. A criticl number c D is such tht either f (c) = 0 or else f (c) does not exist. If f (c) is locl extremum, then c is criticl number of f. Use the Closed Intervl Method to find bsolute extrem of f on [,b].. Find the criticl numbers of f in (,b). 2. Evlute f t criticl numbers nd t the endpoints,b. 3. The lrgest function vlue you obtin is the bsolute mximum; the smllest is the bsolute minimum. 4.2 The Men Vlue Theorem Men Vlue Theorem: Let f be continuous on [,b] nd differentible on (,b). Then f f (b) f () (c) = for some b c (,b). [The slope of the tngent line t c equls the slope of the secnt line between (, f ()) nd (b, f (b)).] If f (x) = 0 for ll x (,b), then f is constnt on (,b). If f (x) = g (x) on (,b), then f (x) = g(x) + c on (,b), where c is constnt. 4.3 How Derivtives Affect the Shpe of Grph Incresing/Decresing Test () If f (x) > 0 on (,b), then f is incresing on (,b). (b) If f (x) < 0 on (,b), then f is decresing on (,b). First Derivtive Test: Let c be criticl number of continuous function f. Let x increse through c. () If f goes from + to, then f (c) is locl mximum. (b) If f goes from to +, then f (c) is locl minimum. (c) If there is no chnge sign in f, then f (c) is not locl extremum. Concve upwrd: where the grph of f lies bove ll its tngents on n intervl I. Concve downwrd: where the grph of f lies below ll its tngents on n intervl I. Concvity Test () If f (x) > 0 on I, then f is concve upwrd on I. (b) If f (x) < 0 on I, then f is concve downwrd on I. Inflection Point: A point P on the grph of f where f is continuous nd concvity chnges from upwrd to downwrd or else from downwrd to upwrd. Second Derivtive Test: Suppose f is continuous ner c. () If f (c) = 0 nd f (c) > 0, then f (c) is locl min. (b) If f (c) = 0 nd f (c) < 0, then f (c) is locl mx. 4.4 Indeterminte Forms nd l Hospitl s Rule Quotients: 0 0, ± ± Products: (0)(± ) Difference: L Hospitl s Rule: Let f nd g be differentible on n open intervl I (except perhps t ) with g (x) 0. Let f /g be n indeterminte quotient. Then lim f (x) g(x) = lim f (x) g (x) provided the ltter limit exists or else is ±. For indeterminte products or differences, first convert to indeterminte quotients, then pply l Hospitl s Rule. Indeterminte power: 0 0 0 For indeterminte powers, proceed s follows.. Lbel the limiting expression: y = f (x) g(x). 2. Tke nturl logs of ech side: lny = g(x)ln f (x). 3. Evlute the indeterminte product. lim(lny) = lim(g(x)ln f (x)) = L 4. Then lim f (x) g(x) = e L. Here s why ( lim f (x) g(x) = limy = lim e lny) = e (limlny) = e L since the exponentil function is continuous.

4.5 Summry of Curve Sketching When sketching curve by hnd, the following concepts re relevnt. (Tht sid, technology enbles you to plot curves nd identify fetures rpidly nd with miniml effort!) A. Domin B. Intercepts C. Symmetry D. Asymptotes E. Intervls of increse or decrese F. Locl extrem G. Concvity nd points of inflection. H. Discontinuities 4.7 Optimiztion Problems Anlyze the problem. Drw digrm nd/or introduce suitble nottion nd vribles. Come up with function of one vrible to be mximized or minimized. Then employ clculus to solve the problem. Grphing my lso be useful. First Derivtive Test for Absolute Extreme Vlues Let f be continuous on n intervl I. Suppose c I is criticl number of f. () If f > 0 for x < c nd f < 0 for x > c, then f (c) is the bsolute mximum of f. (b) If f < 0 for x < c nd f > 0 for x > c, then f (c) is the bsolute minimum of f. 4.9 Antiderivtives A function F is n ntiderivtive of f on n intervl I if F (x) = f (x) for ll x in I. If F is n ntiderivtive of f, the most generl ntiderivtive of f on I is F (x) +C where C is rbitrry constnt. A short tble of ntiderivtives ppers below. For bigger list, see REFERENCE pges 6 0 in your textbook fter the index. (Or use MATLAB s int commnd.) Function Antiderivtive Function Antiderivtive c f (x) cf (x) cosx sinx f (x) + g(x) F (x) + G(x) sinx cosx x n (n ) x n+ n + sec 2 x tnx x ln x secxtnx secx e x e x csc 2 x cotx b x b x lnb cscx cotx cscx x 2 sin x + x 2 tn x 2

MATH 5: Clculus, SET8 SUMMARIES [Belmonte, 207] 5 Integrls 5. Ares nd Distnces Are: In 5.2 (below), if f 0 on [,b], then f (x) dx gives the re of the region bounded by the grph of f, the verticl lines x = nd x = b, nd the x-xis. Distnce: In 5.2 (below), if v = f (t) 0 on [,b] is velocity, then f (t) dt is the distnce trveled between times nd b. 5.2 The Definite Integrl Definite Integrl: Let f be function defined for x b. Split [,b] into n subintervls of equl width x = (b )/n with endpoints x i = + i x, for i = 0,,2,...,n. (Note tht x 0 = nd x n = b.) Let xi [x i,x i ] be ny smple points. The definite integrl of f from to b is defined by the limit f (x) dx = lim n n i=i f (x i ) x, provided the sme rel vlue is obtined for ll possible choices of smple points. If smple points re midpoints of the subintervls, the sum M n = Σ f (xi ) x gives the Midpoint Rule. Left or right endpoints yield left or right sum rules L n nd R n, respectively. Know these terms from your book: integrble, integrl sign, integrnd, integrtion, lower nd upper limits of integrtion. Study Appx E: Sigm Nottion. Use MATLAB s symsum. Functions tht re continuous, piecewise continuous, or monotonic on [,b] re integrble. Properties of the Definite Integrl Let c, m, nd M be constnts. Let f nd g be integrble on [,b].. 2. cdx = c(b ) f (x) + g(x) dx = f (x) dx + g(x) dx 3. c f (x) dx = c f (x) dx 4. f (x) g(x) dx = f (x) dx g(x) dx 5. f (x) dx = c f (x) dx + c f (x) dx 6. If f 0 on [,b], then f (x) dx 0. 7. If f g on [,b], then f (x) dx g(x) dx. 8. m f M on [,b] m(b ) 9. If b =, then f (x) dx = 0. 0. If > b, then f (x) dx = b f (x) dx. f (x) dx M (b ) 5.3 The Fundmentl Theorem of Clculus Let f be continuous on [,b]. The prts below re FTC & FTC2.. The function g(x) = x f (t) dt is continuous on [,b] nd differentible on (,b) with g (x) = f (x). 2. f (x) dx = F (b) F (), where F is n ntiderivtive of f ; tht is, F = f. So differentition nd integrtion re seen to be inverse processes. FTC2 mkes it esier to compute definite integrls by hnd. Or use the int commnd in MATLAB to compute them utomticlly! 5.4 Indefinite Integrls nd the Net Chnge Theorem The indefinite integrl f (x) dx denotes n ntiderivtive of f. So f (x) dx = F (x) mens F (x) = f (x). Net Chnge Theorem: The integrl of rte of chnge is the net chnge: F (x) = F (b) F (). [This is widely used in the nturl nd socil sciences.] For tble of integrls, see REFERENCE pges 6 0 in your textbook fter the index. (Or use MATLAB s int commnd.) 5.5 The Substitution Rule Frmework: Let u = g(x) be differentible function of x whose rnge is n intervl I. Let f be continuous on I. The differentil dx is n independent vrible nd the differentil du is defined s du = g (x) dx. Substitution Rule for Indefinite Integrls: With u nd f s bove, we hve f (g(x))g (x) dx = f (u) du. Substitution Rule for Definite Integrls: If g is continuous on [,b] nd f is continuous on the rnge of u = g(x), then f (g(x))g (x) dx = g(b) g() f (u) du. Integrls of Symmetric Functions: Suppose f is continuous on [,]. Even: f ( x) = f (x); odd: f ( x) = f (x). () If f is even, then f (x) dx = 2 0 f (x) dx. (b) If f is odd, then f (x) dx = 0. 6. Ares Between Curves The re between the curves y = f (x) nd y = g(x) between verticl lines x = nd x = b is A = f (x) g(x) dx. The re between the curves x = f (y) nd x = g(y) between horizontl lines y = c nd y = d is A = d c f (y) g(y) dy. Compute integrls by resolving bsolute vlue s f g or g f on subintervls if necessry.

MATH 5: Clculus, SET8 SUMMARIES [Belmonte, 207] 0 Prmetric Equtions 0. Curves Defined by Prmetric Equtions Terms nd Concepts Vector function r(t) = [x(t), y(t)]: Here the rel-vlued prmeter t vries over suitble domin, either specified or the lrgest subset of rel numbers for which the constituent expressions x(t) nd y(t) re defined. A prmeterized curve in the xy-plne is the geometric reliztion of vector function. Vector eqution of line L(t) = r 0 +tv: Here t cn tke on ny rel vlue. This is specil cse of the preceding item, with the constituent expressions x(t) nd y(t) being liner in t; i.e., first degree polynomils in t. Red on to see this. Prmetric equtions of line: Here x(t) = x 0 + t nd y(t) = y 0 + bt, with (x 0,y 0 ), point on the line identified with the position vector r 0 = [x 0,y 0 ], nd v = [,b] giving direction of the line. Prmetric line segment from A(x 0,y 0 ) to B(x,y ): r(t) = A +t (B A), 0 t. Here A = [x 0,y 0 ] nd B = [x,y ] re the respective position vectors. Clerly when t = 0 we re t A nd when t = we re t B. Moreover, the constituent expressions x(t) = x 0 + (x x 0 )t nd y(t) = y 0 + (y y 0 )t re liner in t, whence line segment. If d 2 y/dx 2 < 0 on n open intervl, the curve is concve down on the intervl. The curve hs n inflection point t P if d 2 y/dx 2 chnges sign t P. Derivtives of Vector Functions Let r(t) = [x(t), y(t)], where x(t) nd y(t) re differentible functions of t. Tngent vector to r(t) t r() is v() = r r(t) r() r( + h) r(0) () = lim = lim. t t h 0 h This is, of course, the derivtive of r t. Regrding the prmeter t s time nd r(t) s position vector, the tngent vector is clled the velocity vector. Hence velocity is the derivtive of position. Speed is the mgnitude of the velocity vector, v = r. Tngent line to curve r(t) t r(): L(t) = r() +tv(). Vector derivtive properties [ f,g] = [ f,g ] (u + w) = u + w (u w) = u w (cw) = cw (u w) = u w + u w 0.2 Clculus with Prmetric Curves Tngents to Prmetric Curves First Derivtive: Let x = f (t) nd y = g(t) be differentible functions of t where y is lso differentible function of x. Vi the Chin Rule, dy dt = dy dx dy, whence dx dt dx = dy/dt dx/dt if dx/dt 0. A horizontl tngent line (the slope of which is zero) occurs where dy/dt = 0 nd dx/dt 0. A verticl tngent line (the slope of which is infinite ) occurs where dx/dt = 0 nd dy/dt 0. Second Derivtive: Similrly, d 2 y dx 2 = d ( ) d dy = dt (dy/dx). dx dx dx/dt If d 2 y/dx 2 > 0 on n open intervl, the curve is concve up on the intervl.

MATH 5: Clculus, SET8 SUMMARIES [Belmonte, 207] 2 Vectors; Geometry of Spce 2.2 Vectors [2-dimensionl] Terms nd Concepts Vector: A quntity hving mgnitude (length) nd direction. Geometriclly, n equivlence clss of directed line segments hving the sme mgnitude nd direction (nlogy: 2 nd 2 4 re equivlent frctions). A prticulr member of this clss hs strting point (the til of the vector) nd n ending point (the hed of the vector). Anlyticlly, n ordered pir of components (rel numbers or symbols): = [, 2 ]. While Stewrt uses ngle brckets... to delimit vectors, most folks use squre brckets; so shll we! We lso write s when writing the nme of vector by hnd. Position vector: The distinguished member of the equivlence clss tht strts t the origin. Displcement vector: AB, the vector from A(, 2 ) to B(b,b 2 ), relized s [b,b 2 2 ], end strt slotwise. Mgnitude: The length of vector = [, 2 ] is given by = 2 + 2 2 vi the Pythgoren Theorem. Therefore, AB = (b ) 2 + (b 2 2 ) 2. Zero vector: The vector 0 = [0,0], whose length is zero nd which hs no specific direction. Vector ddition / vector sum: Add components slotwise. + b = [, 2 ] + [b,b 2 ] = [ + b, 2 + b 2 ]. Tringle Lw / Prllelogrm Lw: The geometric (hed-to-til) interprettion of vector ddition. Sclr: rel number or symbol. Sclr multipliction: Given sclr c nd vector, the sclr multiple of c with is obtined by multiplying ech component of by c; i.e., c = c[, 2 ] = [c,c 2 ]. Vector subtrction / vector difference: Formlly, b = + ( ) b. Just subtrct components slotwise: b = [, 2 ] [b,b 2 ] = [ b, 2 b 2 ]. Unit vector: A vector whose mgnitude is. Given nonzero vector 0, the unit vector in the direction of is - ht., given by â =, often written. Stndrd bsis vectors: In V 2, the set of of ll 2-D vectors, these re the vectors i = [,0] nd j = [0,]. They form n orthonorml bsis; i.e, they re perpendiculr unit vectors in terms of which ny vector in V 2 my be expressed. = [, 2 ] = [,0] + [0, 2 ] = [,0] + 2 [0,] = i + 2 j Resultnt vector: The vector sum of severl vectors; e.g., the resultnt force is the sum of severl forces, wheres the resultnt velocity is the sum of severl velocities. Vector Properties Let c nd d be sclrs nd, b, c be vectors. We hve these rules.. + b = b + (commuttivity of vector ddition) 2. + (b + c) = ( + b) + c (ssocitivity of vector ddition) 3. + 0 = (The zero vector is the dditive identity.) 4. + ( ) = 0 (The dditive inverse of is.) 5. c( + b) = c + cb (Sclr multipliction distributes over vector ddition.) 6. (c + d) = c + d (nother instnce of distributivity) 7. (cd) = c(d) (ssocitivity of sclr multipliction) 8. = ( is multiplictive identity for sclr multiplction) 2.3 The Dot Product Terms nd Concepts Let = [, 2 ] nd b = [b,b 2 ] be 2-D vectors. Dot product (mth definition): b = b + 2 b 2. Dot product (physics definition): b = b cosθ, where θ is the ngle between vectors nd b; equivlent to mth definition by the Lw of Cosines. Angle between nonzero vectors: θ = cos ( b b Orthogonlity: Nonzero vectors nd b re perpendiculr if nd only if b = 0. Sclr projection of b onto : comp b = b. ( ) b Vector projection of b onto : proj b =, the sclr projection times the unit vector in the direction of ; lso known s the prllel projection. Orthogonl projection of b onto : orth b = b proj b, whence b is the sum of vector prllel to nd vector perpendiculr to. Orthogonl complement: = [ 2, ], red - perp since it is perpendiculr to. Work: W = F D = F D cosθ, where F is the (constnt) force vector nd D is the displcement vector. ).