Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x) = L. x Definition of Continuity: The function f(x) is continuous t x = if f() exists nd lim x f(x) = f(). Limits t Infinty: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x increses without bound. We write lim f(x) = L. Similrly x for x. Rules for Computing Limits: Assuming tht ll the limits exist: ( ) lim f(x) ± g(x) = lim f(x) + lim g(x) x x x ( )( ) lim f(x)g(x) = lim f(x) lim g(x) x x x f(x) lim f(x) lim x g(x) = x lim g(x) x provided tht lim x g(x) 0. Sum Product Quotient A Specil Limit: lim θ 0 sin θ θ = 1. f(x) L Hôpitl s Rule: If lim f(x) = 0 nd lim g(x) = 0, then lim x x x g(x) = lim x f(x) Similrly, if lim f(x) = ± nd lim g(x) = ±, then lim x x x [2] Derivtives g(x) = lim x f (x) g (x). f (x) g (x). Definition of Derivtive: The derivtive of the function y = f(x) is defined to be dy = f f(x + h) f(x) (x) = lim. h 0 h The derivtive f (x) mesures the rte t which the function f(x) is chnging. Bsic Differentition Rules: If u nd v re functions of x, nd if c is constnt, then d(u ± v) d(uv) d(u/v) d(u n ) = du ± dv = u dv + v du v du = u dv v 2 du = nun 1 Addition Rule Product Rule Quotient Rule Power Rule Mth 9 Course Summry/Study Guide (1) Fll, 2005 prepred by J.H. Silvermn
The Chin Rule: If y = f(u) nd u = g(x), then y = f(g(x)) my be viewed s function of x. The Chin Rule sys tht dy = dy du du. If we let h(x) = f(g(x)), then nother wy to write the sme rule is h (x) = f (g(x))g (x). Higher Order Derivtives: The second derivtive of y = f(x) is the derivtive of the derivtive of f(x). Common nottions for the second derivtive re d2 y 2 nd f (x). Similrly, the n th derivtive of y = f(x) is obtined by differentiting n times. Nottions for the n th derivtive re dn y n nd f (n) (x). Implicit Differentition: If x nd y re relted by eqution, you cn find formul for dy/ by differentiting both sides of the eqution using differentition rules nd the chin rule, nd then solving for dy/. The nswer will generlly be n expression tht involves both x nd y. Logrithm Differentition: If y = f(x), sometimes it is esier to first write ln(y) = ln(f(x)), use rules of logrithms to simplify ln(f(x)), nd then differentite implicitly. In cses where y hs the form f(x) g(x), this my be the only method tht works. Derivtives of Some Specil Functions: d sin(x) d tn(x) = cos(x) d cos(x) = sin(x) = sec 2 (x) d sec(x) = sec(x) tn(x) Trigonometric Functions d sin 1 (x) = 1 1 x 2 d cos 1 (x) 1 = 1 x 2 d tn 1 (x) = 1 d sec 1 (x) 1 1 + x 2 = x x 2 1 Inverse Trigonometric Functions d ln(x) = 1 de x x = ex d log (x) = 1 d x x ln = x ln() Logs nd Exponentils [3] Further Differentil Clculus Topics The Men Vlue Theorem: The Men Vlue Theorem sys tht if f(x) is differentible, then there is lwys some c between nd b for which f (c) = f(b) f(). b Mth 9 Course Summry/Study Guide (2) Fll, 2005 prepred by J.H. Silvermn
The geometric interprettion of this formul is tht there is point on the curve y = f(x) between x = nd x = b where the tngent line is prllel to the line connecting (, f()) to (b, f(b)). Differentils: The differentil of y = f(x) is dy = f (x). The quntity is n independent vrible which we usully tke to be quite smll. The differentil dy is n pproximtion of how much f chnges when x increses by. This gives the liner pproximtion formul f(x + x) f(x) + f (x) x [4] Applictions of the Derivtive Slopes nd Tngent Lines: The slope of the curve y = f(x) t the point (, f()) is f (). The tngent line to the curve y = f(x) t the point (, f()) is y f() = f ()(x ). Relted Rtes Problems: In relted rtes problem, two or more quntities which re chnging in time re relted by equtions. The first step is to give nmes to quntities nd write down reltions ( sketch is often helpful). You re generlly told how fst some of the quntities re chnging, nd re sked to find how fst the remining qunitity is chnging. This is done by differentiting the reltions with respect to time (using the chin rule), substituting the known vlues into the differentited equtions, nd then solving for the unknown vlue. Mximum/Minimum Problems: In mx/min problem, you re given quntity, sy Q, which is to be mximized for minimized. Do the following steps: (1) Use the given informtion ( sketch is often helpful) to express Q in terms of single vrible, sy x. (2) Find the criticl vlues by computing the derivtive Q (x) nd solving the eqution Q (x) = 0. (3) For ech criticl vlue, determine whether is mximum, minimum, or neither. This cn be done by the chnge-of-sign test: If Q (x) goes from plus to minus s x psses, then gives locl mximum. If Q (x) goes from minus to plus s x psses, then gives locl minimum. If Q (x) doesn t chnge sign s x psses, then is neither mx nor min. Another wy to check is using the second derivtive test: If Q () > 0, then gives locl minimum. If Q () < 0, then gives locl mximum. If Q () = 0, then the second derivtive test gives no informtion. (4) If the vrible x is required to be between nd b, compute Q() nd Q(b), the vlues t the endpoints. (5) Compre the vlues of Q(x) t the criticl points nd t the endpoints nd pick out the boslute mximum nd/or the bsolute minimum. Curve Sketching: To sketch the grph of function y = f(x), do the following steps: (1) Compute f (x) nd determine where: f (x) > 0 the curve is rising. f (x) < 0 the curve is flling. f (x) = 0 the curve my hve locl mx or min. Mth 9 Course Summry/Study Guide (3) Fll, 2005 prepred by J.H. Silvermn
(2) Compute f (x) nd determine where: f (x) > 0 the curve is concve up. f (x) < 0 the curve is concve down. f (x) = 0 the curve my hve n inflection point. (3) Determine vlues of for which lim f(x) = ±. These re verticl symptotes. x ± They often occur t points where the denomintor of f(x) vnishes. (4) Compute lim x f(x) nd lim x f(x). If these limits exist, they give horizontl symptotes. (5) Drw nd lbel ll mxim, minim, inflection points, nd symptotes. Add few other points to your grph (for exmple, the point (0, f(0)) nd points where y = 0). (6) Using these mrked points nd your knowledge of where the grph is rising, flling, concve up, nd concve down, sketch the grph. Newton s Method: Newton s Method is wy of finding solutions to f(x) = 0. Strting with n initil guess x = x 1, one produces list of vlues x 1, x 2, x 3,... by the generl formul x n+1 = x n f(x n) f (x n ). This list of vlues (usully) converges quite rpidly to solution. [5] Integrtion Riemnn Sums nd Definite Integrls: First brek the intervl from to b into n pieces, ech of width x = (b )/n. Next choose n x-vlue in ech little piece, sy x 1 in the first piece, x 2 in the second piece, nd so on. The Riemnn sum of f is then n f(x i ) x. i=1 The definite integrl of f from to b is the limit of the Riemnn sums s the number of pieces goes to infinity nd the width of ech little piece goes to zero: Bsic Integrtion Rules: f(x) = lim n n f(x i ) x. i=1 ( ) b f(x) ± g(x) = f(x) ± f(x) = c f(x) + c g(x) f(x) Anti-Derivtives nd Indefinite Integrls: A function F (x) is clled n nti-derivtive for f(x) if F (x) = f(x). If F (x) is one nti-derivtive of f(x), then every nti-derivtive Mth 9 Course Summry/Study Guide (4) Fll, 2005 prepred by J.H. Silvermn
hs the form F (x) + C for some constnt C. This most generl nti-derivtive is clled the indefinite integrl of f nd is written f(x). Bsic Indefinite Integrls: The differentition formuls for powers, trig functions, logs, exponentils, inverse trig functions, etc. ll give corresponding indefinite integrl formuls. The Fundmentl Theorem of Clculus (one prt) : Let F (x) be ny ntiderivtive for f(x). Tht is, F (x) = f(x). Then f(x) = F (b) F (). The Fundmentl Theorem of Clculus (the other prt): Define function F (x) by F (x) = x f(t) dt. Then F (x) = f(x). More generlly, if g(x) is function of x, define function F (x) = g(x) f(t) dt. Then F (x) = f(g(x))g (x). Evlution of Integrls by Substitution: In n integrl f(x), we cn substitute u = g(x) provided we lso use the reltion du = g (x) to express everything in terms of u. For the definite integrl f(x), the limits of integrtion in the new u integrl re g() nd g(b), becuse s x goes from to b, the vlue of u = g(x) goes from g() to g(b). [6] Applictions of Integrtion Are Between Curves: The re below the curve y = f(x) nd bove the x-xis nd lying between x = nd x = b is A = f(x). (This formul is vlid provided f(x) 0 for ll x between nd b.) The re below the curve y = f(x) nd bove the curve y = g(x) nd lying between x = nd x = b is ( ) A = f(x) g(x). (This formul is vlid provided f(x) g(x) for ll x between nd b.) Volume by Cross Sections: Suppose tht when solid is cut t x, the re of the cross section is A(x). Then the volume of the solid is V = A(x). Mth 9 Course Summry/Study Guide (5) Fll, 2005 prepred by J.H. Silvermn
Volume of Revolution using Disks or Wshers: If the region below the curve y = f(x) nd bove the x-xis nd lying between x = nd x = b is rotted bout the x-xis, the volume swept out is V = πf(x) 2 (This formul is vlid provided f(x) 0 for ll x between nd b.) If the region below the curve y = f(x) nd bove the curve y = g(x) nd lying between x = nd x = b is rotted bout the x-xis, the volume swept out is V = ( πf(x) 2 πg(x) 2) (This formul is vlid provided f(x) g(x) for ll x between nd b.) Work: Work is force pplied over distnce. If the force is constnt, then the work done is simply Work = Force Distnce. If the force vries with position, sy the force t the point x is given by the function F (x), then the work done in moving from x = to x = b is W = F (x). A spring exerts force proportionl to the distnce it is stretched (this is known s Hooke s Lw). So if x = 0 is the unstretched position, then the force is F = kx. The spring constnt k cn be determined if you re told the force t some point x = x 1. The work done in stretching the spring from x = to x = b is W = kx = 1 2 kx2 b = 1 2 kb2 1 2 k2. Averge Vlue of Function: The verge vlue of the function f(x) for x b is f vg = 1 f(x). b Mth 9 Course Summry/Study Guide (6) Fll, 2005 prepred by J.H. Silvermn