Mth 4: Finl Exm Formuls to Know This ocument tells you every formul/strtegy tht you shoul know in orer to o well on your finl. Stuy it well! The helpful rules/formuls from the vrious review sheets my be helpful to review s well. When printing this ocument, you my not nee to print the lst 4 pges, s they re uplictes of other hnouts. Integrtion Techniques:. Integrtion by Prts: u v = uv v u.. For integrls of the form sin m x cos n x x or sec m x tn n x x, choose your u crefully so tht the remining power of the other trigonometric function is even fter using prt of it for u. In the cse where powers of sin x n cos x re ll even, use the power reucing formuls given in the trigonometric ientities. 3. Trigonometric Substitution: Know the following tble n how to use it: Expression: Substitution: Ientity: x x = sin θ sin θ = cos θ + x x = tn θ + tn θ = sec θ x x = sec θ sec θ = tn θ 4. Prtil Frctions: Cse : (x ) (x ) = A x + B x Cse : x (x ) = A x + B x + C (x ) Cse 3: Cse 4: (x ) (x + ) = A x + Bx + C x + (x + ) = Ax + B x + + Cx + D (x + ) Series Strtegies: If the Form Inclues: Use First: Notes: Only Numbers Rise to Powers: Geometric You cn fin vlue when convergent. n Cn be Written s b n b n+k (Telescoping) Write Prtil Sums n Tke Limit You my nee prtil frctions or log rules to rewrite it. Other tests show convergence, but prtil sums gives vlue. n = n p n = (b n ) n Fctoril (n!) or Almost Geometric p-series Root Rtio Almost Geometric is numbers rise to powers times lgebric terms. n = ( ) n b n Alternting Series Sum in Numertor or Denomintor Comprison If this fils, use the Limit Comprison. Sine, Cosine, or Inverse Tngent in Numertor Absolute Convergence sin n, cos n, rctn n π/ Trigonometric Function or Divergence Obvious Limit Not 0 Esy to Integrte Function Integrl Unsure Rtio If this fils, try Integrl
Note: For the Alternting Series n the Integrl, you my nee to show f (n) < 0 for n (some numeric vlue) to show( ecresing. Useful Fcts for Series s: + ) n = e.7, cos (nπ) = ( ) n n Are, Volume, n Averge Vlue:. The re between n g (x) ( g (x) on [, b]) is given by A = b g (x) x. We cn o y-xis integrtion s long s we o the curve to the RIGHT first.. Disk Metho: Let R be region boune by y = ( 0 on [, b]), y = 0, x =, n x = b. The volume of the soli generte by revolving R roun the x-xis is V = b π [] x. The y-xis version is similr. 3. For these, note tht the formul πr is use, so if rotting roun n xis prllel to the x- or y-xis, let r be the istnce from the xis of rottion to n rbitrry point x or y in your intervl. 4. Wsher Metho: Let R be region boune by y =, y = g (x) ( g (x) 0 on [, b]), x =, n x = b. The volume of the soli generte by revolving R roun the x-xis is V = b π ( [] [g (x)] ) x. The y-xis version is similr. 5. For these, note tht the formul π (R r ) is use, so if rotting roun n xis prllel to the x- or y-xis, let R be the istnce from the xis of rottion to n rbitrry point on the the further curve n r be the istnce to tht point on the closer curve. 6. Shell Metho: Let R be region boune by y =, y = g (x) ( g (x) 0 on [, b]), x =, n x = b. The volume of the soli generte by revolving R roun the y-xis is V = b πx [ g (x)] x. The y-xis version is similr. 7. For these, note tht the formul πrh is use, so if rotting roun n xis prllel to the x- or y-xis, let r be the istnce from the xis of rottion to n rbitrry point x or y in your intervl, n let h be the height of the rectngle t tht point. 8. The verge vlue of function is given by f ve = b x. b Polr Coorintes:. Equtions to switch between rectngulr n polr: x = r cos θ y = r sin θ x + y = r tn θ = y x. For region boune by r = f (θ), θ =, n θ = b, the re of the region is given by b b [f (θ)] θ or r θ if you remember tht r = f (θ). 3. For region boune by R = f (θ), r = g (θ), θ =, n θ = b, where R r, the re of the region b ( is given by [f (θ)] [g (θ)] ) b ( θ or R r ) θ. Trigonometric Ientities:. sin x + cos x =. tn x + = sec x 3. cot x + = csc x 4. sin (x) = sin x cos x 5. cos (x) = cos x sin x 6. sin x = ( cos (x)) 7. cos x = ( + cos (x))
Unit Circle For given point (x, y) on the unit circle, the vlues of the 6 trigonometric functions re s follows. sin θ = y cos θ = x tn θ = y x sec θ = x csc θ = y cot θ = x y 3
Derivtive Rules n g (x) re functions, n, c, n n re rel numbers (possibly with the usul restrictions). x (c) = 0 x (x) = x (xn ) = nx n x ( x) = x Constnt Multiple Rule: x (c ) = c f (x) Difference Rule: x ( g (x)) = f (x) g (x) Sum Rule: x ( + g (x)) = f (x) + g (x) Prouct Rule: x ( g (x)) = f (x) g (x) + g (x) Quotient ( Rule: ) Chin Rule: = g (x) f (x) g (x) x g (x) [g (x)] x (f (g (x))) = f (g (x)) g (x) x (ex ) = e x x (x ) = x ln x (ln x) = x x (log x) = x ln Use Logrithmic Differentition for function of the form [f(x)] g(x). (sin x) = cos x x (sec x) = sec x tn x x (cos x) = sin x x (csc x) = csc x cot x x x (tn x) = sec x x (cot x) = csc x ( sin x ) = x x (sec x) = x x x x (cos x) = x (csc x) = x x x x (tn x) = + x x (cot x) = + x 4
Integrtion Rules n g (x) re functions, n, c, n n re rel numbers (possibly with the usul restrictions). All formuls shoul inclue +C t the en. 0 x = C k x = kx x n x = xn+ n + x x = ln x Constnt Multiple Rule: c x = c x Sum Rule: + g (x) x = x + g (x) x Difference Rule: g (x) x = x g (x) x Substitution Rule: b f (u (x)) u (x) x = u(b) u() f (u) u Integrtion by Prts: u v = uv v u e x x = e x x x = x ln sin x x = cos x cos x x = sin x sec x x = tn x sec x tn x x = sec x csc x cot x x = csc x csc x x = cot x tn x x = ln sec x cot x x = ln sin x sec x x = ln sec x + tn x csc x x = ln csc x + cot x ( x ) x x = sin + x x = ( x ) tn x x x = sec ( x ) 5
Series s As long s the conitions of the test re stisfie, this tble gives the result of whether the series converges or iverges. If test is inconclusive, try nother test. n= n : Conitions: Convergence: Divergence: Inconclusive: Geometric n = r n r < ; converges r N/A ( is number) to r p-series n = p > p N/A n p Telescoping Series Divergence Integrl Comprison Limit Comprison Alternting Series Absolute Convergence Rtio n = b n b n+k for n some k; s n = k. k= Works for ll series. n positive, ecresing; f (n) = n continuous n, b n positive n, b n positive; n = c, where b n 0 < c < n = ( ) n b n, b n 0; lso works for ( ) n+, ( ) n, etc. Works for ll series. Works for ll series. s n exists n is finite Cnnot be etermine. x converges n b n n converges n= b n converges n= b n (lso works if c = 0) b n+ b n (ecresing) AND b n = 0 n converges n= n+ n < (ctully gives Root Works for ll series. bs. convergence) n n < (ctully gives bs. convergence) s n DNE or is infinite n 0 or DNE iverges x n b n n iverges n= b n iverges n= (lso works if c = ) b n 0 or DNE (by ivergence test) Cnnot be etermine. (or ) n+ n > n n > (or ) b n N/A n = 0 n not positive, ecresing n b n n n= b n converges; n b n n b n iverges n= n b n exist. oes not b n not ecresing n iverges n= n+ n = n n = 6