Final Review, Math 1860 Thomas Calculus Early Transcendentals, 12 ed

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1 Finl Review, Mth 860 Thoms Clculus Erly Trnscendentls, 2 ed 6. Applictions of Integrtion: 5.6 (Review Section 5.6) Are between curves y = f(x) nd y = g(x), x b is f(x) g(x) dx nd similrly for x = f(y) nd x = g(y). b 6. Volumes by slices: b A(x)dx where A(x) is the re of the cross section perpendiculr to the x-xis. Specil cse is the solid of rottion obtined by rotting the region between curves y = f(x) nd y = g(x), x b bout the x-xis: b π f(x) 2 g(x) 2 dx. Wht hppens in the cse the xis of rottion is y = 3? There is n nlogous formul obtined by interchnging x nd y. Exmple: The region between the curves x = y 2 nd y = x 2 is rotted bout the y-xis. Find the volume of the solid generted. 6.2 The method of cylindricl shells: the region between curves y = f(x) nd y = g(x), x b bout the y-xis: b 2πx[f(x) g(x)]dx. Wht hppens when the xis of rottion is x = 2? Agin interchnging x nd y gives n nlogous formul. Exmple: Rotte the region in the previous exmple round the x-xis. 6.3 Arc Length. The curve y = f(x), x b hs length b +(f (x)) 2 dx 6.4 Are of Surfces of Rottion: The curve y = f(x), x b is totted bout the x xis. Find the re of the surfce. 2π b 7. Trnscendentl Functions: f(x) +(f (x)) 2 dx 7. The Nturl Logrithm lnx = x (/t)dt hs hs the property ln(xy) = lnx+ lny etc. (d/dx)lnu = (/u)du/dx. Wht is tnxdx? 7. e x, the Nturl Exponentil is the inverse of lnx: e lnx = x = ln(e x ). (d/dx)e u = e u du/dx nd we sw exp(x) = e x if x is rtionl nd so for ll 7. Generl logs nd exponentils. Differentite y = 3 secx. or y = log 2 x. 7.3 Hyperbolic Functions: coshx = ex +e x ; sinhx = ex e x Techniques of Integrtion: nd tnhx = sinhx coshx.

2 2 8. Integrtion by prts: udv = uv vdu. On p 407, wht is u nd dv in questions -24? 8.2 Integrtion of powers of Trig functions Identities: () (sinx) 2 +(cosx) 2 = (b) (cosx) 2 = 2 (+cos2x) (c) (sinx) 2 = 2 ( cos2x) (d) (secx) 2 = (tnx) 2 + (e) (cscx) 2 = (cotx) 2 + (f) sin2x = 2sinxcosx (g) cos2x = (cosx) 2 (sinx) 2 (sinx) m (cosx) n dx Cses: m is odd (u = cosx); n is odd (u = sinx); m nd n even (cosx) 2 = 2 (+cos2x) (sinx)2 = 2 ( cos2x) 8.2 Integrtion of powers of Trig functions (tnx) m (secx) n dx If n is even substitute u = tnx. If m is odd then substitute u = secx. Also secxdx = ln secx+tnx +C Also 8.3 Trig Substitution secxdx = ln secx+tnx +C For Integrls Involving Substitute Use the Identity 2 x 2 x = sinθ, dx = cosθdθ 2 x 2 = cosθ 2 +x 2 x = tnθ, dx = (secθ) 2 dθ 2 +x 2 = secθ 8.4 Prtil Frctions.. Divide; 2. Fctor divisor 3. Expnd by prtil frctions 4. Solve for coefficients 5. Integrte.

3 3 8 Evlute xrctnxdx = (cosx)(sinx) 6 dx = (cosx) 4 dx = (tnx) 3 dx = v 2 v 2 dv = +x 2 (+x) 3 dx 8.7 Improper integrls. Type I nd II. dx < if nd only if p >. xp. Comprison Test. Assume 0 f(x) g(x). Then g(x)dx < implies f(x)dx < OR f(x)dx = implies g(x)dx = 0. Series: 0. Sequences lnn () lim n = 0 (b) lim n /n = (c) lim x /n = (d) lim x n = 0 if x <. (e) lim (+ x n x n (f) lim n! = 0 2. Series. ) n = e x 0.2 Geometric series +r+r 2 +r 3 = /( r). 0.2 Telescoping series. 0.2 nth term test for divergence. lim n n Integrl Test 0.3 p-series n /np converges iff p >. 0.4 Comprison Test n /(n2 +n)

4 4 0.4 Limit Comprison Test. n /(n2 n) 0.6 Conditionl or bsolute convergence? 0.5 Alternting Series Test. Is n ( )n / n conditionlly convergent? AST is only for conditionl convergence. 0.6 Rtio Test. lim n n+ / n 0.6 Root Test. lim n n /n 0.7 Power series c n(x ) n. Rdius of convergence nd intervl of convergence. Rtio or root test. Check end points? 0.8 Integrtion nd differentition of power series: f(x) = c n(x ) n f (x) = nc n (x ) n n= f(x)dx = C + c n n+ (x )n+ All three power series hve the sme rdius of convergence. 0.8 The geometric series nd power series. For exmple, /x = /( + (x )) = (x )+(x ) 2 (x ) 3 (Here r = (x ) nd =. Convergence if r = x <. 0.9 Tylor series nd Mclurin series ( = 0) f (n) () (x ) n n! nd Tylor polynomils, T N (x) of degree N is P N (x) = N f (n) () (x ) n n! 0.9 Common Mclurin series: e x = +x+ x2 2! xn n! cosx = x2 2! + x4 4! + ( )n x 2n (2n)! sinx = x x3 3! + x5 5! + ( )n x 2n+ (2n+)! 0.0 Integrtion of power series like tht of e x2 Prmetric nd Polr Curves.. Prmetric Curves. Grphing

5 5.2 Prmetric Curves nd Clculus. dy dx = dy/dt. Tngent line to prmetric curve. dx/dt b Length (compre to Section 6.3) (x (t)) 2 +(y (t)) 2 dt.3 Polr Coordintes.4 Polr Coordintes. Grphing. Circles (r =constnt. r = cosθ r = sinθ) nd crdioids (r = (±cosθ) or r = (±sinθ)) nd the flowers like r 2 = sin3θ..5 Are in polr coordintes R 3 2. Distnce between points. β α 2 f(θ)2 dθ 2. Eqution of sphere centered t (4,-,3) nd rdius Vectors. Length nd direction. Unit vectors. Force, displcement nd velocity. 2.3 Dot product. Algebric nd geometric difinitions. Angle between vectors. 2.3 Component of b long : comp b = ( b)/. Projection of b long : proj b = b Cross product. Algebric nd geometric definitions. b is perpendiculr to nd b nd by the right hnd rule. Length is the re of the prllelogrm. 2.4 Triple vector product ( b c). Tke bsolute vlue of this rel number to get the volume of the prllelepiped determined by the 3 vectors. 2.4 b is rel number; b is vector. 2.5 Eqution of plne through three points P Q nd R. The norml is PQ PR. 2.5 PrmetricequtionofthelinethroughP(,b,c)ndQis x,y,z =,b,c +tpq. Here PQ = d,d 2,d 3 is the direction vector of the line nd we hve x = +td, y = b+td 2 nd z = c+td 3 if we write out the components. 2.5 The symmetric equtions of the sme line re x d = y b d 2 = z c d 3

Disclaimer: This Final Exam Study Guide is meant to help you start studying. It is not necessarily a complete list of everything you need to know.

Disclaimer: This Final Exam Study Guide is meant to help you start studying. It is not necessarily a complete list of everything you need to know. Disclimer: This is ment to help you strt studying. It is not necessrily complete list of everything you need to know. The MTH 33 finl exm minly consists of stndrd response questions where students must