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.


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1 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 justify their work. In ddition to these, the Finl Exm my consist of: fill in the blnk, true/flse, or multiple choice questions. Most instructors gree tht good wy to study for the finl is to do lots of problems to help fmilirize yourself with ll of the concepts covered. Sections contining similr concepts hve been grouped in blue boxes. Most MTH 33 finl exm writers gree tht the items below contin crucil mteril for showcsing MTH 33 knowledge nd re therefore very importnt. Expect t lest one problem from ech group on the finl exm Volumes Work Recll the formul for volume V = b A(x) dx. Where A(x) is the crosssectionl re perpendiculr to xxis. If f(x) is vrible force function then the work done in moving the object from to b is given by: W = b f(x) dx Recll Hooke s Lw for springs f(x) = kx. Where x is the number of units beyond the spring s nturl length
2 6.  Inverse Functions Recll the formul: (f ) () = f (f ()) The Nturl Logrithmic Function Recll the definition nd properties of ln x. Know tht d dx (ln x ) = x. Remember the steps in logrithmic differentition The Nturl Exponentil Function Recll the properties of e x Generl Logrithmic nd Exponentil Functions Know properties such s x = e x ln nd log x = ln x ( + n Recll the limit: lim x 0 ( + x) /x = e = lim n Exponentil Growth nd Decy ln ) n Know tht the only solutions to the differentil eqution dy/dt = ky re: to help you derive generl logrithms nd exponentils. y(t) = y(0)e kt Know the formuls for Rdioctive Decy/Growth, Newton s Lw of Cooling, nd Compound Interest
3 6.6  Inverse Trigonometric Functions Know the lgebric properties of trigonometric inverses. Using the Implicit Function Theorem, know how to derive the formuls: d ( sin x ) = dx, x 2 d ( cos x ) = dx, x 2 d ( tn x ) = dx + x 2 The derivtive formuls bove yield the ntiderivtive formuls: dx = x 2 sin x + C, x dx = ( x ) tn + C Hyperbolic Functions Know the definitions, identities, nd derivtives of the hyperbolic functions Indeterminte Forms nd L Hospitl s Rule Lern to recognize the following indeterminte forms when evluting limits: 0/0 / Understnd how nd when L Hospitl s Rule cn be used to evlute certin indeterminte limits
4 7.  Integrtion by Prts Recll the formul for integrtion by prts:. u dv = uv v du Trigonometric Integrls Know the strtegies for integrting Know the strtegies for integrting sin m x cos n x dx. tn m x sec n x dx Trigonometric Substitution Use the substitutions below to help evlute integrls. 2 x 2 x = sin θ, 2 + x 2 x = tn θ, x2 2 x = sec θ Integrtion of Rtionl Functions by Prtil Frctions Recll how to perform long division of polynomils. Know the different cses for prtil frctions nd how/when to correctly pply ech
5 7.8  Improper Integrls Remember both types of improper integrls nd know how to crefully evlute ech. Recll tht dx is convergent if p > nd divergent if p. xp Know the Comprison Test for nd Limit Comprison Test for improper integrls Arc Length Know the formuls for rc length: L = b d + [f (x)] 2 dx, L = + [g (y)] 2 dy c Seprble Equtions Know tht seprble differentil equtions cn be written s h(y) dy = g(x) dx. Be ble to explicitly solve vriety of seprble differentil equtions
6 0.  Curves Defined by Prmetric Equtions Be ble to trnsform prmetrized curves x = f(t), y = g(t) to Crtesin equtions. Recognize grphs or prmetrized curves nd the direction of movement Clculus with Prmetric Curves Know how to find tngent lines to prmetric curves. β (dx ) 2 ( dy ) 2 Remember the rc length formul: L = + dt dt dt α Polr Coordintes Know how to trnsform between Crtesin nd Polr Coordintes. Know how to sketch polr grphs Ares nd Lengths in Polr Coordintes Know the re formul for polr equtions: A = b Remember the rc length formul for polr equtions: L = 2 [f(θ)]2 dθ. b r 2 + ( ) 2 dr dθ. dθ
7 .  Sequences.2  Series Know the limit properties of sequences (e.g., sum, product, etc.). Remember the theorem: The sequence {r n } is convergent if < r nd divergent for ll other vlues of r. { lim = 0 if < r < n if r = Know the definitions of incresing, decresing, monotonic, nd bounded. Recll tht if r < then the geometric series Know the test for divergence theorem: If r n is convergent nd n=0 r n = r n= lim n 0, then the series n Recll sum nd sclr multiple equtions for summtions..3  The Integrl Test nd Estimtes of Sums Be ble to use The Integrl Test for series convergence. Know the pseries Test for convergence..4  The Comprison Tests n= n is divergent. n= is convergent if p > nd divergent if p. np Know The Comprison Test nd The Limit Comprison Test for series convergence..5  Alternting Series Be fmilir with The Alternting Series Test nd use it to determine when lternting series converge..6  Absolute Convergence nd the Rtio nd Root Tests Recll the definitions of bsolutely convergent nd conditionlly convergent. Remember tht bsolutely convergent series re convergent. Know The Rtio Test nd The Root Test for series convergence odd 7
8 .8  Power Series Recll the definitions for power series, rdius of convergence, nd intervl of convergence. Know the strtegy for finding power series intervl of convergence (typiclly using the rtio test)..9  Representtions of Functions s Power Series Know the formul: x = + x + x2 + x 3 + = x n, x < Know how to integrte or differentite power series using termbyterm differentition nd termbyterm integrtion..0  Tylor nd Mclurin Series Know tht if f hs power series representtion f(x) = n=0 c n (x ) n then the coefficients c n re given by: n=0 c n = f (n) () n! Recll Tylor s Inequlity for finding the mximum error in pproximting function with Tylor Polynomil. Know the formuls from Tble in.0 such s: e x x n =, for ll x. n!.  Applictions of Tylor Polynomils Use Tylor s formul to estimte the ccurcy of the pproximtion f(x) T n (x) in given intervl. n=
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