Main topics for the First Midterm

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1 Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 2-3, Sections , nd Sections (essentilly ll of the mteril covered in clss). Be sure to know the results of the problems on HWs 1-6 nd understnd their proofs so tht you cn reproduce them. Mke sure to void these common mistkes: Remember tht cx = c x only works if c 0. When writing n ε, δ-proof, remember tht you need to consider every possible vlue of ε > 0, not just specific vlue. Remember tht δ cn NEVER be negtive. f(x) If lim f(x) = lim g(x) = 0, it does NOT men tht lim x c x c x c g(x) is 0 or undefined, it simply 0 mens you need to find some other wy to compute the limit. Remember tht criticl point is vlue for x, while n extrem is vlue for y = f(x). Remember tht criticl point might not be either locl mximum or minimum. If c is criticl point with f (c) = 0, then the second derivtive test tells you nothing. c might be locl mximum, locl minimum or neither. The first derivtive test my be useful in situtions like this. 1

2 Mke sure you re comfortble with the following: Proofs: Understnd wht it mens to write mthemticl proof. Understnd proof by contrdiction nd induction. In prticulr, understnd wht proof by induction consists of. Remember to include ll prts of proof by induction (bsic cse, inductive hypothesis, inductive step, conclusion) Specify which cse you re considering (e.g. n = 1, n = k, n = k + 1) Specify wht you re ssuming nd wht you wnt to show Know wht is needed to prove sttements like If A, then B or A iff B. Inequlities: Know how to solve liner nd polynomil inequlities (mke sure to check the boundry vlues!). Know how to solve inequlities involving more complicted functions (bsolute vlues, squre roots, trig functions, etc.). Understnd how to express solutions in intervl nottion, or s sets on number line. Understnd wht the quntity b mens geometriclly. Trigonometry: Be fmilir with the functions sin x, cos x nd tn x. Know how to drw their grphs nd compute their vlues t certin points. Know the bsic trig identities (e.g. sin x = sin x, sin 2 x + cos 2 x = 1, etc.). Understnd their reltion to the unit circle, nd know how to use the unit circle to solve problems involving trig functions. Definition of limit: Understnd the intuitive mening of limit, nd the ε, δ definition (nd understnd why they re relted!). Mke sure tht you cn remember the definition of limit without help, nd cn explin wht it mens (i.e. don t just memorize for every ε > 0 there is δ > 0... understnd wht this mens, nd why we defined it this wy). Also mke sure you understnd the definition of one-sided limits. Know how to modify the definition of lim x c f(x) = L in the cse where one or both of c nd L re ±. ε, δ-proofs: Know wht is needed to prove sttement like lim x c f(x) = L, both when f is specific function (like f(x) = x 2 + 1) nd when f is generl function. 2

3 Know how to give n ε, δ-proof when f is liner function, or when f is (simple) non-liner function (such s x 2 +, 1 x, x, x + b where, b re constnts. ) Remember tht you must prove tht your given vlue of δ works, its not enough to simply give the vlue. Remember tht your vlue for δ should NEVER be negtive! Computing limits: Know how to compute limits of most simple functions. In prticulr: polynomils, rtionl functions, nd functions involving bsolute vlues, squre roots nd trig functions. Be fmilir with the theorems bout limits which we proved in clss (or you proved on your homework), nd know how to use them. In prticulr, know how to use the definition of limit to prove sttements for the limits of f + g nd αg. Understnd how to compute the limits of functions like f + g, f g, αf, f g f(x) Understnd why lim x c g(x) might still exist if lim f(x) = lim g(x) = 0. x c x c nd f g. Know how to use the Pinching Theorem to compute limits of more complicted functions. ( ) ( ) 1 1 Remember the formuls lim f(x) = lim f nd lim f(x) = lim x h 0 + h f x h 0 h Know how to compute lim x P (x) nd lim x P (x) for polynomil, P, by looking t the first term. Continuity: Know wht it mens for function to be continuous (both the intuitive explntion, nd the rigorous definition). Understnd the difference between being continuous t c; continuous on (, b); nd continuous on [, b]. Know how you need to redefine the function to mke it continuous in the cse of removble discontinuity. Know how to show using the definition of continuity tht functions like f + g nd αg re continuous. Know how { to check if given function is continuous. In prticulr, when is the function f(x) x < h(x) = g(x) x continuous? Intermedite Vlue Theorem: Know the sttement of the theorem (in prticulr, under wht conditions is it true?). Know how to use it to estimte solutions to equtions. Understnd how it cn be used in proofs. Extereme Vlue Theorem: 3

4 Know the sttement of the theorem (in prticulr, under wht conditions is it true?). Understnd how it cn be used in proofs. Derivtives: Understnd the intuitive mening of derivtive (both s the slope of the tngent line to grph, nd s the liner pproximtion to function). Know how to find the eqution for the tngent to the grph of function f(x) t x = Know the f (x) nottion nd the dy dx nottion for derivtive. In prticulr, understnd why f(c + h) f(c) + hf (c). Know how to tell if function is differentible (nd when it isn t). In prticulr, when { f(x) x < is the function h(x) = g(x) x differentible? Remember tht differentible function is lwys continuous (but not the other wy round). In prticulr, remember tht f(x) = x is n exmple of everywhere continuous function f(x) is not differentible t x = 0 nd know how to show it. Know the rigorous definition of derivtive, i.e. s the limit of f(c+h) f(c) h. Know the lterntive expression for derivtive s limit, i.e. the limit of f(x) f(c) x c Know the difference between being differentible t point, nd being differentible in n intervl. Know how to directly use the definition to compute derivtive (by just tking limit). Know how to use the definition of derivtive to show functions like f + g nd αg re differentible Computing derivtives: In generl, you should mke sure you know how to compute the derivtive of ny function you know how to write down. Know how to compute the derivtives of simple functions (like polynomils). Know how to compute the derivtives of sums nd products of function. ( ) Know how to compute f. g (If you hve trouble remembering the quotient rule, remember tht you cn compute this just by knowing nd using the product ) rule.) Know how to use the chin rule to compute (f g). When using the chin rule, mke sure to keep trck of which vrible you re differentiting with respect to. Remember tht d dx f(y) f (y). Understnd how you cn use the chin rule multiple times to compute the derivtive of something like f(g(h(x))). Remember tht d dx xr = rx r 1 for ny rtionl r. Know the derivtives of sin x nd cos x, nd know how to use these to compute the derivtives of more complicted trig functions. Know how to use implicit differentition to compute dy dx when you do not necessrily hve formul for y in terms of x. ( 1 g 4

5 Word Problems Know how to trnslte rel world problem into clculus problem. Know how to find the mximum or minimum vlue of some quntity. Do not forget to consider the closed intervl of possible vlues for the rgument when mximzing or minimizing some prticulr, be sure to check wht hppens t the endpoints. Higher order derivties: Know wht it mens to tke higher order derivtive of function. Understnd the nottions f (x), f (n) (x) nd dn y dx n. Know how to compute the second derivtive of function (i.e. by tking the derivtive twice). Understnd how you cn use induction to prove formul for dn y dx n. Minim nd mxim of functions: Know (nd understnd) the definitions of locl nd bsolute mxim/minim. Know wht criticl point of function is, nd know tht ny locl extrem re criticl points. Remember tht NOT ll criticl points need to be locl extrem. Remember tht criticl points need to be inner points for the domin of the functions. I.e. in order for x = c to be criticl point you need c (, b) such tht f is defined on ll of the points of (, b) including c. Know wht it mens for function to be incresing or decresing, nd know how to determine this by looking t the sign of the derivtive. Distinguish between the definition of n inresing function nd theorems helping you determine when function is incresing. Know how to use the first nd second derivtive tests to determine whether criticl point is mximum, minimum or neither, nd know why these tests work. Understnd why the first derivtive test cn be used in more situtions thn the second. Understnd wht endpoint extrem re, nd how to find them. Understnd how to find the bsolute extrem of function (both function defined on [, b] or n open intervl (, b) or n infinite intervl). Do not forger to consider the endpoints (if they exist) when finding the bsolute extrem of function. Know wht to do if f is not continuous t criticl point c. Men Vlue Theorem: Know the sttement (but not the proof). In prticulr, know the necessry conditions for the function. 5

6 Know how to use this theorem to get informtion bout function from its derivtive. Know how to find the point x = c stisfying the conditions of the theorem. Or if there is no such point, to show tht there is no such number x = c nd explin why. Properties of functions relted to grphing: Mke sure tht you know the following properties of functions nd their grphics. Tht is, you should know the definition, how to chech if given function hs these properties using the definition or n uxiliry theorem, nd wht the grph of function with these properties look like. Concve up/concve down functions. Inflection points. (Remember tht point where f (c) = 0 or f (c) DNE does NOT need to be n inflection point.) Verticl nd horizontl symptotes. (Remember which nd how mny limits you need to check in ech cse.) Verticl tngents nd cusps. (Remember these properties re bout the derivtive nd know how to distinguish between them.) Know how properties of f (x) relte to properties of f(x). Tht is, if you know tht f (x) hs some property (e.g. positive, incresing, hs verticl symptote) wht does this tell you bout the function f(x) nd its grph. Mke sure to distinguish between the definition of concve up function nd theorems helping you determine when function is concve up. Know how to drw the grph of function, either if you re given n eqution for the function, or simply told properties tht the function must stisfy. Integrtion: Understnd intuitively wht n integrl is. Tht is, tht of infinitely smll quntities f(t)dt. f(t)dt is n infinite sum Understnd the reltion between integrls nd re. In prticulr, understnd wht hppens when f(x) is not positive everwhere. Also, how to clculte the integrl vi the res of geometric figures. Mke sure you understnd the nottion f(t)dt. In prticulr understnd tht this is number, which depends only on nd b, nd NOT on t. The vrible t only dummy vrible nd the letter t cn be substituted with ny other unused letter. If the dummy vrible t ever ppers outside of the integrl sign, then you hve done something wrong. Estimting Integrls: Know wht prition of n intervl [, b] is. Know wht the norm, P of prtition P is. Understnd why prtitions with smller vlues of P re considered better. 6

7 Know wht the lower nd upper sums, L f (P ) nd U f (P ) re. Tht is, know the definitions, nd how to compute them for given or n rbitrry prtition. Note tht they involve finding miniml nd mximl vlues of function, so you my need to find the criticl points. Know how to define f(t)dt in terms of L f (P ) nd U f (P ). Know how we cn use upper nd lower sums to compute exct vlues of some bsic integrls (e.g. cdt, tdt) Know wht the Riemnn sum Sf (P ) is, nd know how to compute it. b Understnd the definition lim P 0 S f (P ) = Properties of Integrls: Know bsic properties of integrls: b f(t) + g(t)dt = b kf(t)dt = k Understnd how holds for ll, b, c f(t)dt f(t)dt + g(t)dt f(t)dt f(t)dt is defined for = b nd > b so tht f(t)dt = Fundmentl Theorem of Clculus: Understnd the function F (x) = x c c f(t)dt. f(t)dt + c f(t)dt Tht is, understnd wht this function reprsents (i.e. it is n re under the curve.) Know on which intervl F (x) is continous nd on which differentible (i.e.intervls of the type [, b] nd (, b) respectively) nd the vlues of F (x) t the endpoints of the intervl. Understnd tht the function F (x) is function of x, NOT function of t. (It would be useful to keep trck of which vribles re dummy nd which rel vribles. ) Know tht F (x) = f(x) nd understnd the intuition for it. (This reltionship is relly importnt throughout the rest of the qurter.) 7

8 In prticulr, remember tht F (x) should not depend in ny wy on the choice of c. If c ppers in your fomul for F (x), you hve done something wrong. Know how to compute the derivtive F (x) when composed with nother function g(x) i.e. know how to use the Chin rule to find the derivtive of F (x) = f(t)dt. Know how to use tht F (x) = f(x) in order to find the criticl points of F (x), locl extrem vlues of F (x), inflection points, nd etc. Miscellneous Problems from Homework Know wht is the difference between functions f(x) nd g(x) continuous on n intervl I when f = g. Know wht is the reltionship between the limits of f nd g if f(x) g(x) on some intervl I. Know how to prove tht the limit of the bsolute vlue of function is the bsolute vlue of the limit i.e lim f = L if lim f = L. Know how to prove tht if lim x c f(x) = L, then f(x) is bounded ner but different from c i.e. there re constnts m nd M such tht m < f(x) < M. Know how to prove by definition tht if lim x c f(x) = L > 0, then f(x) > 0 for x ner but different from c. Know how to show tht f(x) is liner function if f = 0 for ll x in some intervl I. Know how to prove by induction problems similr to the ones on Problem set 5. Know the simplifying expressions for the sums n nd n 2. h(x) 8