Min topics for the Second Midterm The Midterm will cover Sections 5.4-5.9, Sections 6.1-6.3, nd Sections 7.1-7.7 (essentilly ll of the mteril covered in clss from the First Midterm). Be sure to know the results of the problems on HWs 3-7 nd Quizzes 4-6 nd understnd their proofs so tht you cn reproduce them. In ddition, some mteril from erlier in the term, nd from Mth 15100, my be needed to solve certin problems. In prticulr, mke sure you re comfortble with the following: Understnd the mteril from Mth 15100, nd from the first hlf of the term. In prticulr: Understnd wht it mens to write mthemticl proof (including proofs by contrdiction Know how to tke derivtives Know how to determine the bsic properties of function (incresing/decresing, concvity, min/mx vlues, symptotes, etc.), nd know how to grph functions Understnd wht n integrl is, both intuitively (s sum of infinitely smll quntities), nd the ctul definition (involving upper nd lower sums). Understnd how n integrl cn written s limit of Riemnn sums Mke sure you understnd the nottion this is number, not function of t. b f(t)dt, in prticulr mke sure you understnd why Remember, the dummy vrible should never pper outside of the integrl Remember to lwys include the dt (or, or dy, or du) in your nottion! Know the bsic properties of integrls. In prticulr mke sure you know (nd understnd!) the fundmentl theorem of clculus! Know how to use the fundmentl theorem to compute integrls. Indefinite Integrls: Know wht n ntiderivtive of function is, nd how this cn be used to compute n integrl. Understnd why ny two derivtives of function differ by constnt, Know wht the nottion f(x) mens, nd wht it mens to sy f(x) = F (x) + C (in prticulr, wht does the +C men? Mke sure you remember to include the +C in your nswers. Know how you would find function f, given its derivtive nd its vlue t one point (or its second derivtive nd its vlue t two points) 1
Motion of n Object Moving long coordinte xis Know the reltions mong position, velocity, nd ccelertion in terms of integrls nd derivtives for n object moving long coordinte xis Know the difference between position nd distnce, nd between velocity nd speed Know wht negtive velocity or negtive ccelertion men Know the eqution of motion for n object with constnt ccelertion, initil position x 0, nd initil speed v 0 nd know how to derive it u-substitution: Understnd why f (u(x))u (x) = f(u(x)) + C Know how to use this to compute certin indefinite integrls (remember to substitute the vlue of u bck in t the end - your finl nswer should be in terms of x, not u Know the form for definite integrls b f (u(x))u (x) = u(b) u() f(u)du In prticulr, remember tht the bounds chnge for the integrting vrible! (And understnd why this hppens.) Remember tht it isn t necessry to substitute in the vlue of u(x) when computing definite u(b) integrls, it s enough to just evlute f(u)du s n integrl in u (why?) u() Know the common tricks for u-substitution (letting some complicted expression be u, letting the function in rdicl be u, letting u = x +, finding u tht will mke du pper in the integrl, etc.) It will not lwys be obvious wht the correct choice of u is. Sometimes you might need to be bit cretive to think of it. Don t be frid to try few different things, if you don t immeditely see wht to do. Also remember tht you cn lso you u-substitution bckwrds (i.e. writing x s function of r u,) such s the substitution x = r sin u to evlute r 2 x 2 Properties of integrls: Know the properties of integrls covered in clss (such s f(x) g(x) for f(x) b b g(x), f(x) f(x), or f(x) =? for f n even or odd function. Know how to compute d Men Vlue Theorem: v(x) u(x) r b f(x) for ny functions u, v, f(x). Know the sttement of the men vlue theorem for integrls (including the weighted version.) Know the proof of the unweighted version (i.e. the first men vlue theorem of integrls). b 2
Are Know how to find the verge vlue of function nd the number c from the theorem. Know how to find the mss nd the center of mss of rod by tking weighted verges nd integrting. Know how to compute the re enclosed by the grphs of two (or more ) curves. Remember tht this my require finding the intersection points of the curves nd the curves my intersect more thn twice. Also remember tht re hs to be positive, keep trck of whether you should be computing b f(t) g(t)dt or b g(t) f(t)dt nd don t forget to explin why Remember tht geometry doesn t cre bout xes - you cn just s esily find n re by integrting with respect to y insted of x. Sometimes this will be esier Remember tht you will sometimes need to use more thn one integrl to clculte n re (i.e. you my need to brek your region into two or more pieces to compute the re.) Don t be frid to do this, but lso mke sure there isn t simpler pproch you re missing. To see wht re the left nd right bound/upper nd lower bound, it will be useful to move horizontl segment (when integrting for y) or verticl segment (when integrting for x) long the region to see the curves the segment intersects - they re your bounds. If the curves chnge somewhere long the region, then you need to use more thn 1 integrl for ech set of bounds. Remember tht res (or volumes) show ALWAYS be positive. Volume Know how to compute the volume of solid by finding the res of its cross sections nd then integrting these Understnd why this gives you the volume, both intuitively nd in terms of Riemnn sums Know how to use this to compute volume when the cross sections re simple shpes. In prticulr, know how to find the res of such cross sections using similr tringles or the Pythogoren theorem In prticulr, remember tht if the cross sections re perpendiculr to the x xis, then you need to integrte for x nd if the cross sections re perpendiculr to the y-xis, then you need to integrte for y. (Why?) Solids of Revolution Understnd how to use the disk or wsher method to compute the volume of solid of revolution, nd understnd why this is just specil cse of the cross section method Understnd how to use the shell method to compute the volume of solid of revolution, nd understnd why this is NOT specil cse of the cross section method. (Wht s different here? Why does it still work?) Know how to recognize which method will be esier to use for ny prticulr problem. 3
In theory, you cn solve ny solid of revolution problem using either the wsher or shell method. However, often one of these is significntly more difficult tht the other. Lern how to recognize which one will be less work. Mke sure you cn remember which vrible you should be integrting with respect to with ech method. This depends both on the xis of revolution, nd on the choice of method. For instnce, if you re rottion bout the x xis, then you will integrte with respect to x if you re using the wsher method, but you will integrte with respect to y if you re using the shell method (if done correctly, these will both give you the sme nswer.) Don t forget to explin your set up i.e. for the wsher method - intervl nd rdius (s the difference of the bounds), for the shell method - intervl, rdius, height, thickness Know wht to do when the xis of revolution is still horizontl or verticl line but not necessrily the x-xis or the y-xis i.e. know how the rdius nd the height chnge for ech of the wsher nd shell method Inverses of functions Understnd wht it mens for function to be one-to-one, nd know how to show tht function is one-to-one (this usully involves showing it is incresing or decresing) Remember which functions hve inverses nd how to show tht Wht is the definition of n inverse function f 1 (x) Know how to clculte the function f 1 (x) Remember wht is the reltionship between the grphs of f nd f 1. Wht is the domin nd rnge of f 1 in terms of these of f(x)? Understnd why f 1 (f(x)) = x Know how to compute (f 1 ) (x). Remember tht you cn sometimes compute this (t lest t specific points) without first finding f 1 (x). Inverse Trig Functions Know how to define the bsic inverse trig functions: rcsin nd rctn (in prticulr, wht re their domins nd rnges?) Know how to define other inverse trig functions (e.g. rccos, rcsec etc.) in terms of these Know how to compute certin vlues of these functions (if you know some vlue of trig function, then you lso know vlue of the inverse trig function) Know how to compute things like tn(rcsin x) or rcsin(sin x) (note tht this is not lwys x) If you cn t immeditely see how to do this with trig identity, try drwing picture. Most problems like this cn be solved by drwing the unit circle or n pproprite right tringle Know how to tke the derivtives of inverse trig functions (especilly rcsin nd rctn) Know how to use this to compute 2 x nd 2 x 2 + 2 x 2 +bx+c by completing the squre (keep in mind tht how you hndle these will depend on whether the polynomil x 2 + bx + c hs rel roots or not. More generlly, mke sure you know how to compute things like or x 2 +bx+c 4
Trig identities Know the simple identities like sin( x) = sin x, cos x = sin(π/2 x), sin(π x) = x etc. If you understnd how the unit circle works, you won t need to memorize ll of these! You should be ble to come up with these on your own by just looking t the picture. Know how to find the vlues of trig functions t certin specil vlues (e.g. those you cn get from 45-45-90 or 30-60-90 right tringles) Know the Pythgoren identity: sin 2 x + cos 2 x = 1, nd know how to use this to get relted identities (like 1 + tn 2 x = sec 2 x) Know how to tke the derivtives of the bsic trig functions. In prticulr, mke sure you know tht (sin x) = cos x, (cos x) = sin x, (tn x) = sec 2 x = 1 cos 2 x Know the double ngle identities: sin(2x) = 2 sin x cos x nd cos(2x) = 2 cos 2 x 1 = 1 2 sin 2 x. The second will be especilly helpful when the integrnd is something like sin 2 x or cos 2 x to decrese the degree of the trig function. It my lso help to be somewht fmilir with the sum identities: sin( + b) = sin cos b + cos sin b, cos(+b) = cos cos b sin sin b nd the relted product identities for sin sin b,cos cos b, sin cos b. They re helpful in integrting experessions of the form: sin(nx) cos(mx), sin(nx) sin(mx), cos(nx) cos(mx). Exponentils nd Logrithms: Know how to define ln(x) nd exp(x). In prticulr, understnd why the definition you hve lerned before this course, nmely ln(x) = log e (x) nd exp(x) = e x, re NOT sufficient. Mke sure you understnd the point of lectures on Section 7.2 nd 7.4. Wht ws the issue i.e. why did we hve to derive the properties of the new functions gin? Mke sure you know the bsic properties (such s ln(xy) = lnx+lny, exp(x+y) = exp(x) exp(y)), nd understnd why these formuls follow from these definitions. Also mke sure you cn recognize n incorrect formul (e.g. x+y = x + y, ln ln b = ln ( b ) etc.) so tht you don t get the formuls wrong. Know wht exp (x) nd ln (x) re Know how to use the function ln(x) nd exp(x) to define the functions x nd log (x) for ny > 0. Integrting with exponentils nd logrithms: Understnd why x = ln x + C nd not lnx + C f (x) Know how to recognize n integrl in the form nd how to compute it using logrithms f(x) Know how to use this to clculte tn x nd understnd how you cn use this to clculte some other integrls involving trig functions. Know wht e x is nd wht x is for ny > 0. Know how to use the function e x in u-substitution, either by letting u = e x so du = e x or by letting u = the power of the exponentil function so tht the integrl becomes e u du. 5
Logrithmic Differentition Understnd how to use logrithms to compute derivtives, nmely f (x) = f(x) d ln f(x) Understnd why this is useful. We could lwys hve computed f (x) without doing this, but often this will be esier. Know how to use this to compute d xr nd d x. Know how to clculte the derivtive of something like h(x) = f(x) g(x). In prticulr, know wht the domin of the function h(x) is. Remember tht logrithmic differentition does hve limittion - it does not work t points where the function f(x) is 0. Remember how to get round this limittion. Exponentil growth nd decy Know how to solve the differentil eqution f (x) = kf(x) to get f(x) = Ce kx Understnd how this eqution shows up in lot of nturl situtions (e.g. popultion growth, rdioctive decy, compound interest, etc.) nd therefore why we cn use exponentil functions to understnd these situtions Know the relted terms. In prticulr, know wht doubling time nd hlf life mens Know how to find the constnt C in terms of the initil conditions of the function. 6