Topics for 161.01 finl.1 The tngent nd velocity problems. Estimting limits from tbles. Instntneous velocity is limit of verge velocity. Slope of tngent line is limit of slope of secnt lines.. The limit of function. Intuitive definition: limit is the y vlue pproched s the x vlues go towrds point. Be creful guessing limits numericlly: round off errors or simply picking points too fr wy my cuse trouble. Left-hnd, right-hnd, nd two-sided limits. (If left nd right limits re different, two sided limit does not exist.) ( π ) Undmped oscilltions t rbitrrily high frequencies do not hve limits (e.g. lim sin does x 0 x not exist.) If function vlues get rbitrrily lrge s x vlue is pproched, limit is sid to be +. Similrly for. Identify verticl symptotes bsed on infinite limits. (Verticl symptotes re lines, not numbers.).3 Clculting limits using the limit lws. Limits of sum, difference, or product is respectively the sum, difference, or product of the limits provided they both exist. Limit of constnt times function is the constnt times the limit of function (specil cse of product.) Limit of quotient is the quotient of the limits provided they both exist nd the lower limit is not 0. Squeeze theorem: If f g h, f L nd h L s x, then g L s x..4 The precise definition of limit. lim x f(x) = L mens for every ε > 0, there is δ > 0 such tht if 0 < x < δ, then f(x) L < ε. Similr definitions hold for left nd right limits. Be ble to use the definition to prove the limit for liner problem, e.g. lim x (4x 1)..5 Continuity. A function f is continuous t if f(), the limit s x nd the limit s x + ll exist nd re equl. We cn exchnge the order of limits nd continuous functions. Polynomils, trig functions, rtionl functions, root functions, exponentil functions, nd log functions re continuous on their domins. Composition of continuous functions is continuous. Intermedite vlue theorem. Intuitively: A continuous function cnnot go from one vlue to nother without going through the vlues in between. To show two functions re equl on n intervl, show their difference must be 0 somewhere.
.6 Limits t infinity; horizontl symptotes. Limits t (or ) describe how function behves for lrge (or smll, respectively) x-vlues. Horizontl symptotes re the lines tht function pproches s x ±. (At most two horizontl symptotes.) Do lgebr to resolve indeterminte form problems (, 0/0, /, etc... ). Be ble to find tngent lines nd norml lines..7 Derivtives nd rtes of chnge. The derivtive f () is defined: f f(x) f() f( + h) f() () = lim = lim. x x h 0 h Be ble to find derivtives using the limit definition, especilly of polynomils, rdicls, nd rtionl functions. The derivtive indictes the instntneous rte of chnge. Thus the derivtive of position is velocity, the derivtive of velocity is ccelertion, the derivtive of ccelertion is jerk, the derivtive of function is the slope of its tngent line, etc....8 The derivtive s function. Be comfortble with both Newton s nd Leibnitz s nottions. f f(x ) f(x) f(x + h) f(x) (x) = lim = lim. x x x x h 0 h Not ll functions re differentible t every point. (e.g. f(x) = x is not differentible t 0.) If f is differentible t, then f is continuous t. The second derivtive is the derivtive of the derivtive, the third derivtive is the derivtive of the second derivtive, etc... Be ble to estimte the vlue of derivtive from grph or tble. 3.1 Derivtives of polynomils nd exponentil functions. Power rule, constnt rule, constnt multiple rule, sum nd difference rules. The derivtive of e x is e x. (Except for constnt multiples of e x, no other function is its own derivtive.) 3. The product nd quotient rules. ( ) f (fg) = f g + fg, = gf fg g g. Sometimes it is esier to lgebriclly simplify n expression nd then differentite insted of using the quotient rule. 3.3 Derivtives of trigonometric functions. sin x 1 cos x lim = 1, lim = 0, nd more complicted combintions. x 0 x x 0 x Derivtives of trig functions: sin, cos, tn, csc, sec, cot. 3.4 The chin rule. Suppose F (x) = f(g(x)). Then F (x) = f (g(x))g (x). Suppose y = f(u) where u = g(x). Then dy dx = dy du du dx. d dx (x ) = x ln.
3.5 Implicit differentition. Use implicit differentition to find first nd second derivtives. Derivtives of inverse trig functions: sin 1, cos 1, tn 1, csc 1, sec 1, cot 1. 3.6 Derivtives of logrithmic functions. (ln x) = 1/x. Since log b x = (ln x)/(ln b), it follows tht (log b x) = 1/(x ln b). Logrithmic differentition: Tke logs of both sides, implicitly differentite, then solve for y. Good for complicted products, quotients, nd powers. 3.7 Rtes of chnge in the nturl nd socil sciences. Derivtive is the rte of chnge. Biology, physics, chemistry, etc... ll study things tht chnge. Position/velocity/ccelertion/jerk problems, such s tossed bll. Mrginl revenue/cost problems. Density problems. Be ble to dpt to other problem types. 3.8 Exponentil growth nd decy. If y = ky, then y(t) = y(0)e kt. Mny processes cn be modeled with this reltion, including: growth, temperture chnges, nd compound interest. rdioctive decy, popultion 3.9 Relted rtes. Write generl formuls, tke the derivtive (use the chin rule), then plug in your dt nd solve for the missing pieces. Common tricks include: similr tringles, lw of cosines. Best to lwys work in rdins (becuse the derivtives of sin nd cos re different if using degrees). 3.10 Liner pproximtions nd differentils. Ner the point of tngency, the tngent line pproximtes the grph of the function. The formul for the tngent line is clled the lineriztion of the function t the point. If y = f(x), then the differentil is dy = f (x)dx. Use differentils to estimte error; use lineriztion or differentils to estimte function vlues. 4.1 Mximum nd minimum vlues. Criticl points re plces in the domin where the derivtive is zero or undefined. Locl extrem cn only hppen t criticl points. (A criticl point need not be n extrem.) Absolute extrem hppen t end points or t criticl points. 4. The men vlue theorem. Rolle s theorem: If f is continuous on [, b], differentible on (, b) nd f() = f(b), then there is c in (, b) such tht f (c) = 0. i.e. If two points hve the sme y vlue, then somewhere in between the function hs horizontl tngent. (Under certin bsic ssumptions.) Men vlue theorem: If f is continuous on [, b] nd differentible on (, b) then there is c in (, b) such tht f f(b) f() (c) =. b
i.e. Somewhere the slope of (continuous nd differentible) function is the sme s the slope of the secnt line. Use MVT to show inequlities, prove the number of roots function hs. Rolle s theorem is specil cse of MVT. If f = g, then f(x) = g(x) + C for some constnt C. 4.3 How derivtives ffect the shpe of grph. If f > 0 then f is incresing; if f < 0, then f is decresing. First derivtive test: if f > 0 to the left of criticl point nd f < 0 to the right, then we switched from incresing to decresing, so the criticl point is locl mx. If insted f switches from negtive to positive, then the criticl point is minimum. If f > 0, then f is concve up (i.e. the grph of f lies bove ll its tngents.) If f < 0, then f is concve down. Inflection points re points where the concvity chnges. Second derivtive test: Suppose c is criticl point. If f (c) > 0, then locl min t x = c. If f (c) < 0, then locl mx t x = c. If f (c) = 0, then the second derivtive test provides no informtion; try the first derivtive test. 4.4 L Hospitl s rule. If 0/0 or / in limit, tke the derivtive of the top nd the bottom, then tke the limit, otherwise just plug in if quotient. Turn indefinite multipliction forms into division forms by dividing by the reciprocl insted of multiplying. Turn ddition or subtrction into multipliction or division by getting common denomintor or other lgebric mnipultion. Turn exponentil into multipliction by tking the nturl log, evluting the limit, then tking the exponentil of the result. 4.5 Summry of curve sketching. Look for: domin, intercepts, symmetry, symptotes (you my need to use L Hospitl s rule), intervls of incresing nd decresing, locl mx nd mins, nd concvity nd points of inflection. 4.6 Grphing with clculus nd clcultors. Use clculus to pick suitble window (or windows) to grph in; you wnt to show ll the fetures listed in 4.5 bove. 4.7 Optimiztion problems. Word problems. Strtegy: Drw digrm, introduce vribles for distnces etc in your digrm, find lgebric reltionships between the vribles, express desired quntity in terms of one vrible, check criticl points nd end points for mxes nd mins. 4.9 Antiderivtives. F is n ntiderivtive (or indefinite integrl) of f if F = f. (i.e. function whose derivtive is f.) n ntiderivtive of f is When sked for generl ntiderivtive of function, find prticulr one nd dd n rbitrry constnt C.
5.1 Ares nd distnces. Approximte re using finite number of rectngles, left-, right- nd mid-points. Are under the grph is limit of the sum of the res of pproximting rectngles. Σ-nottion. Distnce is re under velocity function. 5. The definite integrl. Definition of definite integrl. (Riemnn sums: know the difference between left-, right-, nd midpoint sums). Σ-nottion formuls for 1, i, i, i 3. Properties of the integrl. 5.3 The fundmentl theorem of clculus. Derivtive of x f(t) dt, nd more complicted problems involving chin rule prt. f(x) dx = F (b) F (), where F is ny ntiderivtive of f. 5.4 Indefinite integrls nd the net chnge theorem. Indefinite integrls re the sme s ntiderivtives, nd unique up to +C. Don t forget to write the +C. Integrl of rte of chnge is the net chnge: F (t) dt = F (b) F (), this is rephrsing of the second fundmentl theorem. Verify ntiderivtive formuls by differentiting. 5.5 The substitution rule. u-substitution is the reverse of the chin rule: If u = g(x), then f(g(x)) g (x) dx = f(u) du f(g(x)) g (x) dx = g(b) g() For indefinite integrls, chnge bck to the originl vrible. f(u) du Integrls of even functions, of odd functions (might be hidden). 6.1 Ares between curves. 6. Volumes. The re of curves tht don t cross is the integrl of (top-bottom) or (right-left). Brek into multiple pieces if the curves cross. Discs (integrte πr ), wshers (integrte πrbig πr smll ), nd other cross-sections (integrte A(x), the cross-sectionl re). 6.3 Volumes by cylindricl shells. Integrte πrh. ex: y = f(x) from to b rotted bout x = c hs volume V = π (x c)f(x) dx.
6.4 Work. Work is force times distnce, lthough for our purposes t lest one of these is going to vry, so you ll hve to integrte. Hooke s lw (spring problems): f = kx. Note x is the distnce the spring is stretched beyond its nturl length. Common problems: springs, pumping wter, lifting chin. Remember: pounds is unit of force, kilogrms is unit of mss. When working with metric problems, multiply mss by g 9.8 to get force, but do not multiply pounds by grvity; they lredy include it. 6.5 Averge vlues. The verge of f on [, b] is f = 1 b f(x) dx. Men vlue theorem for integrls: If f is continuous on [, b], then there is c in [, b] such tht f(c) = f. (Somewhere the function equls its verge vlue.) 7.1 Integrtion by prts. This is the reverse process of the product rule. u dv = uv v du, or for definite integrls, u dv = uv b v du. You my need to use this process more thn once or in combintion with other techniques. Strtegy: Try LIATE for picking u. Tht is, tke u to be whichever of the following comes first: (L)ogs, (I)nverse trig, (A)legebric functions, (T)rig functions, or (E)xponentil functions. If your choice of u nd dv does not work, try different one. Tricky problems, such s e x sin(x)dx. 7. Trig integrls. When integrting sin m (x) cos n (x)dx, if m is odd, turn ll but one sine into cosine vi sin (x) = 1 cos (x), let u = cos(x). Similrly if n is odd. If both m nd n re even, use hlf-ngle formuls: sin (x) = 1 cos(x), nd cos (x) = 1 + cos(x). For tn m (x) sec n (x) dx, if n is even, keep sec, turn the rest into tngents vi sec (x) = 1 + tn (x), let u = tn(x). If m is odd, keep sec(x) tn(x), turn the remining tngents into secnts vi tn (x) = sec (x) 1, let u = sec(x). Other cses re hrder. For sin(mx) cos(nx)dx, nd similr problems, eliminte the product vi one of: () sin(a) cos(b) = 1 (sin(a B) + sin(a + B)) (b) sin(a) sin(b) = 1 (cos(a B) cos(a + B)) (c) cos(a) cos(b) = 1 (cos(a B) + cos(a + B)) 7.3 Trig substitution. Useful for problems involving squre roots. You my need to complete the squre on the inside. For x, let x = sin(θ). For + x, let x = tn(θ). For x, let x = sec(θ). Use right-tringle trig to convert ntiderivtive bck to originl vribles.
7.4 Prtil frctions. If degree of numertor is bigger thn or equl to degree of denomintor, do long division first. Fctor denomintor completely. For liner terms, use numertor of form A; for irreducible qudrtics use numertor of the form A x + B. If fctor of the denomintor is rised to some power, use multiple terms with ll integer exponents between 1 nd tht power. Sometimes you need to mke u-substitutions before you cn do prtil frctions. If integrnd involves n g(x), try letting u = n g(x). (Rtionlizing substitution.) 7.5 Strtegy for integrtion. Look for lgebric simplifictions or mnipultions t ech step. Primry techniques re u-substitution nd integrtion by prts. Use these to reduce the problem to n esier integrl, repet. Look out for opportunities to use prtil frctions, trig integrls, or trig substitution. 8.1 Arc length. L = 1 + (f (x)) dx = d c 1 + (g (y)) dy. Arc length function (rc length from point): s(x) = 8. Surfce re of solids of revolution. x 1 + (f (t)) dt. When rotting f(x) round x-xis, use S = π f(x) 1 + (f (x)) dx. When rotting g(y) round y-xis, use S = π d c g(y) 1 + (g (y)) dy. You my need to solve for x or y.