M408S Concept Inventory smple nswers These questions re open-ended, nd re intended to cover the min topics tht we lerned in M408S. These re not crnk-out-n-nswer problems! (There re plenty of those in the book, nd you need to prctice them, too.) Plese red the originl questions nd write out nswers before compring to wht is written below. You ll get lot more from the exercise tht wy. 1) If f(x) is function of one vrible, wht does f () men? How do you compute it? f f(x) f() f(+h) f() () = lim x = lim x h 0. This is the rte t which h the function f(x) is chnging s x psses through. It is lso the slope of the line tngent to y = f(x) t (, f()). You compute it with whole bunch of useful formuls, from the derivtives of x n, e x, etc. to the product rule, quotient rule, chin rule, logrithmic differentition nd implicit differentition. The formuls re SO powerful tht it s esy to forget wht derivtive relly is. 2) Wht is n nti-derivtive? The nti-derivtive of f(x) is function whose derivtive is f(x). Every continuous function hs n nti-derivtive, nd ny two nti-derivtives differ by constnt. 3) Wht does the definite integrl b f(x)dx men? It s the limit lim N N i=1 f(x i ) x of sum, ssuming tht the limit exists. It represents the totl mount of stuff represented by f(x) (re, volume, whtever) between x = nd x = b. 4) How do we compute it? 99% of the time we use the Fundmentl Theorem of Clculus. Find n nti-derivtive F (x) of f(x) nd compute F (b) F (). 5) Wht re some of the things tht we cn compute using definite integrls? The min exmples we looked t were res under curves nd volumes of solids, especilly solids of revolution. Other stndrd exmples re rclength, surfce re, nd work. Almost ny bulk quntity cn be expressed s n integrl (lthough not necessrily 1-dimensionl integrl). 6) Wht re some of the strtegies for finding nti-derivtives (which we usully cll integrls out of lziness)? Under wht circumstnces do you 1
use ech one? ) u-substitution is our bred-nd-butter technique nd is the chin rule turned inside-out. We use it so often tht the book doesn t even list it s technique of integrtion. b) Integrtion-by-prts is the product rule turned inside out. We use it when the integrnd is the product of two terms, one of which gets lot simpler when integrted, nd the other of which doesn t get too much worse when integrted. (E.g. x sin(x)dx). c) We lerned bunch of tricks for doing trig integrls, mostly revolving round the identity sin 2 (x) + cos 2 (x) = 1 nd the double-ngle formuls sin 2 (x) = (1 cos(2x))/2 nd cos 2 (x) = (1 + cos(2x))/2. d) Trig substitutions like x = sin(θ) or x = tn(θ) or x = sec(θ) llow us to convert integrls involving expressions like x 2 ± 2 or 2 x 2 or their squre roots into trig integrls. e) Prtil frctions is method for reducing rtio P (x)/q(x) of polynomils into sum of simpler terms, ech of which cn be integrted seprtely. 7) Wht does the differentil eqution dy = 3y tell you bout y s dx function of x? (Answer this WITHOUT solving the differentil eqution. Wht does the differentil eqution ctully sy?) The eqution sys tht the rte t which the function y is chnging (tht s dy/dx) is 3 times the vlue of the function (tht s 3y). The more you hve, the fster it grows. The solution to this differentil eqution is y = Ae 3x. 8) Wht sorts of differentil equtions do you know how to solve? How do you solve them? We know how to solve seprble equtions dy/dx = f(x)/g(y), nd tht s bout it. We cross multiply to get g(y)dy = f(x)dx, integrte both sides (don t forget the constnt of integrtion!) nd then try to solve for y in terms of x. 9) Wht models of growth do you understnd? When do you use exponentil growth vs. logistic growth? Exponentil growth is where dy/dt is proportionl to y. If dy/dt = ry, then y = Ae rt. Logistic (or limited ) growth is where the growth strts to tp out when y gets too big. The two stndrd exmples re popultion growth with limited food supply or the spred of n epidemic. 10) Wht does the sttement lim n 2 n = 2 men? Is the sttement 2
true? The sttment mens tht, s n gets bigger nd bigger, 2 n gets closer nd closer to 2. Of course this sttement is flse. 11) Wht does the sttement n=0 2 n = 2 men? Is the sttement true? It mens tht, s n gets bigger nd bigger, the prtil sum 1+1/2+1/4+ + 2 n gets closer nd closer to 2. This sttement is true. 12) Wht s the difference between sequence nd series? A sequence is (usully infinite) list of numbers. A series is the sum of ll of the numbers in the list. To every series n is ssocited sequence {s n } of prtil sums. The sum of the series n is the limit of the sequence {s n }. 13) Wht does convergence men for () improper integrls, (b) sequences, nd (c) series? There re two kinds of improper integrls. In one kind, the limits of integrtion go to infinity, nd in the other the function itself blows up. In both cses you hve to tke limit s the region of integrtion grows. e x dx 0 b converges mens tht lim b 0 e x dx exists. 1 dx diverges mens tht 0 x 2 1 dx the limit lim b 0 doesn t exist. b x 2 Convergence of sequence { n } mens tht there is limiting number L such tht, whenever n is very lrge, n is very close to L. In other words, lim n n = L. Convergence of series n mens there is number L such tht, whenever n is very lrge, the prtil sum s n = n i=0 i is close to L. 14) Wht re some of the techniques for telling whether series converges? When do you use ech one? ) Divergence test. If { n } does not converge to 0, then n diverges. b) Integrl test. If f(x) is decresing positive function, then n=1 f(n) nd f(x)dx either both converge or both diverge. 1 c) Comprison tests. If n nd b n re both positive series, then we cn compre the two. If the smller series diverges, then the bigger series diverges. If the bigger series converges, then the smller series converges. If lim n /b n is nonzero number, then they either both converge or both diverge. 3
d) Alternting series. If the terms strictly lternte in sign, shrink in size, nd go to zero, then the series converges. e) Absolute convergence. If n converges, then n converges. f) Rtio nd root test. These re utomted wys of compring series to geometric series. 15) Define the rdius of convergence nd the intervl of convergence of power series. For every series c n (x ) n there is number R, clled the rdius of convergence, such tht the series converges bsolutely for x < R nd diverges when x > R. The intervl of convergence is the set of x s for which it converges. This is n intervl from R to + R, nd my include 0, 1 or 2 endpoints. 16) When expressing (known) function s power series, how do you find the coefficients? Tke derivtives nd evlute t x = : c n = f (n) ()/n!. 17) How do you tke the derivtive or integrl of power series? Term by term, s long s you re within the rdius of convergence. 18) Wht re the Mclurin series for e x, sin(x), cos(x), ln(1+x), tn 1 (x) nd (1 + x) k? e x = n=1 xn /n!, sin(x) = ln(1 + x) = ( 1) n+1 x n ) n=1 n x n. ( k n=0 n ( 1) n x 2n+1 n=0 (2n+1)!, tn 1 (x) = n=0, cos(x) = n=0 ( 1) n x 2n (2n)!, ( 1) n x 2n+1 2n+1, nd (1 + x) k = 19) How ccurte is the pproximtion T k (x) f(x), where T k (x) is k-th order Tylor polynomil. The error is f (k+1 (c)(x ) k+1 /(k + 1)!, where c is mystery number somewhere between x nd. 20) Wht is Tylor series good for? Evluting functions, evluting integrls, solving differentil equtions, nd tking limits. 21) If f(x, y) is function of two vribles, then wht does f/ x men? How do you compute it? (Ditto for f/ y, but if you understnd f/ x you probbly understnd f/ y.) f x = f/ x is the rte t which f(x, y) is chnging when we move in the 4
x direction nd hold y fixed. You compute it using the usul formuls for derivtives, treting y s if it were constnt. 22) Wht do f xy nd f yx men? How re they relted? These re second derivtives. f xy mens tke the prtil with respect to x, nd then tke the prtil of tht with respect to y. f yx is the opposite order. Clirut s theorem sys tht they re equl. 23) Wht is double integrl? This is n integrl over 2-dimensionl region. Brek the region into little boxes, compute f(x, y ) A for representtive point (x, y ) in ech box, dd up the contributions, nd tke limit s you chop things into smller nd smller boxes. 24) Wht is n iterted integrl? How is tht different from double integrl? How do you use iterted integrls to compute double integrls? An iterted integrl is n expression like 1 3 0 2 yexy dxdy. It mens do 1-dimensionl integrl with respect to x, treting y s constnt, nd then integrte the result with respect to y. (Or sometimes the other wy round.) Tht s not the sme s integrting over 2D region, but it cn be USED to integrte over 2D region. Fubini s theorem sys tht, if R = [, b] [c, d] is rectngle nd f(x, y) is continuous function, then d b b d f(x, y)da = f(x, y)dxdy = f(x, y)dydx. R c 25) Wht re Type-I nd Type-II regions? How do you do double integrl over Type-I region? Over Type-II region? A type-i region is region bounded by two verticl lines nd two curves of the form y = g(x) nd y = h(x). A type-ii region is bounded by horizontl lines nd curves of the form x = g(y) nd x = h(y). A double integrl over type-i region boils down to the iterted integrl, b c h(x) g(x) f(x, y)dydx, while h(y) g(y) double integrl over type-ii region boils down to d f(x, y)dxdy. c 26) Wht is switching the order of integtion? How does tht work? Some regions (like rectngles) re both Type-I nd Type-II. Integrls over such regions cn be expressed s iterted integrls in two wys, nd these iterted integrls hve to be equl. If you get one of them nd it s too hrd to solve, rewrite it s double integrl, then rewrite THAT s the other kind of iterted integrl, nd try to do it tht wy. 5
27) Wht re some of the things tht you cn compute using double integrls? Volumes, popultions, verge elevtions, rinfll, verge rinfll. 6