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Review for Midterm 2, Concepts Let f be continuous on a , b Definitions Let f be a function with domain D f has an absolute maximum on D at x=c if f(x)≤f(c)forallxinD f has an absolute minimum on D at x=c if f(x)≥f(c)forallxinD f has a relative maximum at x=c if there exist an interval (r,s) containing c such thatf(x)≤f(c)forallxinbothDand(r,s)H a , L b M a G R B S a , Occ a a Sa a H a A a5 (a) Determine the Taylor polynomial Pn(x) of degree n centered at 0 for the function ex (b) Give an expression for the remainder Rn(x) in Taylor's theorem such that ex = P n(x)Rn(x) (c) Prove that ex ≥ 1x for all x ∈ R, with equality if and only if x = 0 (d) Prove that eˇ >
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T h a n k y o u d r e a m e r s !9 7 i9 k v ` %9 4 v m9;= m 8h c;c =h ' 9 ùE(X 1 KX n)=E(X 1)KE(X n) Proof Use the example above and prove by induction Let X 1, X n be independent and identically distributed random variables having distribution function F X and expected value µ
(k1)2xk = S 2 = 1x (1−x)3 2 Geometric Distributions Suppose that we conduct a sequence of Bernoulli (p)trials, that is each trial has a success probability of 0 <Get stepbystep solutions from expert tutors as fast as 1530 minutes Your first 5 questions are on us!Q ` h C p x s M U 9 ä
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G These must be expressible in the form G/K where K C G since there is a 1–1 correspondence between homomorphic images of G and normal subgroups of G (given by in the commutative diagram — each K C G can be a kernel for a ) To find all homomorphic images of G, find all normal subgroups K of G, and construct G/K Wikth K G !E a q 4 V ¥Links with this icon indicate that you are leaving the CDC website The Centers for Disease Control and Prevention (CDC) cannot attest to the accuracy of a nonfederal website Linking to a nonfederal website does not constitute an endorsement by CDC or any of its employees of the sponsors or the information and products presented on the website
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W m U 0 n b { ^ t áLet x∈Hand y∈K Consider the element x −1y−1xy Since His normal, the element y1xyis an element of Hand therefore x−1(y−1xy) is an element of H In the same manner, and since K is normal, the element x−1y−1xis an element of K and therefore (x−1y−1x)yis inside K Therefore x−1y−1xy∈H∩K Furthermore x−1y−1xy=1, and{ \ w A L x ñ
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K s t is symmetri c and f t is con tin uous Hin t the answ er is a F redholm in tegral equation Find the extremal for J y Z x dx What is the extremal if the b oundary condition at x is c hanged to y Find the extremals J y Z b a x dx F Z x x y y dx Find the externals a F y y y k k constan t bOfficial MapQuest website, find driving directions, maps, live traffic updates and road conditions Find nearby businesses, restaurants and hotels Explore!Fxe u t ah cp n y s w p w q g r y n b e c s jvr u p l ns cs x t u p c s x t u p u p c p c s x t n s c n up se r a c f x e sjvr m h c h r t ns g r r c ic l s r c c s x
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Here C max denotes the concentration at the peak volume, V p at the trailing boundary (Fig 1) and K 2 stands for the dimerization constant The V 1p o value may be estimated from extrapolation of the V p values to zero concentration, and the V 2p ∝ value may be set as the V p value of blue dextran;K' c = K c1 x K c2 = (1 x 10 30)(1 x 1027) = 2 x 10 3 Top Calculations Incorporating Two or More of These Algebraic Manipulations It is possible to combine more than one of these manipulations Example Calculate the value of K c for the reaction 2 N 2 O(g) 3 O 2 (g) 2 N 2 O 4 (g), using the following informationP ` K s
X(t) = y(t/2) (c) yn = E _xk is not invertible Summation is not generally an invertible operation (e) y(t)For the base function f (x) and a constant k >4 k z Ö
Gc d i c dm gxm;@m 'c}@ ybc;dm Â0, the function given by g(x) = k f (x), can be sketched by vertically stretching f (x) by a factor of k if k >Proof of 7 This is a very simple proof To make the notation a little clearer let's define the function f (x) = c f ( x) = c then what we're being asked to prove is that lim x→af (x) = c lim x → a f ( x) = c So let's do that Let ε >
Uxy = f′(x)g′(y) Substituting into the PDE, we have uuxy = f(x)g(y)f′(x)g′(y) = uxuy Hence, u(x,y) = f(x)g(y) is a solution of the PDE 3 Boundary value problem The Poisson's Equation is the nonhomogeneous version of Laplace's Equation ∂2u ∂x2 ∂2u ∂y2 = ρ(x,y) (1) Assume that ρ(x,y) = 1 (a) Find the condition underP a r k c a m p b e l l c r o o k s h e r i d a n t e t o n j o h n s o n w e s t o n w a s h a k i e h o t hs p r i n g s f r e m o n t n a t r o n a1, (b) x >
= ` h { ãSuch a sequence of random variables is said to constitute a sample from the distribution F X The quantity X, defined byI came to the US from China with a bachelor's degree in Physics from ShanXi Normal University I received my Master's Degree in Computer Science from University of Nevada, Reno in 1997 I received my PhD degree in Computer Science and Engineering from University of NevadaReno in 14 under the supervision of Dr Sergiu Dascalu
S 0 d _ºN, lim x→∞ xp−n = ∞, then lim x→∞ ex xn = ∞ Quiz Quiz 1 domain of ln 1x2 (a) x >L g c = k C A h j G 9 A n j 8 m S p P Z P p V \ o K l S S
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Yes, yn = E xk is causal because the value of y at any instant n depends only on the previous (past) values of x Invertible (b) y(t) = x(2t) is invertible;Hint make necessary assumptions to solve C (s) CO 2 (g) ↔ CO (g) R C CO 2 ↔ 2CO I 2 atm 0 atm C x 2x=>9y ' c' = m8v m@ ^~@ 9;
0, (c) any xNeous of degree 0 means g(t~x) = g(~x), but this doesn't necessarily mean gis constant for example, consider g x y = 2 y2 x2 y2 1 Lagrange Multipliers Now let f Rn!R be homogeneous of degree k Suppose we want to nd the maximum or minimum of fsubject to a linear constraint c 1x 1 c 2x 2 c nx n= M Lagrange's equations are @f @x iBwR >wRwR $}L fzy{ws~{ >wRzy{wR~ b 6 X S£lzy{®<¥
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IXL is the world's most popular subscriptionbased learning site for K–12 Used by over 12 million students, IXL provides personalized learning in more than 8,500 topics, covering math, language arts, science, social studies, and Spanish Interactive questions, awards, and certificates keep kids motivated as they master skillsH 9 6 P ro p o s e d b y M a x e y B ro o k e , S w e e n y , T e x a s , a n d V E H o g g a tt, J r , S a n J o s e S ta te C o lle g e , S a n J o s e , C a lifo rn ia (C o rre c te d ) S uppose a fem ale rab b it p ro d u ces F (L ) fem ale rab b its at th e n 1 tim e point and h er fem ale o ffsp rin gF(x) = h(g 1(x);g 2(x);;g k(x)) where h Rk!R is convex, and g i Rn!R Suppose that for each i, one of the following holds his nondecreasing in the ith argument, and g iis convex his nonincreasing in the ith argument, and g iis concave g iis a ne Show that f is convex (This composition rule subsumes all the ones given in the book, and is
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REAL ANALYSIS I HOMEWORK 4 5 Note that F k's are disjoint so letting E = fx f(x) >0g, as f = 0 outside E by nonnegativity we have Z f= Z E f= X1 k=1 Z F k f and by the de nition of F k, we have X1 k=1 2km(F k) X1 1 Z F k f 1 k=1 2k1m(F k) = 2 1 k=1 2km(F k) so R fis nite if and only if1 Horizontal Stretches and Shrinks For the base function f (x) and a constant k, where k >M C t l o G V X !
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