4. Let f = ez?y. (a) Find the second-order Taylor's polynomial of f about (1,0). (b)...
I. Let f : R2 → R be defined by f(x)l cos (122) 211 Compute the second order Taylor polynomial of f near the point xo - 0. A Road Map to Glory (On your way to glory, please keep in mind that f is class C) a) Fill in the blanks: The second order Taylor's polynomial at h E R2 is given by T2 (h) = 2! b) Compute the numbers, vectors and matrices that went into the blanks...
Let H=F(x,y) and x=g(s,t), y=k(s,t) be differentiable functions. Now suppose that g(1,0)=8, k(1,0)=4, gs(1,0)=8, gt(1,0)=2, ks(1,0)=1, kt(1,0)=5, F(1,0)=9, F(8,4)=3, Fx(1,0)=13, Fy(1,0)=7, Fx(8,4)=9, Fy(8,4)=2. Find Hs(1,0), that is, the partial derivative of H with respect to s, evaluated at s=1 and t=0.
2 er Let I be an interval of R, and define the function f :I→ R by f(x) 1 +e2z or every z EZ. (a) Find the largest interval T where f is strictly increasing. (b) For this interval Z, determine the range f(T) (c) Let T- f(I). Show that the function f : I -» T is injective and surjective. (d) Determine the inverse function f-i : T → 1. (e) Verify that (fo f-1)()-y for every y E...
Find the range of the function f(x) = x2 A) (0,0) B)[0,00) C)(1,0) D)(-1,00) E)-, -1] U [0,0] F)RUR G)(1,0) H) (1,-1] Select one: a. F b. C C. B d. H e. D f. E g. A h. G
2 over R. Define U g-Ло íj fg for f,0€ y 5. Let V be the vector space of polynomials of degree a. If U is the subspace of scalar polynomials, find U b. Apply Gram-Schmidt to the basis1, t,t2) of V. c. If a E R find ga E V with (f, ga)(a) for all f E V
manitudle of a Vector 5. Reter to Problem 4. Find the following B, e) the scalar product f)內, g) 3 * a) the scalar product b) the magnitude of A, x c) the magnitude of d) the angle between A and B, h) the angle between亡and T 4. Graph the following vectors on the same coordinate system (three dimensions) = -51 + 4)-6k 4. Given the vectors: 궂. 4个.价. Find the following a) A +
12. Let g(x), h(y) and p(z) be functions and define f(x, y, z) = g(x)h(y)p(2). Let R= = {(x, y, z) E R3: a < x <b,c sy <d, eszsf} where a, b, c, d, e and f are constants. Prove the following result SS1, 5100,2)AV = L*()dx ["Mwdy ['Plzdz.
(b) Let f 0, 1-R be a C2 function and let g, h: [0, 00)-R be C1. Consider the initial-boundary value problem kwr w(r, 0) f(a) w(0, t) g(t) w(1, t) h(t) for a function w: [0,1 x [0, 0)- R such that w, wn, and wa exist and are continuous. Show that the solution to this problem is unique, that is, if w1 and w2 [0, 1] x [0, 00)- R both satisfy these conditions, then w1 = w2....
(i) Find a non-zero polynomial in Z3 x| which induces a zero function on Z3. f(x), g(x) R have degree n and let co, c1,... , cn be distinct elements in R. Furthermore, let (ii) Let f(c)g(c) for all i = 0,1,2,...n. g(x) Prove that f(x - where r, s E Z, 8 ± 0 and gcd(r, s) =1. Prove that if x is a root of (iii) Let f(x) . an^" E Z[x], then s divides an. aoa1
(i)...
® Define TELive) T(X,Y,2)=(2x, 48452, 4y+32) by vectors a) Find its characteristic Polynomial b) use a) to find eigenvalues and eigen c) Diagonalize MCT) d) Find adj (MCT) e) Find inverse of MCT)