(c) Let f : IR2 -R2 be given by f(x,)= (a1)2-y1, (-12) Let S, S' be the subsets of R2 as indicated in the picture below. Prove that f maps S onto S' (0,1) v-axis V=1 (2,1) (1,1) y =(x-1)2 у-ахis u 1 v=u-1 u-axis (1,0) (0,0) х-аxis (1,0)
(c) Let f : IR2 -R2 be given by f(x,)= (a1)2-y1, (-12) Let S, S' be the subsets of R2 as indicated in the picture below. Prove that f maps S...
(e) Letf: R2-R2 be given by f(a,y) = (V-y,y) Let A, B be the subsets of R2 as indicated in the picture below. Prove that f maps A onto B. (0,1) (1;1) (-1,1) (0,1) v=1 1/2 y-axis y=x2 v-axis v -u b ets t) ide ods.a notteog (0,0) X-axis u-axis (0,0)
(e) Letf: R2-R2 be given by f(a,y) = (V-y,y) Let A, B be the subsets of R2 as indicated in the picture below. Prove that f maps A onto...
Given a second order linear homogeneous differential equation а2(х)у" + а (х)У + аo(х)у — 0 we know that a fundamental set for this ODE consists of a pair linearly independent solutions yı, V2. But there are times when only one function, call it y, is available and we would like to find a second linearly independent solution. We can find y2 using the method of reduction of order. First, under the necessary assumption the a2(x) F 0 we rewrite...
5. Let f: R2 + R2 be defined by [a2 + 2xy] f(x,y) = | 14,9) | xy2 ] (a) Explain why f we are guaranteed to have an inverse defined on an open neigh- borhood of [31]? = f(1, 1), but not at [1 0]? = f(1,0). (b) Give the derivative of the inverse function at (3,1): DF-1(3,1). e a la IV
4. Consider the functions f : R2 R2 and g R2 R2 given by f(x, y) (x, xy) and g(x, y)-(x2 + y, x + y) (a) Prove that f and g are differentiable everywhere. You may use the theorem you stated in (b) Call F-fog. Properly use the Chain Rule to prove that F is differentiable at the point question (1c). (1,1), and write F'(1, 1) as a Jacobian matrix.
4. Consider the functions f : R2 R2 and...
1. Let L: R2-R2 be defined by L(x.y) (x +2y, 2x - y). Let S be the natural basis of R2 and let T = {(-1,2), (2,0)) be another basis for R2 . Find the matrix representing L with respect to a) S b) S and1T c) T and S d) T e) Find the transition matrix Ps- from T basis to S basis. f) Find the transition matrix Qre-s from S-basis to T-basis. g) Verify Q is inverse of...
Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 | g(z, y) = 0} S Suppose g(a, b)-0 so that (a, b) E S and dg(a, b)メO. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above. 2) Since dg(a,b) 0, argue that it suffices to...
Can someone explain how to do this problem?
2. Let f(t, y) — х +у, 0<x< 1, 0 <y<1 < ,Y < !) (a) Find P (X 1 2 (b) Find P(X < 2Y)
Question 1(a&b)
Question 3 (a,b,c,d)
QUESTION 1 (15 MARKS) Let X and Y be continuous random variables with joint probability density function 6e.de +3,, х, у z 0 otherwise f(x, y 0 Determine whether or not X and Y are independent. (9 marks) a) b) Find P(x> Y). Show how you get the limits for X and Y (6 marks) QUESTION 3 (19 MARKS) Let f(x, x.) = 2x, , o x, sk: O a) Find k xsl and f(x,...
5. Let A =R x R and f: A+ A be given by the rule f(x, y) = (x – y, x + y). (a) Prove f is one-to-one. (b) Prove f is onto A. (Comment: don't forget that if given b E A, you construct a such that f(a) = b, you must also show a E A.) (c) What is the inverse function? (d) Is f a permutation? Explain.