MATH2010 Lecture Notes - Lecture 1: Phase Portrait

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yellowllama248

6 Aug 2018

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Department

Mathematics

Course

MATH2010

Professor

Dietmar

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Classi cation of 2d systems: x = ax, a = (cid:20) a b c d(cid:21): t = a + d, d = ad bc, p( ) = 2 t + d. Case a: t 2 4d > 0. 1,2 = (t qt 2 4d )/2. General solution: (v1, v2: eigenvectors) x(t) = c1e 1tv1 + c2e 2tv2. Half line trajectories: if c2 = 0 x(t) = c1e 1tv1. H1+ = {x = v1 | > 0} if c1 > 0. H1 = {x = v1 | < 0} if c1 < 0. Same for h2 if c1 = 0, c2 > 0 or < 0: the 4 half line trajectories separate. Indicate direction of motion by arrows point- ing in the direction of increasing t. Trajectories: if 1 > 0 then x(t) = c1e 1tv1. Moves out to for t (outwards arrow on h1+) Approaches 0 for t : if 1 < 0 then x(t) = c1e 1tv1.

Related Questions

K(x,y,z) = (x, x + y,x + y + z)

L(x,y,z) = (2x - y, x + 2y)

S(x,y,z) = (z,y,x)

T(x,y) = (2x + y,x + y,x - y,x - 2y)

G: R^3 -> M(2,2) defined by G(x,y,z) = (y z , (-x + y) -x)

H: M(2,2) -> P2 defined by H(a b, c d) = (a + b) + (a - 2d)x+ dx^2

Find the indicated linear transformation if it is defined. ifnot defined explain why it is not:

1) K + S

2) GS

3)2S^3 - S^2 + 3S - 4I

Find the matrix that represent each of the linear transformationK,L,S,and T for the following

1)TL

2)K + S

Find the matrix that represent each of linear transformation intwo ways, first using the result normal way and using therepresention of the matrix.

T:R^3 --> R^4 given by T(X) = Ax, where

A = [1 1 0

2 0 3

-1 1 1

1 1 2]

and v1 = (-3,3,2) v2 = (0,0,0)

2) T: R^4 --> R given by T(x1,x2,x3,x4) = x1 + 2x2 + 3x3 +4x4 and v1 = (1,2,1,-2) v2 = (2,3,0,-2)

3)T: M(2,3) --> P2

P2 = (a11 + a13)x^2 + (a21 - a22)x + a23

v1 = [-1 3 1

2 2 0]

v2 = [2 1 -2

1 1 0]

& determine wheter w1 or w2 is the iamge of thecorresponding linear transformation

w1 = x^2 + 2x + 1,w2 = x - 2.

[1] 1. (a) Solve 2 – 31 < 7. Solution: We have – 7<-3 < 7 which gives -4 0 on (-0, 1] U [2,). Next, we need to see where Vr2 – 3r +2 -1 > 0. If r2 - 3.c + 2 – 1 > 0, then V.22 – 3.0 +2 > . Observe that this is definitely true if r < 0. If r > 0, then squar- ing both sides we get 22 - 3.+2 > 1 2 > 3. V A VICO Thus, the domain of f is (-00, ). [2] (c) Find the exact value of tan(sec-14). Solution: Let y = sec-14. Then, sec y = 4. From the diagram we get that tan(sec-? 4) = tan y = 15 [2] (d) Find all r such that sin(2.x) = COS I. Solution: We have 2 sin r cos r = COS I, SO 0 = 2 sin Cos I -Cos I = cos .r (2 sin c - 1). Hence, sin(2x) = cost when cos x = 0, or when sin r = Thus, r= +ik, kez, I= +2nk, ke Z, or r= +2nk, kez [2] (e) State the precise mathematical definition of lim f(x) = 0. Solution: For every e > 0 there exists a 8 >0 such that if 0 < x-al <, then f(c) >

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If U and V are positive constants, find all critical points of the function. (Enter your answers as a comma-separated list.) F(t) = Ue t + Ve -t t = Find all critical points of f(x) = 20 x 2 e - 0.3x and classify them as local minima of ,maxima. x = (smaller value) This critical point is a . x = (larger value) This critical point is a . Consider f(x) = x 8 (7 - x) 9. Find f '(x) and simplify your answer. f '(x) = Find all critical points of f(x) and then use the first derivation test to determine local maxima and minima. f(x) has a at x = (smallest value) Local Maximum OR Minimum OR Neither f(x) has a at x = Local Maximum OR Minimum OR Neither f(x) has a at x = (largest value) Local Maximum OR Minimum OR Neither Find the inflection points of f(t) = t 4 + t 3 - 18t 2 +8. Give exact answer. t = (smallest value) t = (largest value) Sketch a possible of y = f(x), using the given information about derivation y' = f'(x) and y" = f"(x). Assume that the function is defined and continuous for all real x. Let f be a function f(x) > 0 for all x. Let g = 1/f. If f is increasing in an interval around x 0, what about g? g is increasing in an interval x 0. g is zero in an interval around x 0. g is decreasing in an interval around x 0. We cannot tell whether g is increasing or decreasing in an interval around x 0. g is constant in an interval around x 0. If f has a local maximum at x 1, what about g? g has a local maximum at x 1. g has both a local minimum and a local maximum at x 1. We cannot tell whether g has a local minimum or a local maximum at x 1. g has an inflection point at x 1. g has a local minimum at x 1. If f is a concave down at x 2, what about g? g is increasing at x 2. g is flat at x 2. We cannot tell whether g is concave up of concave down at x 2. g is concave down at x 2. g is concave up at x 2. You are given the graph of the second derivation function f" below. Which of the given have x-values that are inflection points of the function f? (Select all the apply.) A B C D E F G H I none of the above

MATH2010 Lecture Notes - Lecture 1: Phase Portrait

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