Question

Answer all parts of question 1.

1a.) Show that the solutions of x' = arc tan (x) + t cannot have maxima

1b.) Show that the solution x(t) of the Cauchy problem x' = 2 + sin(x), x(0) = 0, cannot vanish for t>0

1c.) Let Φ(t) be the solution of the ivp x' = tx - t^3, x(0) = a^2 with a not equal to 0. Show that Φ has a minimum at t=0.

Answer #1

Please answer all parts of the following question. Please show
all work and all steps.
1a.) Let Xn be a Marcov chain with the states S = {0,1} starting
from 0. The transition probability is given by
p = ( 1/3 2/3)
1/2 1/2
Compute P(X2=1) and compute P(X3=0 given
X2=1)
1b.) Suppose that T1 and T2 are stopping times. Determine
whether the following are stopping time or not: (1) T1 + T2, (2) T1
− 1 (assuming T1 ≥...

Show that
x(t) =c1cosωt+c2sinωt, (1)
x(t) =Asin (ωt+φ), (2) and x(t) =Bcos
(ωt+ψ) (3)
are all solutions of the differential equation d2x(t)dt2+ω2x(t)
= 0. Show that thethree solutions are identical. (Hint: Use the
trigonometric identities sin (α+β) =sinαcosβ+ cosαsinβand cos (α+β)
= cosαcosβ−sinαsinβto rewriteEqs. (2) and (3) in the form of Eq.
(1). To get full marks, you need to show the connection between the
three sets of parameters: (c1,c2), (A,φ), and (B,ψ).)
From Quantum chemistry By McQuarrie

1a.
Find the domain and range of the function. (Enter your answer
using interval notation.)
f(x) = −|x + 8|
domain=
range=
1b.
Consider the following function. Find the composite
functions
f ∘ g
and
g ∘ f.
Find the domain of each composite function. (Enter your domains
using interval notation.)
f(x) =
x − 3
g(x) = x2
(f ∘ g)(x)=
domain =
(g ∘ f)(x) =
domain
are the two functions equal?
y
n
1c.
Convert the radian...

use
the squeeze theorum to show that
*** please show work
limx→0 cos(x)x^8 sin(1/x)=0
limx→0 tan(x)x^4 cos(2/x)=0

In this problem, x = c1 cos t + c2 sin t is a two-parameter
family of solutions of the second-order DE x'' + x = 0. Find a
solution of the second-order IVP consisting of this differential
equation and the given initial conditions.
x(π/6) = 1 2 , x'(π/6) = 0
x=

(a) verfiy that y=tan(x+c) ia a one parameter family of
solutions of the differential equation y'= 1+x^2
(b)since f(x,y)= 1+y^2 and df/dy= 2y are continuous everywhere,
the region R can be taken to be the entire xy-plane. Use the family
of solutions in part A to find an explicit solution of the first
order initial value problem y'= 1+y^2, y(0)=0. Even though x0=0 is
in the interval (-2,2) explain why the solution is not defined on
its interval
(c)determine the...

1) In the interval [0,2π) find all the solutions possible (in
radians ) :
a) sin(x)= √3/2
b) √3 cot(x)= -1
c) cos ^2 (x) =-cos(x)
2)The following exercises show a method of solving an equation
of the form: sin( AxB C + ) = , for 0 ≤ x < 2π . Find ALL
solutions .
d) sin(3x) = - 1/2
e) sin(x + π/4) = - √2 /2
f) sin(x/2 - π/3) = 1/2

1.Solve sin(x)=−0.61sin(x)=-0.61 on 0≤x<2π0≤x<2πThere are
two solutions, A and B, with A < B
2.Solve 5cos(5x)=25cos(5x)=2 for the smallest three positive
solutions.Give your answers accurate to at least two decimal
places, as a list separated by commas
3.Solve 5sin(π4x)=35sin(π4x)=3 for the four smallest positive
solutions
4.Solve for tt, 0≤t<2π0≤t<2π
21sin(t)cos(t)=9sin(t)21sin(t)cos(t)=9sin(t)
5.Solve for the exact solutions in the interval [0,2π)[0,2π). If
the equation has no solutions, respond with DNE.
2sec2(x)=3−tan(x)
6.Give the smallest two solutions of sin(7θθ) = -0.6942 on [...

(Part 1) Find all of the solutions of the given differential
equations:
a.) y' = -2y (answer should be y = -(1 / ln2) * ln(t
* ln(2) + c))
(Part 2) Find the solution of the IVP:
b.) y' = -2y3, y(0) = 0
c.) y' = 1 + cos(y), y(0) = pi / 2 (answer should be y(t) =
2arctan(1 + t))
d.) y' = sqrt(1 - y2), y(0) = 0 (Hint: y' > 0)
Please show work!

1. The absolute maximum value of f(x) = x 3 − 3x 2 + 12 on the
interval [−2, 4] occurs at x =? Show your work.
2.t. Let f(x) = sin x + cos2 x. Find the absolute maximum, and
absolute minimum value of f on [0, π]. Show your work.
Absolute maximum:
Absolute minimum:
3.Let f(x) = x √ (x − 2). The critical numbers of f are_______.
Show your work.

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