Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Chapter 13.3, Problem 51E
(a)
To determine
To explain: Whether the given statement “
projvu=projuv“ is true or false.
If false, give a counterexample.
(b)
To determine
To explain: Whether the given statement “If nonzero vectorsu and v have the same magnitude they make equal angles with
u+v“ is true or false. If false, give a counterexample.
(c)
To determine
To explain: Whether the given statement “
(u⋅i)2+(u⋅j)2+(u⋅k)2=|u|2“ is true or false. If false, give a counterexample.
(d)
To determine
To explain: Whether the given statement “If
u is orthogonal to
v and
v is orthogonal to
w, then
u is orthogonal to
w.“ is true or false. If false, give a counterexample.
(e)
To determine
To explain: Whether the given statement “The vectors orthogonal to
〈1,1,1〉 lie on the same line.“ is true or false. If false, give a counterexample.
(f)
To determine
To explain: Whether the given statement “If the
projvu=0,then u and v (both nonzero) are orthogonal.“ is true or false. If false, give a counterexample.
The OU process studied in the previous problem is a common model for interest rates.
Another common model is the CIR model, which solves the SDE:
dX₁ = (a = X₁) dt + σ √X+dWt,
-
under the condition Xoxo. We cannot solve this SDE explicitly.
=
(a) Use the Brownian trajectory simulated in part (a) of Problem 1, and the Euler
scheme to simulate a trajectory of the CIR process. On a graph, represent both the
trajectory of the OU process and the trajectory of the CIR process for the same
Brownian path.
(b) Repeat the simulation of the CIR process above M times (M large), for a large
value of T, and use the result to estimate the long-term expectation and variance
of the CIR process. How do they compare to the ones of the OU process?
Numerical application: T = 10, N = 500, a = 0.04, x0 = 0.05, σ = 0.01, M = 1000.
1
(c) If you use larger values than above for the parameters, such as the ones in Problem
1, you may encounter errors when implementing the Euler scheme for CIR. Explain
why.
#8 (a) Find the equation of the tangent line to y = √x+3 at x=6
(b) Find the differential dy at y = √x +3 and evaluate it for x=6 and dx = 0.3
Q.2 Q.4 Determine ffx dA where R is upper half of the circle shown below.
x²+y2=1
(1,0)