Resistors in parallel Two resistors wired in parallel in an electrical circuit give an effective resistance of R ( x , y ) = x x + y , where x and y are the positive resistances of the individual resistors (typically measured in ohms). a. Graph the resistance function using the window [0, 10] × [0, 10] × [0, 5]. b. Estimate the maximum value of R , for 0 < x ≤ 10 and 0 < y ≤ 10. c. Explain what it means to say that the resistance function is symmetric in x and y.
Resistors in parallel Two resistors wired in parallel in an electrical circuit give an effective resistance of R ( x , y ) = x x + y , where x and y are the positive resistances of the individual resistors (typically measured in ohms). a. Graph the resistance function using the window [0, 10] × [0, 10] × [0, 5]. b. Estimate the maximum value of R , for 0 < x ≤ 10 and 0 < y ≤ 10. c. Explain what it means to say that the resistance function is symmetric in x and y.
Solution Summary: The author illustrates the effective resistancefunction R(x,y)=xy
Resistors in parallel Two resistors wired in parallel in an electrical circuit give an effective resistance of
R
(
x
,
y
)
=
x
x
+
y
, where x and y are the positive resistances of the individual resistors (typically measured in ohms).
a. Graph the resistance function using the window [0, 10] × [0, 10] × [0, 5].
b. Estimate the maximum value of R, for 0 < x ≤ 10 and 0 < y ≤ 10.
c. Explain what it means to say that the resistance function is symmetric in x and y.
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)
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