Implicit differentiation Given the following equations, evaluate dy/dx. Assume that each equation implicitly defines y as a differentiable function of x . 35. x 2 + 2 x y + y 4 = 3
Implicit differentiation Given the following equations, evaluate dy/dx. Assume that each equation implicitly defines y as a differentiable function of x . 35. x 2 + 2 x y + y 4 = 3
Solution Summary: The author explains the value of dx by using the theorem.
Implicit differentiationGiven the following equations, evaluate dy/dx. Assume that each equation implicitly defines y as a differentiable function of x.
35.
x
2
+
2
x
y
+
y
4
=
3
With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.
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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