Question

Compute the first-order forward, backward, and central difference approximations to estimate the first derivative of y = x3 +4x−15 at x = 0 with h = 0.25. Compare your results with the analytical solution.

Answer #1

Use forward and backward difference approximations of O(h) and a
centered difference approximation of O(h^2 ) to estimate the first
and second derivatives of
f(x)= 0.4x^5 ‐0.2x^3 +6x^2 ‐13 at x=2 using a step size h=1.
Repeat the computation using h values of 0.5, 0.25, and 0.1.
Compare your results with the exact derivative value
at x=2.

find the derivative of cos x at x=1 by first order forward 2nd
order and 4th order central schemes with h=0.01

Derive or prove the three point backward difference
approximation of the first derivative, aka Dh--f, of a
function f(x), using either polynomial interpolant method (Newton's
or Lagrange's) here is the first derivative approximation: f'(x) =
(3* f(x) - 4*f(x-h)+ f(x-2h)) /2h , which is congruent to
Dh--f

Consider the linear first order system [16]
x′ = x + y (1) y′ =4x−2y. (2)
(a) Determine the equilibria of System (1)-(2) as well as their
stability. [6]
(b) Compute the general solution of System (1)-(2). [6]
(c) Determine the solution of the initial value problem
associated with System (1)-(2), with initial condition x(0) = 1,
y(0) = 2.

3) If f=f(x,
y), derive a forward finite difference approximation of
3rd order accuracy, O(h³), for
∂f/∂x.

1218) y=Ax+Cx^B is the general solution of the first-order homogeneous DEQ: (x-y) dx - 2x dy = 0. Determine A and B. Also, include a manual solution in your portfolio. ans:2
1220) y=Ax+Dx^B is the particular solution of the first-order homogeneous DEQ: (x-y) = 2xy'. Determine A,B, & D given the boundary conditions: x=7 and y=5. Include a manual solution in your portfolio. ans:3

1. (5pts.) Compute the derivative dy/dx for y = 7√ 9π +
x ^5 /6 + 27e^x .
3. (5pts.) Write the equation of the tangent line to the
graph of y = 3 + 8 ln x at the point where x = 1.
4. (5pts.) Determine the slope of the tangent line to
the curve 2x^3 + y^3 + 2xy = 14 at the point (1, 2).
5. (5pts.) Compute the derivative dw/dz of the function
w =...

6. Consider the initial value problem
y' = ty^2 + y, y(0) = 0.25,
with (exact) solution y(t).
(a) Verify that the solution of the initial value problem is
y(t) = 1/(3e^(-t) − t + 1)
and evaluate y(1) to at least four decimal places.
(b) Use Euler’s method to approximate y(1), using a step size of
h = 0.5, and evaluate the difference between y(1) and the Euler’s
method approximation.
(c) Use MATLAB to implement Euler’s method with each...

46. Use Newton's Method to approximate the zero(s) of the
function. Continue the iterations until two successive
approximations differ by less than 0.001. Then find the zero(s) to
three decimal places using a graphing utility and compare the
results.
f(x) = 2 − x3
Newton's method:
Graphing utility:
x =
x =
48. Find the differential dy of the given function.
(Use "dx" for dx.)
y = x+1/3x-5
dy =
49.Find the differential dy of the given function.
y...

Assume that we are working with an aluminum alloy (k = 180
W/moC) triangular fin with a length, L = 5 cm, base thickness, b =
1 cm, a very large width, w = 1 m. The base of the fin is
maintained at a temperature of T0 = 200oC (at the left boundary
node). The fin is losing heat to the surrounding air/medium at T? =
25oC with a heat transfer coefficient of h = 15 W/m2oC. Using the...

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