Question

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

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.

Taylor series
f(xi+1)=f(xi)+f'(xi)h+(f''(xi)/2!)h2+(f'''(xi)/3!)h3+.....
given
f''(xi)=(-f(xi+3)+4f(xi+2)-5f(xi+1)+2f(xi))/h2
derive the second derivative of forward finite difference
n=3
what is the order of error

Taylor series
f(xi+1)=f(xi)+f'(xi)h+(f''(xi)/2!)h2+(f'''(xi)/3!)h3+.....
given
f''(xi)=(-f(xi+3)+4f(xi+2)-5f(xi+1)+2f(xi))/h2
derive the second derivative of forward finite
difference

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 function f(x) = sin(x). Suppose we want to
approximate f 0 (0) by using a forward difference approximation and
a stepsize of h. How small must h be in order to guarantee that the
absolute error in our approximation is less than 0.01?

Let f(x,y)=xcos(πy)−ysin(πx)f(x,y)=xcos(πy)−ysin(πx). Find the
second-order Taylor approximation for ff at the point (1, 2).

-find the differential and linear approximation of f(x,y) =
sqrt(x^2+y^3) at the point (1,2)
-use tge differential to estimate f(1.04,1.98)

1) Use finite approximation to estimate the area under the graph
of f(x) = x^2 and above the graph of f(x) = 0 from Xo = 0 to Xn= 2
using
i) a lower sum with two rectangles of equal width
ii) a lower sum with four rectangles of equal width
iii) an upper sum with two rectangles of equal width
iv) an upper sum with four rectangles if equal width
2) Use finite approximation to estimate the area under...

Prove the following :
(∃x)(F(x)⋀(G(x)⋁(H(x)))→(∃x)(∃y)(F(x)⋀(G(y)⋁H(y)))

Let f(x) = 3x^2+x. Find the difference quotient.
f(x+h)-f(x) / h

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