Taylor Approximation of y'=t/y y(0)=1?

Question

I dont fully understand the taylor approx from calc 3c...could you help me out with this problem...y'=t/y y(0)=1 at y(.5) and y(1) when h=.1

thanks work will help a lot...any little bit cause i have more problems involving this

Answer 1

If f is a differentiable function, then we can define the Taylor series around a point a as follows:

T(f,a,x) = f(a) + (1/1!)f'(a)x + (1/2!)f''(a)x^2 + (1/3!)f'''(a)x^3 + ...

If f is well behaved at a, then the Taylor series near a converges, and for all x that is "small enough",

f(a+x) = T(f,a,x)

In your case, you have a = 0, y(a), and a formula for y'(t), so you can compute all the coefficients you need for T(y, 0, x)

It is often the case that you don't need to (or can't) compute the value of a function exactly. In that case, you can often get a good approximation by using only the first few terms of the Taylor series. The number of terms to use depends on the function, the accuracy you need, etc.

Now just how this relates to the specifics of your problem is not at all clear to me. The bit about h = 0.1 suggests that you are trying to solve a differential equation and are asking for the values at 0.5 and 1.0. There is a method of using Taylor series for solving differential equations, but it isn't something one would want to use by hand.

How about more information about your problems?

Answer 2

you like f(a million+h,a million+ok) accelerated as much as 2nd order words in h and ok. This demands first and 2nd order partial differentiation. f_x, f_y, f_xx,f_xy and f_yy so f_x=-2/(3y-2x), f_y=3/(3y-2x), f_xx=-4/(3y-2x)^2, f_xy=6/(3y-2x)^2, f_yy=-9/(3y-2x)^2 the final 2nd order boost of f(x+h,y+ok) = f(x,y)+{hf_x+kf_y)+(a million/2){h^2f_xx+2hkf_xy... all partial derivatives on your case would be calculated at x=a million,y=a million and that i make f(a million+h,a million+ok)=-2h+3k-2h^2+6hk-4.5k^2 and substituting a million+h=x, h=x-a million and ok=y-a million you get Q(x,y)=-2(x-a million)+3(y-a million)-2(x-a million)^2+6(x-a million)(... - 4.5(y-a million)^2 yet verify my paintings word that f_x is partial df/dx, f_xx= partial d^2f/dx^2 and so on.