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(Background: I'm learning about divided differences from this YouTube video).

What I'm looking for is to understand the steps to rearrange a particular equation from one form to another. The original equation is: $$f(x_2) = f(x_0) + \frac{f(x_1)-f(x_0)}{(x_1-x_0)}(x_2-x_0) + b_2(x_2-x_0)(x_2-x_1)$$

The equation when solved for $b_2$ is: $$b_2 = \frac{\frac{f(x_2)-f(x_1)}{x_2-x_1}-\frac{f(x_1)-f(x_0)}{x_1-x_0}}{x_2-x_0}$$

I've tried to solve for $b_2$ but keep messing up! Perhaps someone would be kind enough to show me the steps?

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Let's replace $f(x_i)$ with $y_i$, to reduce clutter. We start with $$ y_2 = y_0 + \frac{y_1-y_0}{x_1 - x_0}(x_2 - x_0) + b_2(x_2-x_0)(x_2-x_1). $$ We want the factor involving $b_2$ on one side, and the rest on the other side. So let's move the first two terms from the right-hand side to the left-hand side: $$ y_2 - y_0 - \frac{y_1 - y_0}{x_1 - x_0}(x_2 - x_0) = b_2(x_2 - x_0)(x_2 - x_1). $$ In order to isolate $b_2$, we need to divide by $(x_2 - x_0)(x_2 - x_1)$. Let's do it in two steps. First divide by $x_2 - x_0$: $$ \frac{y_2 - y_0}{x_2 - x_0} - \frac{y_1 - y_0}{x_1 - x_0} = b_2(x_2 - x_1). $$ Now divide by $x_2 - x_1$: $$ \frac{\frac{y_2 - y_0}{x_2 - x_0} - \frac{y_1 - y_0}{x_1 - x_0}}{x_2 - x_1} = b_2. $$ We got a different formula.

In order to get the formula you stated, we need to somehow come up with the expression $y_2 - y_1$. So let's subtract $y_1$ from both sides of the original equation: $$ y_2 - y_1 = y_0 - y_1 + \frac{y_1-y_0}{x_1 - x_0}(x_2 - x_0) + b_2(x_2-x_0)(x_2-x_1). $$ We have $$ y_0 - y_1 + \frac{y_1-y_0}{x_1-x_0}(x_2-x_0) = \frac{y_1-y_0}{x_1-x_0}(x_2 - x_0 - (x_1 - x_0)) = \frac{y_1-y_0}{x_1-x_0}(x_2 - x_1). $$ For this calculation, we wrote $$ y_0 - y_1 = \frac{(y_0-y_1)(x_1-x_0)}{x_1-x_0}. $$ Substituting this simplification into the equation, we get $$y_2 - y_1 = \frac{y_1-y_0}{x_1-x_0}(x_2-x_1) + b_2(x_2-x_0)(x_2-x_1).$$ Let's move everything to the correct side: $$y_2 - y_1 - \frac{y_1-y_0}{x_1-x_0}(x_2-x_1) = b_2(x_2-x_0)(x_2-x_1).$$ Now divide first by $x_2 - x_1$ (looking toward our goal): $$\frac{y_2 - y_1}{x_2 - x_1} - \frac{y_1-y_0}{x_1-x_0} = b_2(x_2-x_0).$$ Next divide by $x_2 - x_0$: $$\frac{\frac{y_2 - y_1}{x_2 - x_1} - \frac{y_1-y_0}{x_1-x_0}}{x_2-x_0} = b_2.$$ And we're done.

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    I'm astounded that you managed to come up with that so quickly. Took me a while to work through it. I appreciate your help.2011-06-05