find the first three non-zero terms of the Maclaurin expansion of y = f (x)
where the function is defined implicitly by x^2 + xy + y^2 = 1. Also estimate
the error approximating f(0.1) using the first two non–zero terms of this
series.
To find the first three non-zero terms of the Maclaurin expansion of y = f(x), we need to differentiate the given implicit equation with respect to x.
The given equation is x^2 + xy + y^2 = 1.
Differentiating with respect to x, we get:
2x + x(dy/dx) + y + 2yy' = 0
Now, we can solve this equation for dy/dx to get an expression for the derivative of y with respect to x.
dy/dx = (-2x - y) / (x + 2y)
To find the first non-zero term, we substitute x = 0 and y = 0 (since it is a Maclaurin expansion). From dy/dx, we get the first non-zero term:
dy/dx = (-2(0) - 0) / (0 + 2(0))
dy/dx = 0
So the first non-zero term of the Maclaurin expansion is 0.
To find the second non-zero term, we need to find the derivative of dy/dx with respect to x and substitute x = 0 and y = 0 again.
Differentiating dy/dx with respect to x, we get:
d^2y/dx^2 = (-2 - (dy/dx)) / (x + 2y) = (-2 - 0) / (0 + 2(0)) = -2
So the second non-zero term of the Maclaurin expansion is -2.
To find the third non-zero term, we need to find the derivative of d^2y/dx^2 with respect to x and substitute x = 0 and y = 0 again.
Differentiating d^2y/dx^2 with respect to x, we get:
d^3y/dx^3 = d^2y/dx^2 / (x + 2y) = -2 / (0 + 2(0)) = -2
So the third non-zero term of the Maclaurin expansion is -2.
Therefore, the first three non-zero terms of the Maclaurin expansion of y = f(x) are:
0 - 2x - 2x^2.
To estimate the error approximating f(0.1) using the first two non-zero terms of the series, we can use the Lagrange error bound formula.
The Lagrange error bound formula states that the absolute value of the error is bounded by:
|E| <= (M * |x - a|^(n+1)) / (n!)
Where M is the maximum value of the (n+1)th derivative on the interval [a, x].
Since we only have the first two non-zero terms, we only need to consider the error bound for the third non-zero term.
The third non-zero term is -2x^2. The coefficient of x^2 is -2, so M = 2.
Substituting the values into the formula, we get:
|E| <= (2 * |0.1 - 0|^(3)) / (3!)
Simplifying,
|E| <= (2 * 0.1^3) / 6
|E| <= 0.000333
Therefore, the error approximating f(0.1) using the first two non-zero terms of the series is estimated to be less than or equal to 0.000333.