Which of the following is the best linear approximation for f(x)equals sin(x) near x equals =π seconds?
y=-x+pie-1
y=-1
y=-x-pie
y=-x+pie
we all know that near x=0, sin(x) = x (or nearly)
Since sin(pi-x) = sin(x),
sin(pi-x) = pi-x
To find the best linear approximation for f(x) = sin(x) near x = π, we can use the formula for linear approximation:
L(x) = f(a) + f'(a) * (x - a)
where a is the point at which we want to approximate the function, and f'(a) is the derivative of the function evaluated at a.
In this case, a = π and f(a) = sin(π) = 0. The derivative of f(x) = sin(x) is f'(x) = cos(x). Evaluating f'(a) at a = π gives us f'(π) = cos(π) = -1.
Substituting these values into the linear approximation formula, we get:
L(x) = 0 + (-1) * (x - π)
= -x + π
Therefore, the best linear approximation for f(x) = sin(x) near x = π is y = -x + π.
Out of the given answer choices:
y = -x + π - 1 is not the best linear approximation because the constant term should be π instead of π - 1.
y = -1 is not the best linear approximation because it does not depend on the value of x.
y = -x - π is not the best linear approximation because the constant term should be positive instead of negative.
y = -x + π is the best linear approximation based on the explanation above.