1) Find the linearization of
f(x) = sin(x)
at x = 0 AND use it to estimate
sin(pi/87.129)
since sin(x) = x - x^3/6! + ...
the linear approximation at x=0 is sin(x) = x
so, sin(pi/87.129) = pi/87.129
In fact,
pi/87.129 = 0.03605679
sin(pi/87.129) = 0.03604898
To find the linearization of a function at a specific point, we need to use the formula for linear approximation:
L(x) = f(a) + f'(a)(x - a)
where L(x) represents the linearization at x, f(a) represents the value of the function at the point a, f'(a) represents the derivative of the function at the point a, and (x - a) represents the difference between the given x and the point a.
In this case, we want to find the linearization of f(x) = sin(x) at x = 0. Let's calculate it step by step:
Step 1: Find f(0)
Plug in x = 0 into the function f(x) = sin(x):
f(0) = sin(0) = 0
Step 2: Find f'(x)
Take the derivative of f(x) = sin(x):
f'(x) = cos(x)
Step 3: Find f'(0)
Plug in x = 0 into the derivative:
f'(0) = cos(0) = 1
Step 4: Use the linear approximation formula
Plug the values we found into the formula:
L(x) = f(0) + f'(0)(x - 0)
= 0 + 1(x - 0)
= x
Therefore, the linearization of f(x) = sin(x) at x = 0 is L(x) = x.
Now, let's use this linearization to estimate sin(pi/87.129):
Step 5: Plug in the value into the linearization
L(pi/87.129) = pi/87.129
The estimated value of sin(pi/87.129) using the linearization is pi/87.129.