Question

Find the linearization L(x) of the function at a.

f(x)=sin x, a=pi/6

Answer 1

To find the linearization of the function f(x) = sin(x) at a = π/6, we use the formula for linear approximation and find f(a) and f'(a). The linearization L(x) is then expressed as L(x) = 1/2 + (√3/2)(x - π/6).

Step-by-step explanation:

To find the linearization of the function f(x) = sin(x) at a = π/6, we need to use the formula for linear approximation. The linearization L(x) is given by L(x) = f(a) + f'(a)(x-a), where f'(x) is the derivative of f(x) with respect to x.

First, we find f(a) by plugging a into the function: f(π/6) = sin(π/6) = 1/2.

Next, we find f'(a) by taking the derivative of f(x): f'(x) = cos(x). So f'(π/6) = cos(π/6) = √3/2.

Finally, we can write the linearization L(x) as L(x) = 1/2 + (√3/2)(x - π/6).

Answer 2

As is it tangent line
by using slope formula
y-y1 = m(x-x1)
given a=pi/6=x1
f(x)= sinx
above equation can be wriiten as
f(x)-f(a) = f’(a)(x – a)
so L(x)=f’(a) (x-a) + f(a)
f(x) = sinx =f(pi/6) = 1/2= y1
f’(x) = cosx f’(pi/6)= 3/2 = m
now putting values
L(x) = √ 3/2( x-pi/6)+1/2
L(x) = (√ 3/2)x+6-pi√ 3/12