Find the linearization of the function below at x = π/3.?

y=sin(x)/x

Your answer should be a linear function of x where the coefficients are accurate to at least two decimal places. You can enter π in your answer as "pi" (without the quotes).

I am not able to figure out how to find the answer. I tried couple times but it is showing incorrect. Please help me.. Thank you.

To find the linearization of the function y = sin(x)/x at x = π/3, we can use the concept of linear approximation. The linearization is an approximation of the function near a specific point using a linear function.

To begin, we need to find the value of the function at x = π/3, which will be the y-coordinate of the point on the graph we want to approximate.

Substitute x = π/3 into the function:
y = sin(π/3)/π/3

We know that sin(π/3) is equal to √3/2, so:
y = (√3/2)/(π/3)

To simplify, multiply the numerator and denominator by 2:
y = (2√3)/(2π/3)

Divide by π/3, which is equivalent to multiplying by its reciprocal (3/π):
y = (2√3)(3/π)

Simplify further:
y = (6√3)/π

Now that we have the value of the function at x = π/3, let's find its derivative with respect to x. This derivative will represent the slope of the linear function:

To differentiate y = sin(x)/x, we can use the quotient rule:
dy/dx = (x*cos(x) - sin(x))/x^2

Evaluate this derivative at x = π/3:
dy/dx = (π/3 * cos(π/3) - sin(π/3))/(π/3)^2

cos(π/3) is equal to 1/2 and sin(π/3) is equal to √3/2, so:
dy/dx = (π/3 * 1/2 - √3/2)/(π/3)^2

To simplify, multiply the numerator by 2 and the denominator by π/3:
dy/dx = (2π/6 - √3/2)/(π^2/9)

Further simplify by multiplying by the reciprocal of π^2/9, which is 9/π^2:
dy/dx = (2π/6 - √3/2)*(9/π^2)

Simplify further:
dy/dx = (3π - 3√3)/(π^2)

Now we have the slope of the linear function. To complete the linearization, we need the equation of the line in the form y = mx + b, where m is the slope (dy/dx) and b is the y-coordinate at x = π/3.

Substituting the known values into the equation:
y = mx + b
y = (3π - 3√3)/(π^2) * x + (6√3)/π

This is the linearization of the function y = sin(x)/x at x = π/3.

linearization.

f(x)=f(a)+f'(a)(x-a)

Well, f(a)=cos PI/3
f'(a)=-sin(PI/3)

f(x)=cosPI/3-sin(PI/3)*(x-PI/3)

find with your calculator cosPI/3, and sin PI/3 and you have your linear equation.