Find the linearization of the function below at x = π
y=sin(x)/x
Your answer should be a linear function of x where the coefficients are accurate to at least two decimal places.
Don't understand this problem. please help..
To find the linearization of a function at a specific point, we can use a technique called linear approximation. The linearization of a function at a point is given by the equation of the tangent line to the graph of the function at that point.
In this problem, we need to find the linearization of the function y = sin(x)/x at x = π.
Step 1: Calculate the derivative of the function
To find the derivative of y = sin(x)/x, we can use the quotient rule. The quotient rule states that if we have a function of the form f(x) = g(x)/h(x), then its derivative is given by f'(x) = (g'(x)h(x) - g(x)h'(x))/[h(x)]^2.
Let's calculate the derivative of y = sin(x)/x:
y' = [(cos(x) * x) - sin(x)] / x^2
Step 2: Evaluate the derivative at x = π
Now, we substitute x = π into the derivative to find the slope of the tangent line at x = π:
y'(π) = [(cos(π) * π) - sin(π)] / π^2
= [-π] / π^2
= -1/π
Step 3: Find the equation of the tangent line using the point-slope form
Now that we have the slope, -1/π, and the point (π, y(π)), we can use the point-slope form of a linear equation to write the equation of the tangent line:
y - y(π) = m(x - π)
Substituting the values, we have:
y - y(π) = (-1/π)(x - π)
Step 4: Simplify the equation
We simplify the equation by distributing the -1/π:
y - y(π) = (-1/π)(x - π)
= (-1/π)x + 1
Now we have the linearization of the given function y = sin(x)/x at x = π as:
y = (-1/π)x + 1
This linearization is an approximation of the original function near x = π, accurate to at least two decimal places.