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 show the answer as well because I was not able to get the correct answer last time either. Please help me.. Thank you.
show me what you entered as an answer.
.5-(sqrt(3)/2)(x-pi/3)
To find the linear approximation (linearization) of a function at a given point, you need to use the tangent line to the function at that point. The equation for a tangent line is in the form y = mx + b, where m is the slope of the line and b is the y-intercept.
To find the linearization of the function y = sin(x)/x at x = π/3, we need to follow these steps:
1. Calculate the derivative of the function.
2. Evaluate the derivative at x = π/3 to find the slope of the tangent line (m).
3. Plug in the given x-value (π/3) and the slope (m) in the equation y = mx + b and solve for the y-intercept (b).
4. Write down the linearization in the form y = mx + b.
Let's go through each step:
Step 1: Calculate the derivative of the function.
For y = sin(x)/x, we can use the quotient rule:
dy/dx = (x*cos(x)-sin(x))/x^2
Step 2: Evaluate the derivative at x = π/3 to find the slope (m).
Substitute x = π/3 into the derivative equation:
dy/dx = (π/3*cos(π/3)-sin(π/3))/(π/3)^2
We can calculate the value of the slope using π ≈ 3.14159:
m = (π/3*cos(π/3)-sin(π/3))/(π/3)^2
Step 3: Find the y-intercept (b).
Plug in the x-value (π/3) and the slope (m) into the equation y = mx + b and solve for b:
sin(π/3)/π - (π/3*cos(π/3)-sin(π/3))/(π/3)^2 = m(π/3) + b
Simplify further to solve for b.
Step 4: Write down the linearization in the form y = mx + b.
Using the slope (m) and the y-intercept (b) from the previous steps, we can write the linearization:
y = m*x + b
Now you can substitute the accurate values you calculated in the previous steps to get the final answer.