Determine the equation of the tangent line to the function f(x) = x + Sinx at x = pie
first find the slope at x = pi
dy/dx = 1 + cos x = 1+cos pi = 1-1 = 0
so the line is horizontal
at x = pi
y = pi + sin pi = pi + 0 = pi
so the equation of our line is
y = pi
Thank you Damon!
To determine the equation of the tangent line to the function f(x) = x + Sin(x) at x = π, you can follow these steps:
1. Find the derivative of the function f(x) with respect to x. In this case, the derivative is obtained by taking the derivative of each term separately. The derivative of x is 1, and the derivative of Sin(x) is Cos(x). So, the derivative of f(x) = x + Sin(x) is f'(x) = 1 + Cos(x).
2. Substitute x = π into the derivative function f'(x) to find the slope of the tangent line. The slope of the tangent line at x = π is f'(π) = 1 + Cos(π). Since Cos(π) equals -1, the slope becomes f'(π) = 1 - 1 = 0.
3. Now that you have the slope of the tangent line, you need a point on the line to determine the equation fully. The given point is (π, f(π)) = (π, π + Sin(π)). Since Sin(π) is equal to 0, the point becomes (π, π).
4. Use the slope-intercept form of a line, y = mx + b, where m is the slope and b is the y-intercept. Since the slope is 0, the equation becomes y = 0x + b. Since the line passes through the point (π, π), we can substitute x = π and y = π to solve for b. The equation becomes π = 0(π) + b, which simplifies to b = π.
5. Finally, write the equation of the tangent line using the slope-intercept form. The equation becomes y = 0x + π, which further simplifies to y = π.
Therefore, the equation of the tangent line to the function f(x) = x + Sin(x) at x = π is y = π.