Find an equation of the tangent line to the curve at the given point
y = x + cosx, (0,1)
To find the equation of the tangent line to the curve at a given point, you need to find the derivative of the function and evaluate it at the given point. The derivative will give you the slope of the tangent line at the point, and then you can use the point-slope form of a line to write the equation.
First, let's find the derivative of the function y = x + cos(x). To do this, we can apply the rules of differentiation. For the term x, the derivative is simply 1. For the term cos(x), we need to use the chain rule, which states that the derivative of cos(u) is -sin(u) times the derivative of u. In this case, u = x, so the derivative of cos(x) is -sin(x). Combining these results, the derivative of y is:
dy/dx = 1 - sin(x)
Now let's find the slope of the tangent line at the point (0,1). We need to evaluate the derivative at x = 0:
dy/dx = 1 - sin(0) = 1 - 0 = 1
So, the slope of the tangent line at (0,1) is 1.
Now we can use the point-slope form of a line to find the equation of the tangent line. The point-slope form is:
y - y₁ = m(x - x₁)
where (x₁, y₁) is the given point and m is the slope.
The given point is (0,1), and we found that the slope is 1. Plugging these values into the point-slope form, we get:
y - 1 = 1(x - 0)
y - 1 = x
Thus, the equation of the tangent line to the curve y = x + cos(x) at the point (0,1) is y = x.