How do you find the equation of the tangent line to the curve #f(x)= x + cos (x)# at x = 0?

Question

Scarlett Allison · Accepted Answer

Equation of tangent line is #x-y+1=0#

To find equation of tangent line of #f(x)=x+cosx# at #x=0#, we should first find the slope of the tangent and value of function at #x=0#. Then, we can get the equation of the tangent from point slope form of the equation.

At #x=0#, #f(x)=0+cos0=1#

Slope of tangent at #x=0# is given by value of #(dy)/(dx)# at #x=0#.

#(df)/(dx)=1-sinx#, the slope at #x=0# will be

#1-sin0=1-0=1#

Hence slope of tangent is #1#

Hence, equation of tangent line is

#(y-1)=1(x-0)# or #x-y+1=0#

Paisley Johnson · Answer

undefined To find the equation of the tangent line to the curve f(x) = x + cos(x) at x = 0, we need to find the slope of the tangent line at that point and then use the point-slope form of a line to write the equation.

To find the slope, we take the derivative of f(x) with respect to x. The derivative of x is 1, and the derivative of cos(x) is -sin(x). Therefore, the derivative of f(x) is 1 - sin(x).

Plugging in x = 0 into the derivative, we get 1 - sin(0) = 1 - 0 = 1. So, the slope of the tangent line at x = 0 is 1.

Using the point-slope form of a line, which is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope, we can substitute x1 = 0, y1 = f(0) = 0 + cos(0) = 1 into the equation.

Therefore, the equation of the tangent line to the curve f(x) = x + cos(x) at x = 0 is y - 1 = 1(x - 0), which simplifies to y = x + 1.