How do you find the equation of the tangent line to the curve #y= x + cos x# at (0,1)?

Question

Isaiah Allen · Accepted Answer

#y=x+1#

First, find the derivative of the equation

#f'(x)=1-sinx#

Input #x# to find the value of the slope

#f'(0)=1-sin(0)#

Simplify

#f'(0)=1-0=1#

The slope, #m#, is #1#.

Make the substitutions into the slope intercept formula, #y=mx+b#

#1=(1)(0)+b#

Solve for #b#. Simplify

#1=b#

Write the equation of the tangent line.

#y=1x+1#

#or#

#y=x+1#

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Christopher Allers · Answer

undefined To find the equation of the tangent line to the curve y = x + cos(x) at (0,1), we need to find the slope of the tangent line at that point.

To find the slope, we take the derivative of the function y = x + cos(x) with respect to x.

The derivative of x with respect to x is 1, and the derivative of cos(x) with respect to x is -sin(x).

So, the derivative of y = x + cos(x) is dy/dx = 1 - sin(x).

To find the slope at (0,1), we substitute x = 0 into the derivative:

dy/dx = 1 - sin(0) = 1 - 0 = 1.

Therefore, the slope of the tangent line at (0,1) is 1.

Using the point-slope form of a line, y - y1 = m(x - x1), where (x1, y1) is the point (0,1) and m is the slope (1), we can substitute the values:

y - 1 = 1(x - 0)

Simplifying, we get the equation of the tangent line:

y - 1 = x

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