To find the equation of the line tangent to the graph of the function [tex]\( f(x) = 4 - \cos(x) \)[/tex] at [tex]\( x = 0 \)[/tex], we'll follow these steps:
1. Find the derivative of [tex]\( f(x) \)[/tex] to determine the slope of the tangent line.
The function given is:
[tex]\[
f(x) = 4 - \cos(x)
\][/tex]
To find the slope of the tangent line, we need to compute the derivative [tex]\( f'(x) \)[/tex]. The derivative of [tex]\( \cos(x) \)[/tex] is [tex]\( -\sin(x) \)[/tex]. Therefore:
[tex]\[
f'(x) = 0 + \sin(x) = \sin(x)
\][/tex]
2. Evaluate the derivative at [tex]\( x = 0 \)[/tex] to find the slope at that point.
We need to find [tex]\( f'(0) \)[/tex]:
[tex]\[
f'(0) = \sin(0) = 0
\][/tex]
Therefore, the slope of the tangent line at [tex]\( x = 0 \)[/tex] is [tex]\( 0 \)[/tex].
3. Determine the y-coordinate of the function at [tex]\( x = 0 \)[/tex].
We calculate [tex]\( f(0) \)[/tex]:
[tex]\[
f(0) = 4 - \cos(0) = 4 - 1 = 3
\][/tex]
So the function passes through the point [tex]\( (0, 3) \)[/tex].
4. Use the point-slope form of the equation of a line to write the equation of the tangent line.
The point-slope form of a line is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, [tex]\( m \)[/tex] is the slope, and [tex]\( (x_1, y_1) \)[/tex] is the point of tangency. We have [tex]\( m = 0 \)[/tex] and the point [tex]\( (x_1, y_1) = (0, 3) \)[/tex].
Substituting in these values, we get:
[tex]\[
y - 3 = 0(x - 0)
\][/tex]
Simplifying, we find:
[tex]\[
y = 3
\][/tex]
Therefore, the equation of the line tangent to the graph of [tex]\( f(x) = 4 - \cos(x) \)[/tex] at [tex]\( x = 0 \)[/tex] is:
[tex]\[
\boxed{y = 3}
\][/tex]
