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Find the equation of the line tangent to the graph of [tex]f(x) = 4 - \cos(x)[/tex] at [tex]x = 0[/tex].

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GinnyAnswer
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]
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