find all x values on the graph of f(x) = x-2cos(x) where the tangent line has slope 1

To find all \( x \) values where the tangent line to the graph of the function \( f(x) = x - 2\cos(x) \) has a slope of 1, we follow these steps:

### 1. Take the derivative of \( f(x) \)
The derivative of a function gives the slope of the tangent line at any point \( x \). The derivative of \( f(x) \) is:

\[
f'(x) = \frac{d}{dx} \left( x - 2\cos(x) \right)
\]

Using basic differentiation rules:

\[
f'(x) = 1 + 2\sin(x)
\]

### 2. Set the derivative equal to 1
Since the slope of the tangent line is given as 1, we set \( f'(x) = 1 \):

\[
1 + 2\sin(x) = 1
\]

### 3. Solve for \( x \)
Simplify the equation:

\[
2\sin(x) = 0
\]

\[
\sin(x) = 0
\]

Now, solve for \( x \). The general solutions for \( \sin(x) = 0 \) are:

\[
x = n\pi \quad \text{where} \quad n \in \mathbb{Z}
\]

### 4. Conclusion
The values of \( x \) where the tangent line to the graph of \( f(x) = x - 2\cos(x) \) has a slope of 1 are:

\[
x = n\pi \quad \text{where} \quad n \in \mathbb{Z} \ (\text{any integer}).
\]

Would you like more details or have any questions on this?

### Related questions:
1. How would the solution change if the slope of the tangent line were different?
2. What is the significance of \( n\pi \) in terms of periodicity on the graph?
3. How do we determine the concavity of \( f(x) = x - 2\cos(x) \)?
4. What does the second derivative tell us about the behavior of \( f(x) \)?
5. How can you graphically interpret the result \( x = n\pi \)?

### Tip:
The sine function \( \sin(x) = 0 \) repeats periodically, which is why the solution is valid for any integer \( n \). Keep this in mind when solving trigonometric equations.

Find all x values on the graph of f(x) = x - 2cos(x) where the tangent line has slope 1.

Learn how to find the x values on the graph of f(x) = x - 2cos(x) where the tangent line has a slope of 1. This problem uses calculus concepts such as differentiation and trigonometry. The solution involves finding the derivative and solving trigonometric equations. Suitable for Grades 11-12 or college-level calculus students.

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