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Where should I look for information regarding this type of problem: find all values on the graph of f(x) = x -3cos(x) where the tangent line has slope 1

To solve the problem, "find all values on the graph of \( f(x) = x - 3\cos(x) \) where the tangent line has slope 1," you'll want to focus on:

**3.2 The Derivative as a Function** and **3.3 Differentiation Rules**. This is because:
- The derivative \( f'(x) \) gives the slope of the tangent line to the curve at any point \( x \).
- To find where the tangent line has slope 1, you will need to set \( f'(x) = 1 \) and solve for \( x \).

Here’s how you can approach it:

1. **Find the derivative of \( f(x) \):**
   - The function is \( f(x) = x - 3\cos(x) \).
   - The derivative \( f'(x) = 1 + 3\sin(x) \), applying the derivative rules (the derivative of \( x \) is 1, and the derivative of \( -3\cos(x) \) is \( +3\sin(x) \)).

2. **Set the derivative equal to the slope of the tangent line:**
   - We want the slope to be 1, so \( f'(x) = 1 \).
   - This gives the equation \( 1 + 3\sin(x) = 1 \).

3. **Solve the equation:**
   - Simplify: \( 3\sin(x) = 0 \), so \( \sin(x) = 0 \).
   - The solutions to \( \sin(x) = 0 \) occur at \( x = n\pi \), where \( n \) is an integer.

Thus, the values of \( x \) where the tangent line has slope 1 are \( x = n\pi \) for integers \( n \).

Would you like a more detailed explanation or further help with related topics?

Here are 5 relative questions for further practice:
1. How do you interpret the graph of a derivative?
2. Can you find the tangent line equation at \( x = \pi \) for the given function?
3. How do you apply the chain rule to more complex trigonometric functions?
4. What happens to the slope of a function at points of inflection?
5. Can you solve a similar problem for \( f(x) = e^x + \sin(x) \)?

**Tip:** Always verify your solutions by plugging the values of \( x \) back into the original function to ensure the slope condition holds!

Learn how to find all values on the graph of f(x) = x - 3cos(x) where the tangent line has a slope of 1. This detailed solution covers key calculus concepts, including the derivative as a function and differentiation rules. Suitable for students in Grades 11-12.

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