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It looks like you have provided an image related to a Julia programming exercise involving tangent lines to a function \( f(x) = \sin(x) \) at \( x = \frac{\pi}{4} \). 

The code example and instructions suggest the following:

1. **Definition of the function \( f(x) = \sin(x) \)**.
2. **Using `tangent(f, c)` in Julia**, where \( c = \frac{\pi}{4} \), to calculate the tangent line to \( f(x) \).
3. You are asked to:
   - **(c)** Use the tangent line `tline` to find the intercept \( b \) by selecting an appropriate value of \( x \).
   - **(d)** Use `tline` to find the slope \( m \) of the tangent line.

### Explanation:

- The **slope \( m \)** of the tangent line is the derivative of \( f(x) = \sin(x) \) at \( x = \frac{\pi}{4} \), which is \( f'(x) = \cos(x) \).
- At \( x = \frac{\pi}{4} \), \( f'(x) = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \). So, the slope \( m \) is \( \frac{\sqrt{2}}{2} \).
- The equation of the tangent line is \( y = mx + b \). To find \( b \), use the fact that the line passes through \( x = \frac{\pi}{4} \) and \( f\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \).

#### For Julia:

To calculate this:

```julia
using MTH229

f(x) = sin(x)
c = π/4
tline = tangent(f, c)

# Part (c): To find intercept b, choose any value of x
# Let's try x = c (π/4):
x = π/4
y = tline(x)

# Part (d): The slope m is found directly as the output from tline
```

This will calculate the intercept and slope for the tangent line.

Do you want any further details or clarifications?

Here are five related questions you might explore:

1. How does the tangent line change for different values of \( x \)?
2. Can you derive the equation of the tangent line manually for \( f(x) = \sin(x) \)?
3. What happens if you choose a different function, like \( f(x) = \cos(x) \)?
4. How does the Julia `tangent` function compute the tangent line?
5. How can you graph the tangent line and the function together in Julia?

**Tip**: When working with tangents, always remember that the slope is simply the derivative at the point of interest.

Use Julia to find the tangent line to the function f(x) = sin(x) at x = π/4. Use tline to calculate the slope (m) and intercept (b).

Learn how to calculate the tangent line for the sine function f(x) = sin(x) at x = π/4 using Julia. This exercise demonstrates how to compute the slope (m) and intercept (b) of the tangent line using calculus. Suitable for students in Grades 10-12.

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