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In the given exercise 2.D1.55, you're asked to find the equation of the tangent line to the graph of the function \( f \) at the point \( (2, f(2)) \). The information provided includes:

- \( f(2) = 5 \)
- \( f'(2) = -4 \)

### 1st Method:
The equation of a line is given by \( y = ax + b \), where \( a \) is the slope (in this case \( f'(2) \)) and \( b \) is the y-intercept.

1. The slope of the tangent is \( f'(2) = -4 \).
2. Substituting \( (x_0, f(x_0)) = (2, 5) \) into the equation \( y = ax + b \):
   \[
   5 = -4(2) + b
   \]
   \[
   5 = -8 + b
   \]
   Solving for \( b \), we get:
   \[
   b = 13
   \]
   Therefore, the equation of the tangent line is:
   \[
   y = -4x + 13
   \]

### 2nd Method:
Using the point-slope formula for the equation of a tangent line:
\[
y = f'(x_0) \cdot (x - x_0) + f(x_0)
\]
Substituting the values \( f'(2) = -4 \), \( x_0 = 2 \), and \( f(2) = 5 \):
\[
y = -4(x - 2) + 5
\]
Simplifying:
\[
y = -4x + 8 + 5
\]
\[
y = -4x + 13
\]
Thus, the equation of the tangent line is the same:
\[
y = -4x + 13
\]

Would you like more details on the methods used or have any questions?

Here are 5 related questions to think about:
1. How do you find the slope of a tangent line using a derivative?
2. What is the significance of \( f'(2) = -4 \) in this context?
3. How do you apply the point-slope form to different types of functions?
4. Why is it important to find the y-intercept in the first method?
5. Can the second method be generalized to any function at any point?

**Tip:** The derivative gives the slope of the tangent line to a function at a specific point, which is key to understanding the behavior of the function at that point.

Bestem en ligning for tangenten til grafen for f i punktet (2, f(2))

Learn how to determine the equation of a tangent line to a function's graph at a specific point using derivatives. This step-by-step solution covers key calculus concepts, including the use of the derivative to find the slope of the tangent and applying both slope-intercept and point-slope forms. Suitable for Grades 10-12.

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