	If the tangent line to y = f (x) at (−5, 4) passes through the point (11, 11), find  f (−5) and  f '(−5).


We are given that the tangent line to the function \( y = f(x) \) at the point \( (-5, 4) \) passes through another point \( (11, 11) \). We need to find \( f(-5) \) and \( f'(-5) \).

### Step 1: Find \( f(-5) \)
The point \( (-5, 4) \) is explicitly stated to be on the curve, meaning \( f(-5) = 4 \).

### Step 2: Find \( f'(-5) \)
The derivative \( f'(-5) \) represents the slope of the tangent line at the point \( (-5, 4) \). Since the tangent line passes through both \( (-5, 4) \) and \( (11, 11) \), we can calculate the slope of the line passing through these two points.

The slope \( m \) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substituting the points \( (-5, 4) \) and \( (11, 11) \):
\[
m = \frac{11 - 4}{11 - (-5)} = \frac{7}{16}
\]

Therefore, the slope of the tangent line at \( (-5, 4) \) is \( f'(-5) = \frac{7}{16} \).

### Final Answer:
- \( f(-5) = 4 \)
- \( f'(-5) = \frac{7}{16} \)

Would you like more details or have any questions?

Here are some related questions:
1. How do we interpret the slope of a tangent line in a function graph?
2. What is the geometric meaning of a derivative at a point?
3. How do we use the point-slope form of a line equation?
4. Can the slope of a tangent line be used to estimate nearby values of the function?
5. How would this process change if the function was provided explicitly?

**Tip:** The slope of the tangent line at a point on a curve gives the rate of change of the function at that point.

If the tangent line to y = f (x) at (−5, 4) passes through the point (11, 11), find  f (−5) and  f '(−5).

Learn how to find f(−5) and f'(-5) for the function y = f(x) given that the tangent line at the point (-5, 4) passes through (11, 11). This solution covers core calculus concepts like derivatives, tangent lines, and slope calculation. Suitable for students in Grades 11-12.

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