https://file.aiquickdraw.com/mathbot/file/1727635096044qnk0euyj.jpg

Let's analyze the problem and solve for both \( f(5) \) and \( f'(5) \).

### Step 1: Understanding the problem

We are given:
- The function \( y = f(x) \), where \( f(5) = 3 \). 
- The tangent line to \( y = f(x) \) at the point \( (5, 3) \) passes through another point \( (0, 2) \).

We are tasked with finding:
- \( f(5) \), which is given directly as \( f(5) = 3 \).
- \( f'(5) \), which is the slope of the tangent line at \( x = 5 \).

### Step 2: Finding the slope \( f'(5) \)

The slope of the tangent line can be calculated using the formula for the slope between two points. The tangent line passes through the points \( (5, 3) \) and \( (0, 2) \). The slope \( m = f'(5) \) is given by:
\[
f'(5) = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 3}{0 - 5} = \frac{-1}{-5} = \frac{1}{5}
\]

So, \( f'(5) = \frac{1}{5} \).

### Step 3: Final answer

- \( f(5) = 3 \)
- \( f'(5) = \frac{1}{5} \)

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Would you like any more details or have further questions?

Here are 5 related questions to expand this topic:
1. How do we find the equation of the tangent line using the point-slope form?
2. What is the geometric meaning of the derivative at a point?
3. How would the tangent line equation change if \( f'(5) \) was negative?
4. What is the difference between finding a secant line and a tangent line slope?
5. How would the process differ if you were given different points through which the tangent passes?

**Tip:** Remember that the slope of the tangent line at a point gives you the rate of change of the function at that point.

If the tangent line to y = f(x) at (5, 3) passes through the point (0, 2), find f(5) and f'(5).

Solve the calculus problem to find f(5) and f'(5) for the function y = f(x), where the tangent line at (5, 3) passes through (0, 2). Learn how to determine the slope of the tangent line using the slope formula. Suitable for Grades 11-12.

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