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To estimate \( f(94) \) given that \( f(90) = 55 \) and \( f'(90) = 2 \), we can use a linear approximation (or the tangent line approximation) around \( x = 90 \).

### Solution:
The linear approximation formula around \( x = 90 \) is:
\[
f(94) \approx f(90) + f'(90) \cdot (94 - 90)
\]

1. Substitute the given values:
   \[
   f(94) \approx 55 + 2 \cdot (94 - 90)
   \]

2. Calculate \( 94 - 90 \):
   \[
   94 - 90 = 4
   \]

3. Substitute and calculate:
   \[
   f(94) \approx 55 + 2 \cdot 4
   \]
   \[
   f(94) \approx 55 + 8
   \]
   \[
   f(94) \approx 63
   \]

### Answer:
\[
f(94) \approx 63
\]

Would you like more details on this? 

Here are some related questions to explore:

1. How would the estimate change if \( f'(90) \) were a different value?
2. What if we needed to estimate \( f(92) \) instead?
3. How accurate is this linear approximation compared to the actual function?
4. Could we use a quadratic approximation to get a more accurate result?
5. How would the result differ if we used other methods like the Mean Value Theorem?

**Tip:** Linear approximation is especially useful for small intervals around the point where the derivative is given.

Suppose that f(x) is a function with f(90) = 55 and f'(90) = 2. Estimate f(94).

Learn how to estimate f(94) using linear approximation, given f(90) = 55 and f'(90) = 2. This calculus problem explains the step-by-step process of applying the linear approximation formula, making it suitable for students in Grades 11-12.

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