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.