Use linear approximation to estimate f left parenthesis 3.9 right parenthesis given that f left parenthesis 4 right parenthesis equals 5 and f prime left parenthesis 4 right parenthesis equals 2.
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Part 1
f left parenthesis 3.9 right parenthesisalmost equals
  
enter your response here ​(Simplify your​ answer.)

To use linear approximation to estimate \( f(3.9) \), we can apply the formula for linear approximation:

\[
f(x) \approx f(a) + f'(a)(x - a)
\]

Here, we take \( a = 4 \), \( x = 3.9 \), \( f(4) = 5 \), and \( f'(4) = 2 \).

Plugging these values into the formula:

\[
f(3.9) \approx f(4) + f'(4)(3.9 - 4)
\]

Calculating it step-by-step:

1. Find \( (3.9 - 4) = -0.1 \).
2. Then, compute \( f'(4)(3.9 - 4) = 2 \cdot (-0.1) = -0.2 \).
3. Now, add this to \( f(4) \):

\[
f(3.9) \approx 5 - 0.2 = 4.8
\]

Thus, the linear approximation gives:

\[
f(3.9) \approx 4.8
\]

If you have any questions or need further details, let me know! Here are some related questions you might consider:

1. How does the linear approximation compare to the actual function value if known?
2. What are the limitations of using linear approximation?
3. Can you explain how to apply linear approximation to a different function?
4. How would the estimate change if \( f'(4) \) was a different value?
5. What other methods can be used for estimating function values? 

**Tip:** Always verify the proximity of \( x \) to \( a \) for better accuracy in linear approximations.

Use linear approximation to estimate f(3.9) given that f(4) = 5 and f'(4) = 2.

Explore how to use linear approximation to estimate f(3.9) with given values f(4) = 5 and f'(4) = 2. This problem highlights the concept of linear approximation in calculus, suitable for high school students in Grades 11-12.

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