Questions: Use linear approximation to estimate f(3.85) given that f(4)=3 and f'(4)=2. f(3.85) ≈ (Simplify your answer.)

Solution Steps

To estimate \( f(3.85) \) using linear approximation, we can use the formula for the linear approximation of a function at a point. The formula is given by: \[ f(x) \approx f(a) + f'(a)(x - a) \] where \( a \) is the point at which we know the value of the function and its derivative. Here, \( a = 4 \), \( f(4) = 3 \), and \( f'(4) = 2 \). We need to estimate \( f(3.85) \).

1. Identify the point \( a \) and the values \( f(a) \) and \( f'(a) \).
2. Use the linear approximation formula to estimate \( f(3.85) \).

We are given the following values:

• \( a = 4 \)
• \( f(a) = 3 \)
• \( f'(a) = 2 \)
• We want to estimate \( f(3.85) \).

Using the linear approximation formula: \[ f(x) \approx f(a) + f'(a)(x - a) \] we substitute the known values: \[ f(3.85) \approx 3 + 2(3.85 - 4) \]

Now, we perform the calculation: \[ f(3.85) \approx 3 + 2(-0.15) = 3 - 0.3 = 2.7 \]

Final Answer