Use a linear approximation (differentials) to estimate the value of $e^{-0.01}$. Which of the following is the best estimate?

• A. $0.999$
• B. $0.9901$
• C. $0.990$
• D. $0.99005$

Answer: (B)

Verified step by step guidance

1. Identify the function to approximate: The function is \( f(x) = e^x \). We want to approximate \( f(-0.01) = e^{-0.01} \).
2. Choose a point \( a \) near \( x = -0.01 \) where the function is easy to compute. Here, \( a = 0 \) is a good choice because \( e^0 = 1 \).
3. Use the formula for linear approximation: \( f(x) \approx f(a) + f'(a)(x - a) \). For \( f(x) = e^x \), the derivative is \( f'(x) = e^x \).
4. Substitute \( a = 0 \) and \( x = -0.01 \) into the formula: \( f(-0.01) \approx f(0) + f'(0)(-0.01 - 0) \). Since \( f(0) = e^0 = 1 \) and \( f'(0) = e^0 = 1 \), the approximation becomes \( f(-0.01) \approx 1 + 1(-0.01) \).
5. Simplify the expression: \( f(-0.01) \approx 1 - 0.01 \). This gives the linear approximation for \( e^{-0.01} \).