교과서 질문
Dependence on Initial Point
8. Using the function shown in the figure, and, for each initial estimate x_0, determine graphically what happens to the sequence of Newton’s method approximations
c. x_0=2
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4. Applications of Derivatives
Differentials
Multiple Choice
Use a linear approximation (differentials) to estimate the value of . Which of the following is the best estimate?
A
B
C
D
검증된 단계별 안내1
Identify the function to approximate: The function is \( f(x) = e^x \). We want to approximate \( f(-0.01) = e^{-0.01} \).
Choose a point \( a \) near \( x = -0.01 \) where the function is easy to compute. Here, \( a = 0 \) is a good choice because \( e^0 = 1 \).
Use the formula for linear approximation: \( f(x) \approx f(a) + f'(a)(x - a) \). For \( f(x) = e^x \), the derivative is \( f'(x) = e^x \).
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 \( f(-0.01) \approx 1 + 1(-0.01) \).
Simplify the expression: \( f(-0.01) \approx 1 - 0.01 \). This gives the linear approximation for \( e^{-0.01} \).
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