Question

use a linear approximation to estimate the following quantity. choose a value of a that produces a small error and does not require the use of a calculator.

the linear approximation of ( e^{0.01} ) is

(type an exact answer.)

Explanation:

Step1: Recall the linear approximation formula

The linear approximation formula for a function \(y = f(x)\) at \(x = a\) is \(L(x)=f(a)+f^{\prime}(a)(x - a)\). For the function \(y = e^{x}\), \(f(x)=e^{x}\), \(f^{\prime}(x)=e^{x}\). Let \(a = 0\) and \(x=0.01\).

Step2: Calculate \(f(a)\) and \(f^{\prime}(a)\)

When \(a = 0\), \(f(0)=e^{0}=1\) and \(f^{\prime}(0)=e^{0}=1\).

Step3: Apply the linear - approximation formula

Substitute into \(L(x)=f(a)+f^{\prime}(a)(x - a)\), we get \(L(0.01)=e^{0}+e^{0}(0.01 - 0)\).

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Answer:

\(1.01\)