Question

a. write the equation of the line that represents the linear approximation to the following function at the given point a. b. use the linear approximation to estimate the given quantity. c. compute the percent error in the approximation, ( \frac{100| \text{approximation} - \text{exact}|}{| \text{exact}|} ), where the exact value is given by a calculator ( f(x)=ln(1 + x),a = 0,f(0.6) ) a. ( l(x)=square )

Explanation:

Step1: Find \( f(a) \) and \( f^{\prime}(a) \)

Given \( f(x)=\ln(1 + x) \), when \( a = 0 \), \( f(0)=\ln(1+0)=0 \).
Differentiate \( f(x) \) using the formula \( \frac{d}{dx}\ln(u)=\frac{u^{\prime}}{u} \) (where \( u = 1 + x \), \( u^{\prime}=1 \)), so \( f^{\prime}(x)=\frac{1}{1 + x} \). Then \( f^{\prime}(0)=\frac{1}{1+0}=1 \).

Step2: Use the linear - approximation formula \( L(x)=f(a)+f^{\prime}(a)(x - a) \)

Substitute \( a = 0 \), \( f(0)=0 \), and \( f^{\prime}(0)=1 \) into the formula \( L(x)=f(a)+f^{\prime}(a)(x - a) \).
We get \( L(x)=0+1\times(x - 0)=x \).

Answer:

\( L(x)=x \)