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, ( 100 cdot \frac { | \text { approximation - exact } | } { | \text { exact } | } ), where the exact value is given by a calculator
( f ( x ) = \frac { 2 } { x + 1 } ), ( a = 0 ), ( \frac { 2 } { 1.2 } )
a. ( l ( x ) = )

Explanation:

Step1: Find the derivative of the function

The function is \( f(x)=\frac{2}{x + 1}=2(x + 1)^{-1} \).
Using the power rule \((x^n)^\prime=nx^{n - 1}\) and the chain rule \((u^{-1})^\prime=-u^{-2}u^\prime\) (where \(u=x + 1\) and \(u^\prime = 1\)), we have \(f^\prime(x)=-2(x + 1)^{-2}=-\frac{2}{(x + 1)^2}\).

Step2: Evaluate the function and its derivative at \(a = 0\)

When \(x=a = 0\), \(f(0)=\frac{2}{0 + 1}=2\).
When \(x=a = 0\), \(f^\prime(0)=-\frac{2}{(0 + 1)^2}=-2\).

Step3: Use the linear - approximation formula \(L(x)=f(a)+f^\prime(a)(x - a)\)

Substitute \(a = 0\), \(f(0)=2\) and \(f^\prime(0)=-2\) into the formula \(L(x)=f(a)+f^\prime(a)(x - a)\).
We get \(L(x)=2-2(x - 0)=2-2x\).

Answer:

\(L(x)=2-2x\)