Question

please help me find one of the parts with linearization it would be extremely helpful, not the part with the box, thank you!

Answer 1

Given:

`\begin{gathered} y=tan(x) \\ x=(\pi)/(4) \end{gathered}`

To Determine: The Linearization of the given

Solution

The formula for finding the linearization of a function is given below

`L(x)=f(x_0)+f^(\prime)(x_0)(x-x_0)`
`\begin{gathered} x_0=(\pi)/(4)(given) \\ f(x_0)=tan((\pi)/(4))=1 \\ f^(\prime)(x_0)=f^(\prime)(x)=sec^2x \\ f^(\prime)(x_0)=(1)/(cos^2x) \end{gathered}`
`\begin{gathered} f^(\prime)(x_0)=(cos^2x_0)^(-1) \\ f^(\prime)((\pi)/(4))=(cos(\pi)/(4))^(-2) \\ =((1)/(√(2)))^(-2) \\ =(√(2))^2 \\ =2 \end{gathered}`

Therefore

`L(x)=1+2(x-(\pi)/(4))`
`L(x)=2x+1-(\pi)/(2)`