Answer:
[tex]f\left( {2.1} \right) \approx 5.22360[/tex].
Step-by-step explanation:
The linear approximation is given by the equation
[tex]{f\left( x \right) \approx L\left( x \right) }={ f\left( a \right) + f^\prime\left( a \right)\left( {x - a} \right).}[/tex]
Linear approximation is a good way to approximate values of [tex]f(x)[/tex] as long as you stay close to the point [tex]x= a[/tex], but the farther you get from [tex]x=a[/tex], the worse your approximation.
We know that,
[tex]a=2\\f(2) = 5\\f'(x) = \sqrt{3x-1}[/tex]
Next, we need to plug in the known values and calculate the value of [tex]f(2.1)[/tex]:
[tex]{L\left( x \right) = f\left( 2 \right) + f^\prime\left( 2 \right)\left( {x - 2} \right) }=5+\sqrt{3(2)-1}(x-2) =5+\sqrt{5}(x-2)[/tex]
Then
[tex]f\left( {2.1} \right) \approx 5+\sqrt{5}(2.1-2)\approx5.22360[/tex].
