Question

For the function $e^{-x}$, the linear approximation around x = 3 is

• A. $e^{-3}$
• B. $(2-x)e^{-3}$
• C. $(4-x)e^{-3}$
• D. None of the above

Answer 1

The correct option is C $(4-x)e^{-3}$

$e^{-x}=e^{-3-x+3}$

$=e^{-3}\times e^{-(x-3)}$

$=e^{-3}[1-(x-3)+\frac{(x-3{)}^2}{2}....]$

For linear approximation, neglecting $(x-3{)}^2$ and its higher powers, we have

$e^{-x}=e^{-3}[1-(x-3\left)\right]$

$=e^{-3}[4-x]$