Question

The approximate value of $5^{2.01}$, where $\ln 5=1.6095$, is

• A. $25.4125$
• B. $25.2525$
• C. $25.5025$
• D. $25.4024$

Answer 1

The correct option is D $25.4024$

Here, $f\left(x\right)=5^x,x=2,\Delta x=0.01$
We know that, $f(x+\Delta x)=f\left(x\right)+f^{\prime }\left(x\right)\Delta x$
$f^{\prime }\left(x\right)=5^x\ln 5$
$\Rightarrow f\left(2.01\right)=f\left(2\right)+5^2\ln 5\times \left(0.01\right)$
$=25+25\left(1.6095\right)\left(0.01\right)$
$=25+0.402375=25.402375\approx 25.4024$