Question

using differentials find the approximate value of (1.999)^5​

Answer 1

Let,

• x = 2

• △x = -0.001    [∵ 2-0.001 = 1.999]

Let,

• y = x⁵

On differentiating both sides with respect to x , we get  

$\sf{\dfrac{dy}{dx}=5x^4}$

$\boxed{\sf{\Delta y=\dfrac{dy}{dx}\Delta x}}$

Applying the value to the equation,

$\implies\sf{\Delta y=5x^4\times \Delta x}$

Here,

• x = 2

• △x = -0.001

Applying the values to the equation,

$\implies\sf{\Delta y = 5\times 2^4 \times (-0.001)}$

$\implies\sf{\Delta y = 5\times 16 \times (-0.001)}$

$\implies\sf{\Delta y = 80\times (-0.001)}$

$\implies\sf{\Delta y = -0.080}$

We know,

$\boxed{\sf{(1.999)^5=y + \Delta y}}$

Applying the values to the equation,

$\implies\sf{(1.999)^5=2^5-(0.080)}$

$\implies\sf{(1.999)^5=32-0.080}$

$\implies\sf{(1.999)^5\approx 31.920}$

Therefore,

$\boxed{\bold{\sf{(1.999)^5\approx 31.920}}}$