Question

31-36Use a linear approximation (or differentials) to estimate the given number.

31. \({\left( {1.999} \right)^4}\)

Short answer

The required value is: \({\left( {1.999} \right)^4} = 15.968\)

Step-by-step solution

Linear Approximation

The Linear approximation is basically a method in calculus that uses operators such that:

\(L\left( x \right) = f\left( a \right) - f'\left( a \right)\left( {x - a} \right)\).

Calculating values using Linear Approximation:

The givennumber is:

\(f\left( x \right) = {\left( {1.999} \right)^4} = {x^4}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,..........\left( {{\rm{say}}} \right)\)

Now, solving the linear approximation at \(a = 2\):

\(\begin{aligned}L\left( x \right) & = f\left( 2 \right) + f'\left( 2 \right)\left( {x - 2} \right)\\{x^4} & = {\left( 2 \right)^4} + \left( {4{{\left( 2 \right)}^3}} \right)\left( {x - 2} \right)\\{x^4} & = 16 + 32\left( {x - 2} \right)\end{aligned}\)

So, for\(x = 1.999\):

\(\begin{aligned}{\left( {1.999} \right)^4} & = 16 + 32\left( {1.999 - 2} \right)\\ & = 16 - 0.032\\ & = 15.968\end{aligned}\)

Hence, the required value is:

\({\left( {1.999} \right)^4} = 15.968\)