Question

Use the differential to find a decimal approximation of the radical expression. Round to four decimal places. 101 10.0500 O 10.2000 O 9.9500 O 10.1500

Answer 1

We can apply the Taylor expansion for the function:

`f(x)=\sqrt[]{x}`

around x0=100.

The Taylor expansion with 2 terms (one differential) can be written as:

`f(x)\approx f(x_0)+f^(\prime)(x_0)(x-x_0)`

The value of f(x0) is 10.

We have to calculate the first derivative of f(x):

`f^(\prime)(x)=(d)/(dx)\lbrack x^{(1)/(2)}\rbrack=(1)/(2)x^{-(1)/(2)}=\frac{1}{2\sqrt[]{x}}`

Now, we can calculate f'(x0):

`f^(\prime)(x_0)=f^(\prime)(100)=\frac{1}{2\sqrt[]{100}}=(1)/(2\cdot10)=(1)/(20)`

We then can calculate:

`\begin{gathered} f(x)\approx10+((x-100))/(20) \\ f(101)\approx10+((101-100))/(20)=10+(1)/(20)=10.0500 \end{gathered}`

Answer: 10.0500.