# Linear approximation

#### Elize

Using linear approximation calculate sqrt(99)?

#### topsquark

Math Team
Using linear approximation calculate sqrt(99)?
Let $$\displaystyle y(x) = \sqrt{x}$$. Expand this in a Taylor series (with x small compareed to a):
$$\displaystyle y(a - x) \approx y(a) + \left . \dfrac{dy}{dx} \right | _{x = a}\cdot (a - x)$$

So let a = 100 and x = 1.

-Dan

#### Denis

Math Team
Using linear approximation calculate sqrt(99)?
I'm lazy: slightly less than 10 :ninja:

1 person

#### idontknow

It is between 9 and 10 and approximation will change for different values of x and a
$$\displaystyle y(x)\approx y(a)+yâ€™(a)(x-a)$$

#### v8archie

Math Team
Let $$\displaystyle y(x) = \sqrt{x}$$. Expand this in a Taylor series (with x small compareed to a):
$$\displaystyle y(a - x) \approx y(a) + \left . \dfrac{dy}{dx} \right | _{x = a}\cdot (a - x)$$

So let a = 100 and x = 1.

-Dan
Shouldn't that be
$$\displaystyle y(a - x) \approx y(a) - \left . \dfrac{dy}{dx} \right | _{x = a}\cdot x$$

1 person

#### topsquark

Math Team
Shouldn't that be
$$\displaystyle y(a - x) \approx y(a) - \left . \dfrac{dy}{dx} \right | _{x = a}\cdot x$$

Thanks for the catch.

-Dan