Question

Write the linearization of the function at the points indicated. (Enter your answer as an equation. Let x be the independent variable and y be the dependent variable.) x= \(\sqrt{?}25+x\) (0, 5) and (75, 10)

Answer 1

`\displaystyle y=5+(x)/(10)`

`\displaystyle y=10+((x-75))/(20)`

Explanation:

Linearization

It consists of finding an approximately linear function that behaves as close as possible to the original function near a specific point.

Let y=f(x) a real function and (a,f(a)) the point near which we want to find a linear approximation of f. If f'(x) exists in x=a, then the equation for the linearization of f is

`y=f(x)=f(a)+f'(a)(x-a)`

Let's find the linearization for the function

`y=√(25+x)`

at (0,5) and (75,10)

Computing f'(x)

`\displaystyle f'(x)=(1)/(2√(25+x))`

At x=0:

`\displaystyle f'(0)=(1)/(2√(25+0))=(1)/(10)`

We find f(0)

`f(0)=√(25+0)=5`

Thus the linearization is

`\displaystyle y=f(0)+f'(0)(x-0)=5+(1)/(10)x`

`\displaystyle y=5+(x)/(10)`

Now at x=75:

`\displaystyle f'(75)=(1)/(2√(25+75))=(1)/(20)`

We find f(75)

`f(75)=√(25+75)=10`

Thus the linearization is

`\displaystyle y=f(75)+f'(75)(x-75)=10+(1)/(20)(x-75)`

`\displaystyle y=10+((x-75))/(20)`