Question

Use linear approximation, i.e. the tangent line, to approximate 10.1 2 10.12 as follows: Let f ( x ) = x 2 f(x)=x2 and find the equation of the tangent line to f ( x ) f(x) at x = 10 x=10.

Answer 1

The linear approximation of a function
`f(x)` centered at
`x=a` is

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

We have

`f(x)=x^2\implies f'(x)=2x`

and we want to find
`L(10.1)` for
`a=10`.

Since
`f(10)=100` and
`f'(10)=20`, we get

`10.1^2\approx L(10.1)=100+20(10.1-10)=102`

Compare this to the actual value of
`10.1^2=102.01`.