Question

Let ?(?)= (sqrt ?)

The local linearization of ? at 49 is ? (subscript) 49(?) = ? (subscript)0 +?(?−49) is __________
where ?= __________ and ? (subscript) 0= _____________

Using this, we find our approximation (sqrt 49.3) = ___________

Answer 1

let y = f(x) is the function to find local linear approximation at x= x0 we need to find equation of tangent to the given curve

at x = x0

equation of tangent line at x= x0  

y - y0 = f '(x) ( x - x0)

when x goes toward x0 , y from the line goes towards y from the curve

so when x is almost equal to x0 , y from the line almost equal to y from the curve

so we can conclude at x= x0  approximate value of y [y= f(x) ] from the curve is equal to value of y from the line

i.e. y = y0 + f'(x0) ( x- x0 )

so local linear approximation of the curve y = f(x) at x= x0 can be written as

L(x) = y0 + f'(x0) ( x- x0 ) [ f ' (x0)=m ]...................................(1)