Question

Compute Δy and dy for x = 16 and dx = Δx = 1. (Round the answers to three decimal places.)? y = √x

what is Δy

Answer 1

See below.

Explanation:

`(Delta y) /(Delta x)=( f(x+Delta x)-f(x))/(Delta x)`

and

`(dy)/(dx) = lim_(Delta x->0)(f(x+Delta x)-f(x))/(Delta x)`

so `dx` cannot be equal to `1`

Applying it to `f(x) = sqrt(x)`

`(Delta y) /(Delta x)=( sqrt(x+Delta x)-sqrt(x))/(Delta x)=(x+Delta x - x)/((sqrt(x+Deltax)+sqrt(x))Deltax) = 1/(sqrt(x+Deltax)+sqrt(x))`

For `Delta x=1` and `x=16` we have

`Delta y = 1/(sqrt(17)+sqrt(16)) = 1/(sqrt(17)+4) approx 0.123106`

Also we have

`(dy)/(dx) = lim_(Delta x->0)1/(sqrt(x+Deltax)+sqrt(x))=1/(2sqrt(x))`
at `x=16` we have

`(dy)/(dx)=1/8 = 0.125`