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

Question

what is Δy

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Answer 1 · 2016-09-18T09:10:20

See below.

#(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#