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.

#(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#
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Answer 2

Given ( y = \sqrt{x} ), compute ( \Delta y ) and ( dy ) for ( x = 16 ) and ( \Delta x = dx = 1 ).

[ \Delta y = y(x + \Delta x) - y(x) ]

[ dy = y'(x) \cdot dx ]

  1. Calculate ( y(x) ) when ( x = 16 ): [ y(16) = \sqrt{16} = 4 ]

  2. Calculate ( y(x + \Delta x) ) when ( x = 16 ) and ( \Delta x = 1 ): [ y(16 + 1) = y(17) = \sqrt{17} ]

  3. Calculate ( \Delta y ): [ \Delta y = \sqrt{17} - 4 ]

  4. Calculate ( y'(x) ): [ y'(x) = \frac{1}{2\sqrt{x}} ]

  5. Calculate ( y'(16) ): [ y'(16) = \frac{1}{2\sqrt{16}} = \frac{1}{8} ]

  6. Calculate ( dy ): [ dy = \frac{1}{8} \cdot 1 = \frac{1}{8} ]

So, ( \Delta y \approx \sqrt{17} - 4 ) and ( dy = \frac{1}{8} ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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