Use differentials to approximate the arc length of the graph of the equation from $\mathbf { P }$ to $\mathbf { Q }$. Round answer to four decimal places. Select the correct answer.
$$y = x ^ { 2 } + 3 ; \quad \mathrm { P } ( 3,12 ) , \quad \mathrm { Q } ( 3.2,13.24 )$$
A) $6.0828$
B) $3.6497$
C) $1.2166$
D) $2.4331$

Question

Quizplus · Accepted Answer

The arc length can be approximated using the differential formula $ ds = \sqrt{1 + (dy/dx)^2} \, dx $. For $ y = x^2 + 3 $, $ dy/dx = 2x $. At $ x = 3 $, $ dy/dx = 6 $. Thus, $ ds \approx \sqrt{1 + 6^2} \, dx = \sqrt{37} \, dx $. The change in $ x $ is $ 0.2 $, so the approximate arc length is $ \sqrt{37} \times 0.2 \approx 1.2166 $.

Use Differentials to Approximate the Arc Length of the Graph P\mathbf { P }P

Use Differentials to Approximate the Arc Length of the Graph $\mathbf { P }$