compute $delta y$ and $dy$ for the given values of $x$ and $dx=delta x$. (round your answers to three…

compute $delta y$ and $dy$ for the given values of $x$ and $dx=delta x$. (round your answers to three decimal places.)\n$y = sqrt{x - 3}$, $x = 4$, $delta x=0.8$\n$delta y=\n$dy=\nsketch a diagram showing the line - segments with lengths $dx$, $dy$, and $delta y$.

compute $delta y$ and $dy$ for the given values of $x$ and $dx=delta x$. (round your answers to three decimal places.)\n$y = sqrt{x - 3}$, $x = 4$, $delta x=0.8$\n$delta y=\n$dy=\nsketch a diagram showing the line - segments with lengths $dx$, $dy$, and $delta y$.

Answer

Explanation:

Step1: Find the value of $y$ at $x = 4$

$y=\sqrt{x - 3}$, when $x = 4$, $y=\sqrt{4 - 3}=1$

Step2: Find the value of $y$ at $x+\Delta x$

$x+\Delta x=4 + 0.8=4.8$, $y+\Delta y=\sqrt{(4.8)-3}=\sqrt{1.8}\approx1.342$

Step3: Calculate $\Delta y$

$\Delta y=y+\Delta y - y\approx1.342-1 = 0.342$

Step4: Differentiate $y$ with respect to $x$

$y=\sqrt{x - 3}=(x - 3)^{\frac{1}{2}}$, using the power - rule $\frac{dy}{dx}=\frac{1}{2}(x - 3)^{-\frac{1}{2}}$

Step5: Evaluate $\frac{dy}{dx}$ at $x = 4$

When $x = 4$, $\frac{dy}{dx}=\frac{1}{2}(4 - 3)^{-\frac{1}{2}}=\frac{1}{2}$

Step6: Calculate $dy$

$dy=\frac{dy}{dx}\Delta x$, since $\frac{dy}{dx}=\frac{1}{2}$ and $\Delta x = 0.8$, $dy=\frac{1}{2}\times0.8 = 0.400$

Answer:

$\Delta y\approx0.342$ $dy = 0.400$