compute $delta y$ and $dy$ for the given values of $x$ and $dx=delta x$.\n$y = x^{2}-2x$, $x = 6$, $delta…

compute $delta y$ and $dy$ for the given values of $x$ and $dx=delta x$.\n$y = x^{2}-2x$, $x = 6$, $delta x=0.5$\n$delta y=square$\n$dy=square$\nsketch a diagram showing the line - segments with lengths $dx$, $dy$ and $delta x$.

compute $delta y$ and $dy$ for the given values of $x$ and $dx=delta x$.\n$y = x^{2}-2x$, $x = 6$, $delta x=0.5$\n$delta y=square$\n$dy=square$\nsketch a diagram showing the line - segments with lengths $dx$, $dy$ and $delta x$.

Answer

Explanation:

Step1: Recall the formula for $\Delta y$

$\Delta y=f(x + \Delta x)-f(x)$. Given $y = f(x)=x^{2}-2x$, $x = 6$ and $\Delta x=0.5$. First find $f(x+\Delta x)$: $f(6 + 0.5)=f(6.5)=(6.5)^{2}-2\times6.5=42.25-13 = 29.25$. $f(6)=6^{2}-2\times6=36 - 12=24$. So $\Delta y=f(6.5)-f(6)=29.25 - 24=5.25$.

Step2: Recall the formula for $dy$

First, find the derivative of $y=f(x)=x^{2}-2x$. Using the power - rule, $y^\prime=f^\prime(x)=2x - 2$. When $x = 6$, $y^\prime=f^\prime(6)=2\times6-2=10$. Since $dy=f^\prime(x)dx$ and $dx=\Delta x = 0.5$, then $dy=f^\prime(6)\times0.5=10\times0.5 = 5$.

Answer:

$\Delta y = 5.25$ $dy = 5$