Question

Compute Δy and dy for the given values of x and dx = Δx. (Round your answers to three decimal places.) y = 6 x , x = 4, Δx = 1

Answer 1

The function is

y = 6 x

Differentiating w.r.t  x

$\frac{\mathrm{d} y}{\mathrm{d} x}$=6

As given , dx = Δx, and dy=Δy

$\frac{\mathrm{d} y}{\mathrm{d} x}=\frac{\Delta y}{\Delta x}$

So,above equation becomes

$\frac{\Delta y}{\Delta x}=6$

$\Delta y= \Delta x \times 6$

As given , $\Delta x=1$

$\Delta y= 1 \times 6=6$

When  x=4, y=6×4=24

So, When x+ $\Delta x$=4+1=5 then,  y +$\Delta y$=24+6=30

Answer 2

The value of $\Delta y$ and $dy$ for the given value of $x$ and $\Delta x$ is $\boxed6.$

Further explanation:

Given:

The value of $y$ is $6x.$

$y = 6x$

The value of $x$ is $4.$

Explanation:

The given function can be expressed as follows,

$y = 6x$

Differentiate the above equation with respect to $x.$

$\begin{aligned}\frac{{dy}}{{dx}}&= \frac{d}{{dx}}\left( {6x} \right)\\&= 6 \times\frac{d}{{dx}}\left( x \right)\\&= 6\times1\\&= 6\\\end{aligned}$

$dx = \Delta x{\text{ and }}dy = \Delta y$

The value of $\Delta y$ can be obtained as follows,

$\begin{aligned}\frac{{\Delta y}}{{\Delta x}} &= 6\\\Delta y &= 6 \times \Delta x \\\Delta y&= 6 \times 1\\\Delta y&= 6\\\end{aligned}$

The value of $x$ is $4.$

Substitute $4$ for $x$ in equation $y = 6x.$

$\begin{aligned}y&= 6 \times 4\\&= 24\\\end{aligned}$

The value of $x + \Delta x$ can be obtained as follows,

$\begin{aligned}x + \Delta x &= 4 + 1 \\&= 5\\\end{aligned}$

The value of $y + \Delta y$ can be obtained as follows,

$\begin{aligned}y + \Delta y &= 24 + 6\\&= 30\\\end{aligned}$

The value of $\Delta y$ and $dy$ for the given value of $x$ and $\Delta x$ is  $\boxed6.$

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Answer details:

Grade: High School

Subject: Mathematics

Chapter: Derivatives

Keywords: Derivative, value of $x$, function, differentiate, minimum value, dy, compute, given value of $x, y=6x, x=4, x=1$