Estimate how much the function f (x) = (x / x +1 ) -3 will change as x decreases from 4 to 3.8

If anyone could explain this, that would be great..practice problem for a test coming up.

I got -0.008 would like to check it.

Also have another question,

find the derivative f(x) = 2x + 5 / (1 - 2x)^3

2. ## Re: Business Calc Question

Originally Posted by mkp23
Estimate how much the function f (x) = (x / x +1 ) -3 will change as x decreases from 4 to 3.8

If anyone could explain this, that would be great..practice problem for a test coming up.

I got -0.008 would like to check it.

agree

Also have another question,

find the derivative f(x) = 2x + 5 / (1 - 2x)^3

quotient rule and chain rule ...

$f'(x) = \frac{(1-2x)^3 \cdot 2 - (2x+5) \cdot 3(1-2x)^2 \cdot (-2)}{(1-2x)^6}$

pull out the common factors from the two terms in the numerator and simplify

3. ## Re: Business Calc Question

Hello, mkp23!

Estimate how much the function $f(x)\,=\,\frac{x}{x +1}-3$ will change
as $x$ decreases from 4.0 to 3.8

If anyone could explain this, that would be great . . .
. . practice problem for a test coming up.

I got -0.008 would like to check it. . Correct!

The key word is "estimate" ... meaning we're dealing with differentials.

We have: . $f(x) \:=\:\frac{x}{x+1}$

Then: . $df \:=\:\left[\frac{(x+1)\!\cdot\!1 - x(1)}{(x+1)^2} - 0\right]\,dx \quad\Rightarrow\quad df \:=\:\frac{dx}{(x+1)^2}$

We are given: . $x = 4,\;\;dx = -0.2$

Therefore: . $df \;=\;\frac{-0.2}{(4+1)^2} \;=\;-0.008$ .(approximate change)

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We can compare it to the actual change of $f(x).$

$f(4) \:=\:\frac{4}{5} - 3 \;=\;-2.2$

$f(3.8) \:=\:\frac{3.8}{4.8} -3 \;=\;-2.208333\hdots$

The actual change is: . $\Delta f \;=\;(-2.208333) - (-2.2) \;=\;-0.008333\hdots$