Answer:
(a) dp/dx = 90/(x+2)
(b) $2.25
(c) approx = $2.25
exact = $2.22
Step-by-step explanation:
[tex]p = 20 + 90 \ln(2x + 4)[/tex]
(a) To find the rate of change, differentiate [tex]p[/tex] with respect to [tex]x[/tex] :
[tex]\implies \frac{d}{dx}(p) = \frac{d}{dx}(20) + \frac{d}{dx}(90 \ln(2x + 4))[/tex]
[tex]\implies \frac{dp}{dx}=0+90\times\dfrac{1}{2x+4}\times2[/tex]
[tex]\implies \frac{dp}{dx}=\dfrac{180}{2x+4}[/tex]
[tex]\implies \frac{dp}{dx}=\dfrac{90}{x+2}[/tex]
(b) Find [tex]\frac{dp}{dx}[/tex] when [tex]x=38[/tex] :
[tex]\implies \frac{dp}{dx}=\dfrac{90}{38+2}[/tex]
[tex]\implies \frac{dp}{dx}=\dfrac{90}{40}[/tex]
[tex]\implies \frac{dp}{dx}=2.25[/tex]
(c) Approximate price increase associated with the number of units supplied changing from 38 to 39 is [tex]\frac{dp}{dx}[/tex] when [tex]x=38[/tex]
[tex]\implies \$2.25[/tex]
Exact price increase:
[tex]\implies \left. p \right_{x=39}-\left. p \right_{x=38}\\\\\implies [20 + 90 \ln(2 \times 39 + 4)]-[20 + 90 \ln(2\times38 + 4)]\\\\\implies 416.6047323...-414.3823972...\\\\\implies 2.222335133...\\\\\implies \$2.22[/tex]
