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1. What are the asymptotes of the graph of f(x)?

y = 3

y = 0

y = 20

y = 9

 

 

 

2. What is the point of maximum growth rate for the logistic function f(x)? Round to the nearest hundredth.

 

(0.42, 3)

(0.85, 12)

(0.85, 24)

(0, 4)

jbouyer  Feb 24, 2018

Best Answer 

 #1
avatar+7266 
+2

1.

 

\(y=\frac{20}{1+9e^{3x}} \\~\\ y(1 + 9e^{3x})=20 \\~\\ 1 + 9e^{3x}=\frac{20}{y} \\~\\ 9e^{3x}=\frac{20}{y}-1 \\~\\ 9e^{3x}=\frac{20-y}{y} \\~\\ e^{3x}=\frac{20-y}{9y} \\~\\ 3x=\ln(\frac{20-y}{9y}) \\~\\ x=\frac{\ln(\frac{20-y}{9y})}{3}\)

 

There is an asymptote when   9y  =  0   which is when   y = 0

 

There is an asymptote when   \(\frac{20-y}{9y}\)  =  0   which is when   y = 20

 

Here's a graph: https://www.desmos.com/calculator/9czjghjp9s

hectictar  Feb 24, 2018
 #1
avatar+7266 
+2
Best Answer

1.

 

\(y=\frac{20}{1+9e^{3x}} \\~\\ y(1 + 9e^{3x})=20 \\~\\ 1 + 9e^{3x}=\frac{20}{y} \\~\\ 9e^{3x}=\frac{20}{y}-1 \\~\\ 9e^{3x}=\frac{20-y}{y} \\~\\ e^{3x}=\frac{20-y}{9y} \\~\\ 3x=\ln(\frac{20-y}{9y}) \\~\\ x=\frac{\ln(\frac{20-y}{9y})}{3}\)

 

There is an asymptote when   9y  =  0   which is when   y = 0

 

There is an asymptote when   \(\frac{20-y}{9y}\)  =  0   which is when   y = 20

 

Here's a graph: https://www.desmos.com/calculator/9czjghjp9s

hectictar  Feb 24, 2018

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