Creating a rational with given tangent slope

casherr

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develop one rational function that demonstrates this slope (m = - 2) at a specific value x = a

I've tried so many different eqns but they all end up in tangent lines that pass through 2 points on the function very close to each other. Please help!
 

skeeter

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\(\displaystyle f(x) = \frac{1}{x} \implies f’\left(\frac{1}{\sqrt{2}}\right) = -2\)
 

casherr

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it has to be somewhat complex, like x+3/x+2 for example
 

skeeter

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it has to be somewhat complex, like x+3/x+2 for example
your initial post says "one rational function" ... nothing about its complexity.

so, how about \(\displaystyle f(x) = \dfrac{4}{x^2+1}\) ?

does that meet the "somewhat complex" criteria?

If so, determine the unique value \(\displaystyle x=a \) where \(\displaystyle f'(a) = -2\)
 

casherr

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Screen Shot 2021-02-28 at 5.27.17 PM.png

IT WORKED AHHHHHH!!!! thank you so much!!
 

casherr

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wait actually i calculated that wrong, the tangent line equation should be -2x+4, but when i do that, the tangent goes through (0,4) and (1,2) so I'm not sure what I'm doing wrong. I solved for x and it gave me 1 and 0.29559 so used 1 to find the eqn.
 

skeeter

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it’s ok if the line tangent at x=1 intersects the curve again ... it’s not tangent at (0,4), it is tangent at (1,2)
 

casherr

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will there ever be a point on the graph with the slope -2 that only intersects once?
Also, unrelated but if I were to find the instantaneous rate of change for one point on an income per person per year graph, what would my units be?
thank you for all your help so far by the way!
 

Subhotosh Khan

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will there ever be a point on the graph with the slope -2 that only intersects once?
Also, unrelated but if I were to find the instantaneous rate of change for one point on an income per person per year graph, what would my units be?
thank you for all your help so far by the way!
You say:

will there ever be a point on the graph with the slope -2 that only intersects once?​

Are you referring to the graph - with slope -2 (i.e. the graph has a slope of -2 at some given point)?​

You say - only intersects once → what intersects with what?
 

casherr

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You say:

will there ever be a point on the graph with the slope -2 that only intersects once?​

Are you referring to the graph - with slope -2 (i.e. the graph has a slope of -2 at some given point)?​

You say - only intersects once → what intersects with what?

tangent line that only intersects with the graph once at a specific point with the slope -2
 

casherr

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"Are you referring to the graph - with slope -2 (i.e. the graph has a slope of -2 at some given point)?"
im referring to the graph we just discussed, y= 4/(x^2+1), and what point would have tangent -2 that's line only intersects with the graph once
 

skeeter

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will there ever be a point on the graph with the slope -2 that only intersects once?
not with the function I proposed ... not saying it isn't possible with some other function

Also, unrelated but if I were to find the instantaneous rate of change for one point on an income per person per year graph, what would my units be?
(income per person) per year, or (income) per (person per year) ? there is a difference

I suspect you may mean the first ... if so, the derivative would be (income per person) per year2
 

casherr

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is there any reason for the year being squared?
I really appreciate your help!
 

skeeter

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to make sure we get this correct, please verify what both axes of the graph represent with regard to units ...

x axis has what units?

y axis has what units?
 

casherr

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x is years

y is the income per person in dollars
 

skeeter

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x is years

y is the income per person in dollars
then the slope would be (income per person) per year, \(\displaystyle m = \dfrac{\text{income/person}}{\text{year}}\)
 
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