limits problem

Student123

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Apr 12, 2020
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The avg annual salary of a professional baseball player can be modelled by the function S(x) = 246 + 64x - 8.9x^2 + 0.95x^3, where S represents the number of years since 1982. Determine the rate at which the avg salary was changing in 2005.

I keep getting stuck on every question so I'm behind on homework. This is supposed to use the difference quotient with limits so I did lim h->0 of [S(23+h) - S(23)]/h and kept going to plug in the x-value of 2005-1982=23 but then the numerator doesn't have all h's since the constant doesn't cross out.

I got until -11558.65 - 345.4h - 8.9h^2 + 0.95x^3 + 2.85x^2h + 2.85xh^2 + 0.95h^3. I feel like there's something I'm not understanding in the theory behind this problem but I'm not sure what.
 

lex

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Mar 3, 2021
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There shouldn't be any \(\displaystyle x\) in your expression for [S(23+h) - S(23)]/h] as you are substituting 23 for \(\displaystyle x\)
You are going to find the limit as h->0, of [S(23+h) - S(23)]/h]

and [S(23+h) - S(23)]/h] is:

\(\displaystyle \frac{[0.95(23+h)^3-8.9(23+h)^2+64(23+h)+246] - [0.95(23)^3-8.9(23)^2+64(23)+246]}{h}\)
You can simplify this unpleasant expression and then find the limit as h->0
It all should work.
(An alternative to simplifying the expression is to realise that the limit as h->0, will be the coefficient of the h term in the top line of the expression and just work that out).


where S represents the number of years since 1982
(You mean x, not S?)
 

Student123

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Apr 12, 2020
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That helped, thank you so much.
 
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Jomo

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Dec 30, 2014
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Use the difference of cubes and the difference of squares formulas to make this unpleasant expression a bit less unpleasant.
 
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