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March 16th, 2017, 09:41 PM   #1
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Derivative word problem

I am no good at word problems and can't really tell where to start. I believe it's based on the chain rule. Here is the problem:

The rate of government spending (in billions of dollars) in a country is related to the country's unemployment by the function g(u) = 196+u^6/5 where g is the level of government spending and u is the percentage of unemployment. The percentage unemployment rate, in turn, is related to the inflation rate as u(i) = 80(i+10)^-1 where i is the inflation rate. Find and interpret g(i) when i = 7%. Find and simplify g'(i). Find and interpret g'(i) when i = 7%. Round your answer to the nearest million.

Thanks for any help offered.

I don't understand the concept of the question, is it basically asking to take the second derivative of g(i)? I'm not too good at derivatives. I understand all the rules, but when it comes to combining functions, I get lost.
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March 16th, 2017, 10:34 PM   #2
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Is 6/5 the exponent of u?
skipjack is offline  
March 17th, 2017, 04:17 AM   #3
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Originally Posted by skipjack View Post
Is 6/5 the exponent of u?
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March 17th, 2017, 07:40 AM   #4
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You are asking two completely different questions, but nothing in this problem has to do with second derivatives.

One asks for 4 mechanical results.

$g(u) = g = 196 + u^{(6/5)} \implies g'(u) = \dfrac{dg}{du} = what?$

$u(i) = u = \dfrac{80}{i + 10} \implies u'(i) = \dfrac{du}{di} = what?$

You are also asked to find

$g(u(i))\ and\ \dfrac{dg}{di}.$

There is no thought involved, just the application of rules. You don't have to know what any of the necessary manipulations mean: a computer can do them.

But then you are also asked what do these results mean. A computer cannot do that: it requires understanding. I suggest doing the mechanical part first and then asking yourself what it means when you have reduced it all to numbers.

Composition of functions is fundamental to calculus. You MEMORIZE a bunch of simple rules: for example the addition rule of

$\alpha ( \lambda ) = \beta ( \lambda ) \pm \gamma ( \lambda ) \implies \alpha '( \lambda ) = \beta '( \lambda ) \pm \gamma '( \lambda ).$

But suppose you are faced with some more complex function like

$\epsilon ( \lambda ) = sin \left ( \dfrac{1}{ \beta (\lambda) } + \dfrac{1}{ \gamma ( \lambda )}\right ) \implies \epsilon '( \lambda ) = what?$

There is no direct rule to tell you what to do. You use composition of functions and the chain rule to break the problem down to where you can use the rules you have memorized.

The idea behind a function is this

f(x) = some formula

f(y + z) = the formula with y + z replacing x everywhere x appears in the formula.

h(x) = g(f(x)) means you have an f formula and a g formula. Their composition gives you the h formula. But what is the h formula. Calculate the f formula. Whatever you get as an answer, replace x in the g formula with whatever came out of the f formula. That gives you the h formula. It is PEMDAS from first year algebra: you do what is inside parentheses first so you calculate first calculate the f formula and then the g formula.

$f(x) = 2x + 1\ and\ g(x) = x^2 \implies$

$h(x) = g(f(x)) = g(2x + 1) = (2x + 1)^2 = 4x^2 + 4x + 1\ and$

$j(x) =f(g(x)) = f(x^2) = 2x^2 + 1.$
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