If your solution to stated problem does not match the given solution, you should self-critique per instructions at http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm.

Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.

018. `query 18

** Query problem 7th edition 2.5.48  2.5.44 der of 3/(x^3-4)^2 **** What is your result?

Your solution:

Confidence Rating:

Given Solution:

`a This function can be expressed as f(g(x)) for g(x) = x^3-4 and f(z) = 3 / z^2. The 'inner' function is x^3 - 4, the 'outer' function is 3 / z^2 = 3 z^(-2).

So f ' (z) = -6 / z^3 and g'(x) = 3x^2.

Thus f ' (g(x)) = -6/(x^3-4)^3 so the derivative of the whole function is

[3 / (x^3 - 4) ] ' = g ' (x) * f ' (g(x)) = 3x^2 * (-6/(x^3-4)^3) = -18 x^2 / (x^3 - 4)^3.

DER**

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Question: `q **** Query problem 2.5.62 tan line to 1/`sqrt(x^2-3x+4) at (3,1/2) **** What is the equation of the tangent line?

Your solution:

Confidence Rating:

Given Solution:

`a The derivative is (2x - 3) * -1/2 * (x^2 - 3x + 4) ^(-3/2) .

At (3, 1/2) we get -1/2 (2*3-3)(3^2- 3*3 + 4)^(-3/2) = -1/2 * 3 (4)^-(3/2) = -3/16.

The equation is thus ( y - 1/2) = -3/16 * (x - 3), or y = -3/16 x + 9/16 + 1/2, or y = -3/16 x + 17/16.

DER**

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Question: `q **** Query problem 2.5.68 rate of change of pollution P = .25 `sqrt(.5n^2+5n+25) when pop n in thousands is 12 **** At what rate is the pollution changing at the given population level?

Your solution:

Confidence Rating:

Given Solution:

`a The derivative is .25 [ (n + 5) * 1/2 * (.5 n^2 + 5 n + 25) ^(-1/2) )

= (n+5) / [ 8 `sqrt(.5n^2 + 5n + 25) ]

When n = 12 we get (12+5) / ( 8 `sqrt(.5*12^2 + 5 * 12 + 25) ) = 17 / 100 = .17, approx.

DER**

ADDITIONAL COMMENT

Details of calculating P ':

P is of the form f(g(x)) with g(x) = .5 n^2 + 5 n + 25 and f(z) = .25 z^(1/2).

g ' (x) = n + 5 and f ' (z) = .25 ( 1/2 z^(-1/2) ) = 1 / (8 z^(1/2)), or -1 / (8 sqrt(z) ).

Thus

P ' = g ' (x) * f ' (g(x)) = (n + 5) * (1/ (8 sqrt(n^2 + 5 n + 25) ) ) =
(n+5) / (8 sqrt(n^2 + 5 n + 25).

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