open qa 18

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course Mth 271

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

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Your solution:

3/(x^3 -4)^2 = 3 * (x^3 - 4) ^ -2 = -6 * ( x^3 - 4) (3x^2) = -18x^2/(x^3 - 4)

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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.

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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?

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Your solution:

d/dx [1/(sqrt(x^2 - 3x + 4)] = 1/(x^2 - 3x + 4) ^ 1/2 = (x^2 - 3x + 4) ^ -1/2 = -1/2 (x^2 - 3x + 4) ^ -3/2 * ( 2x - 3) = -1/2 (2x - 3) * (x^2 -3x + 4) ^ -3/2

X = 3

d/dx = - 3/2 * 4^ -3/2 = -3/16

Equation of tangent line:

(y - ½) = -3/16 (x - 3) / * 16

16y - 8 = -3 (x-3)

16y - 8 = -3x + 9

16y = - 3x + 17

Y = -3/16 x + 17/16

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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?

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Your solution:

Y’ = 0.25 * (0.5n^2 + 5n + 25) ^ ½ = 0.25 * [1/2 (0.5n^2 + 5n + 25) ^ -1/2 * (n+ 5)]

Y’ = 0.25 * (0.5 (n+5))/(sqrt(0.5n^2 + 5n + 25))

Y’ = (0.125 (n+5))/(sqrt(0.5n^2 + 5n + 25)

n = 12

y’ = 0.17

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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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&#Very good responses. Let me know if you have questions. &#