Assignment 18

course Mth 271

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018. `query 18

Applied Calculus I

03-26-2007

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19:18:13

** Query problem 2.5.48 der of 3/(x^3-4) **** What is your result?

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RESPONSE -->

y=3(x^3-4)^-2

y=-9x^2/(x^3-4)^2

self critique assessment: 2

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19:18:45

This function can be expressed as f(g(x)) for g(x) = x^3-4 and f(z) = 3 / z. The 'inner' function is x^3 - 4, the 'outer' function is 1 / z.

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

Thus f'(g(x)) = -3/(x^3-4)^2 so the derivative of the whole function is

[3 / (x^3 - 4) ] ' = g'(x) * f'(g(x)) = 3x^2 * (-3/(x^3-4)^2) = -9 x^2 / (x^3 - 4)^2.

DER**

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RESPONSE -->

I got this answer correct.

self critique assessment: 2

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19:21:30

**** Query problem 2.5.66 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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RESPONSE -->

(-1/2)(2(3)-3)*(3^2-3(3)+4)^-3/2

-1/2(3)(9-6+4)^-3/2

-1/2(3)(7)^-3/2

(-1.5)(.001457)

=-.0021855

confidence assessment: 2

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19:23:15

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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RESPONSE -->

I solved by keying the points into the equation. I didn't take it another step to get the equation of the tangent line. I have made a note of this answer.

self critique assessment: 2

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19:27:14

**** Query problem 2.5.72 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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RESPONSE -->

.25(n+5)*.50*(.5n^2+5n+25)^-1/2

12+5/8sqrt(.5)(12^2)+5(12)+25)=17/100.23=.1696 is the rate of pollution change.

confidence assessment: 2

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19:28:44

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**

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RESPONSE -->

I got the same answer I just didn't round it off to .17

self critique assessment: 2

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Good work. Let me know if you have questions.