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February 21st, 2011, 07:01 PM   #1
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Derivatives: Differential and Linear approximation HELP

Hi,

I got a review sheet, and I'm having trouble solving these particular questions on differentials and linear approximations...I need some assistance please.

1) In a manufacturing process, ball bearings must be made with radius of 0.5 mm, with a maximum error in the radius of (+or-)0.018 mm. Estimate the maximum error in the volume of the ball bearing.

The formula for the volume of the sphere is_______. If an error (delta)r is made in measuring the radius of the sphere, the maximum error in the volume is (delta)V=______________.

Rather than calculating (delta)V, approximate (delta)V with dV, wher dV=_________.

Replacing r with _________ and dr=(delta)r with (+ or -)_________ gives dV= (+ or -)_________

The maximum error in the volume is about____________mm(cubed).

2)The demand function for a product is given by p=f(q)=90?sqrt(q). Where p is the price per unit in dollars for q units. Use the linear approximation to approximate the price when 2024 units are demanded.

Solution: We want to approximate f(2024). From: f(q)*approximate symbol*=L(q)=f(a)+f'(a)(q-a)

and the fact that f'(a)___________(-1/sqrt(a) or 1/2(sqrt(a)) or -1/2(sqrt(a)) or 1/sqrt(a) ) we choose a=_________.

From f(2025)=_______ and f'(2025)=__________ we get f(2024)*approximate symbol*_________.
hence, the price per unit when 2024 units are demanded is approximately $________.

IF anything is unclear on my part, describing the problems, please let me know.. Thank you.
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February 22nd, 2011, 08:07 PM   #2
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Re: Derivatives: Differential and Linear approximation HELP

I have figured out the second question that is given. So I'm only still stuck on the first question. Thanks!
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February 22nd, 2011, 09:29 PM   #3
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Re: Derivatives: Differential and Linear approximation HELP

The formula for the volume V of a sphere of radius r is:



Thus,



and

Replacing r with 0.5 mm and dr = ?r with ▒0.018 mm gives:



Thus, the maximum error is about 0.0565 mm│.
Thanks from Yuliecw
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February 23rd, 2011, 04:41 PM   #4
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Re: Derivatives: Differential and Linear approximation HELP

Thanks a lot!
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February 24th, 2011, 06:08 PM   #5
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Re: Derivatives: Differential and Linear approximation HELP

Quote:
Originally Posted by ProJO
I have figured out the second question that is given. So I'm only still stuck on the first question. Thanks!
How did you do the 2nd one?? thanks
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February 24th, 2011, 06:28 PM   #6
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Re: Derivatives: Differential and Linear approximation HELP

We may use the linear approximation to f(q) by the point-slope formula:



With we have so



and since and we choose a = 2025. Thus with:

and

we have:



This is very close to the actual value of f(2024).
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