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Using Differentials to Estimate

1. In the manufacturing process, ball bearings must be made with radius of 0.5mm, with a maximum error in the radius of +/- 0.016 mm. Estimate the maximum error in the volume of the ball bearing.

Solution: The formula for the volumen of the sphere is _____.

If an error deltaR is made in measuring the radius of the sphere, the maximum error in the volume is deltaV = _____.

Rather than calculating deltaV, approximate deltaV with dV, where dV = _____.

Replacing r with ___ and dr = deltar with =/- _____ gives dV = +/- ______.

The maximum error in the volume is about _____ mm^3.

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) ~ L(q) = f(a) + f'(a)(q - a)

and the fact that f'(a) = _____.

we chose a = ______.

From f(2025) = ______ and f'(2025) = ______ we get f(2024) ~ _____.

Hence, the price per unit when 2024 units are demanded is approximately $_____.

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Solution Summary

The maximum error in the volume is determined.

$2.19