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Local Linearity and Linearization

AP Calculus AB· difficulty 3/5

A measurement of r=2 cmr = 2~\text{cm} has uncertainty Δr=0.1 cm\Delta r = 0.1~\text{cm}. Using differentials, the resulting error in A=πr2A = \pi r^2 is approximately

  • A

    2π cm22\pi~\text{cm}^2

  • B

    0.4π cm20.4\pi~\text{cm}^2

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  • C

    π cm2\pi~\text{cm}^2

  • D

    0.2π cm20.2\pi~\text{cm}^2

Explanation

dA=2πr dr=2π(2)(0.1)=0.4π cm2dA = 2\pi r\,dr = 2\pi(2)(0.1) = 0.4\pi~\text{cm}^2.

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