Differential Calculus

(1) Definition (Differential Calculus) If is a differentiable function then the differential is defined by the equation where is an independent variable.

(2) Example (Differential Calculus) Find the differential for

    Solution. For the derivative is

and so the differential of is

(3) Example (Differential Calculus) Find the differential for .

    Solution. For the derivative is

and so the differential of is

(4) Example (Differential Calculus) Find the differential for

    Solution. For the derivative is

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and so the differential of is

differential calculus _gr_19.gif]

(5) Example (Differential Calculus) Using differential calculus, approximate

    Solution. If then and using the linear approximation is,

(6) Example (Differential Calculus) Using differential calculus, approximate

    Solution. If then and using the linear approximation is,

(7) Example (Differential Calculus) Using differential calculus, approximate

    Solution. If then and using the linear approximation is,

(8) Definition (Differential Calculus) The approximation

is called the linear approximation of at and the function

is called the linearization of at

The equation of the tangent line to the curve at is

which is so that in fact we have Thus when using differentials to approximate, that is, when using to approximate we are using the tangent line at as an approximation to the curve when is near

(9) Example (Differential Calculus) Find the linearization of   at

    Solution. The linearization of the function at is

Therefore, we have the linear approximation for when is near 0.

(10) Example (Differential Calculus) Find the linearization of   at

    Solution. The linearization of the function at is

Therefore, we have the linear approximation for when is near 0.

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Differential Calculus
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/differential-calculus.html