4.8 - 1 4.8 Antiderivatives A physicist, who knows the...

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Unformatted text preview: 1 4.8 Antiderivatives A physicist, who knows the position x ( t ) of a particle, can easily calculate the velocity of it by differentiation. Assume now, that a different physicist can only measure the velocity v ( t ), but he wishes to know both x ( t ) and v ( t ). Is it possible to find x ( t ) given its derivative? Definition A function is called an of f ( x ) on an interval I if: for all x in I . In other words, F is an antiderivative of f if the derivative of F is f . Example 1 Find an antiderivative of f ( x ) = 2 x Theorem If F ( x ) is an antiderivative of f ( x ) on an interval I then the of f on I is where is an constant. The table of antiderivatives of most common elementary functions is given in Table 4.2 on page 297 in University Calculus. Example 2 Find an antiderivative of f ( t ) = 1 t 4 = t- 4 2 Example 3 Find an antiderivative of h ( s ) = sin(2 s ) Example 4 Find an antiderivative of f ( x ) = x n and g ( x ) = 4 x 2 / 7 Initial Value Problems (IVP) Typical IVP asks you to find a function...
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4.8 - 1 4.8 Antiderivatives A physicist, who knows the...

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