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Popular Calculus >

taylor cos(x),\at pi/2

Solution

taylor

cos (x), at \frac{π}{2}

Solution

\frac{- sin ( at \frac{π}{2} )}{1 !} (x - at \frac{π}{2}) + \frac{- cos ( at \frac{π}{2} )}{2 !} (x - at \frac{π}{2})^{2} + \frac{sin ( at \frac{π}{2} )}{3 !} (x - at \frac{π}{2})^{3} + \frac{cos ( at \frac{π}{2} )}{4 !} (x - at \frac{π}{2})^{4} + \dots

Solution steps

Apply the Taylor Formula

= cos (at \frac{π}{2}) + \frac{\frac{d}{d x} ( cos ( x ) ) ( at \frac{π}{2} )}{1 !} (x - at \frac{π}{2}) + \frac{\frac{d ^{2}}{d x ^{2}} ( cos ( x ) ) ( at \frac{π}{2} )}{2 !} (x - at \frac{π}{2})^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( cos ( x ) ) ( at \frac{π}{2} )}{3 !} (x - at \frac{π}{2})^{3} + \dots

Evaluate Derivatives

= cos (at \frac{π}{2}) + \frac{- sin ( at \frac{π}{2} )}{1 !} (x - at \frac{π}{2}) + \frac{- cos ( at \frac{π}{2} )}{2 !} (x - at \frac{π}{2})^{2} + \frac{sin ( at \frac{π}{2} )}{3 !} (x - at \frac{π}{2})^{3} + \frac{cos ( at \frac{π}{2} )}{4 !} (x - at \frac{π}{2})^{4} + \dots

Refine

= \frac{- sin ( at \frac{π}{2} )}{1 !} (x - at \frac{π}{2}) + \frac{- cos ( at \frac{π}{2} )}{2 !} (x - at \frac{π}{2})^{2} + \frac{sin ( at \frac{π}{2} )}{3 !} (x - at \frac{π}{2})^{3} + \frac{cos ( at \frac{π}{2} )}{4 !} (x - at \frac{π}{2})^{4} + \dots

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