Lesson25_-_The_Fundamental_Theorem_of_Calculus_ws-sol

# Lesson25_-_The_Fundamental_Theorem_of_Calculus_ws-sol -...

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Section 5.4 The Fundamental Theorem of Calculus V63.0121, Calculus I Spring 2010 Compute the derivatives of the following functions. 1. g ( x ) = Z x 0 sin tdt Solution. By the Fundamental Theorem of Calculus, g 0 ( x ) = sin x . 2. g ( x ) = Z 3 x 0 sin tdt Solution. Write this as F ( g ( x )), where F ( u ) = Z u 0 sin tdt , and g ( x ) = 3 x . Then d dx F ( g ( x )) = F 0 ( g ( x )) g 0 ( x ) = sin( g ( x )) · 3 = 3 sin 3 x. Alternatively, we could evaluate the integral directly, and get g ( x ) = - cos x | 3 x 0 = - cos 3 x + 1 So g 0 ( x ) = 3 sin 3 x . 3. g ( x ) = Z 0 2 x sin tdt (Hint: reverse the order of the integral.) Solution. We have g ( x ) = - Z 2 x 0 sin tdt , and we can diﬀerentiate this normally as in the last part to get g 0 ( x ) = - 2 sin 2 x . 4. g ( x ) = Z 3 x 2 x sin tdt (Hint: use the two previous problems.) Solution. Write g ( x ) = Z 0 2 x sin tdt + Z 3 x 0 sin tdt. Then, using the last two parts, g 0 ( x ) = - 2 sin 2 x + 3 sin 3 x .

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Lesson25_-_The_Fundamental_Theorem_of_Calculus_ws-sol -...

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