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Let \ g(y) = \int^y_4 -5t^2\ sin\ (t) \ dt. \ Use \ the \ Fundamental \ Theorem \ of \ Calculus \...

Question:

{eq}Let \ g(y) = \int^y_4 -5t^2\ sin\ (t) \ dt. \ Use \ the \ Fundamental \ Theorem \ of \ Calculus \ to \ find \ g'(y).\\ g'(y) = {/eq}

Fundamental Theorem of Calculus 1 (FTC1):

If f is continuous on {eq}{/eq} then the function F defined by {eq}{/eq}

is continuous on {eq}{/eq} differentiable on (c,d) and F'(x) = f(x) for each {eq}x \in (c,d) {/eq}. In alternate notation, {eq}\displaystyle \dfrac{d}{dx} \left[ \int_a^x f(t) \, dt \right] = f(x). {/eq}

Roughly speaking, FTC1 says that if we first integrate f and then differentiate the result, we get back to the original function f.

Answer and Explanation: 1

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Note that {eq}-5t^2 \sin t {/eq}

is a polynomial times a sine function, which are both continuous functions on {eq}\mathbb R {/eq}, so the...

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The Fundamental Theorem of Calculus

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Chapter 12 / Lesson 10
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The fundamental theorem of calculus is one of the most important points to understand in mathematics. Learn to define the formula of the fundamental theorem of calculus and explore examples of it put into practice.


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