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Question from Test 1 Liquid drains into a tank at the rate 21e-3t units per minute. If the tank starts empty and can hold 6 units, at what time will it overflow? A. log(7)/3 B. (1/3)log(13/7) C. 3 log (13/7) D. 3log(7) E. Never

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**Fundamental Theorem of Calculus (Part 1) (Chain Rule)**

Chapter 5.3 & 5.5 February 6, 2007 Fundamental Theorem of Calculus (Part 1) (Chain Rule) If f is continuous on [a, b], then the function defined by is continuous on [a, b] and differentiable on (a, b) and

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**Fundamental Theorem of Calculus (Part 1)**

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**Fundamental Theorem of Calculus (Part 2)**

If f is continuous on [a, b], then : Where F is any antiderivative of f. ( ) Helps us to more easily evaluate Definite Integrals in the same way we calculate the Indefinite!

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Example

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Example

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Evaluate:

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Evaluate:

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Evaluate:

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Given: Write a similar expression for

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Evaluate: Multiply out:

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**Chain Rule for Derivatives:**

What if instead? Chain Rule for Derivatives: Chain Rule backwards for Integration: Look for:

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Back to Our Example Let

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**The same substitution holds for the higher power!**

With

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**Our Original Example of a Definite Integral:**

To make the substitution complete for a Definite Integral: We make a change of bounds using:

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**Substitution Rule for Indefinite Integrals**

If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then Substitution Rule for Definite Integrals If g’(x) is continuous on [a,b] and f is continuous on the range of u = g(x), then

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In-Class Assignment Integrate using two different methods: 1st by multiplying out and integrating 2nd by u-substitution Do you get the same result? (Don’t just assume or claim you do; multiply out your results to show it!) If you don’t get exactly the same answer, is it a problem? Why or why not?

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