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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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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
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The Fundamental Theorem of Calculus Part 2 & U-Sub
Chapter 5.3 & 5.5 February 6, 2007
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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 We have to find an antiderivative; evaluate at 3;
subtract the results.
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Example
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Example This notation means: evaluate the function at 3 and 2, and subtract the results.
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Example
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Example
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Example
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Example Don’t need to include “+ C” in our antiderivative, because any antiderivative will work.
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the “C’s” will cancel each other out.
Example the “C’s” will cancel each other out.
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Example
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Example
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Example Alternate notation
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Example
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Example
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Example = –1
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Example = –1 = 1
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Example
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Example
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Example
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Evaluate:
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Evaluate: Improper = 0)
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Evaluate:
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Given: Write a similar expression for
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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. ( )
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Evaluate: Multiply out: Use FTC 2 to Evaluate:
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What if instead? It would be tedious to use the same multiplication strategy! There is a better way! We’ll use the chain rule (backwards)
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Chain Rule for Derivatives:
Chain Rule backwards for Integration:
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Look for:
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Back to Our Example Let
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Our Example as an Indefinite Integral
With AND Without worrying about the bounds for now: Back to x (Indefinite):
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The same substitution holds for the higher power!
With Back to x (Indefinite):
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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: When x = -1, u = 2(-1)+1 = -1 When x = 2, u = 2(2) + 1 = 5 The x-interval [-1,2] is transformed to the u-interval [-1, 5]
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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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