5.4 Fundamental Theorem of Calculus Objectives SWBAT: 1)apply the fundamental theorem of calculus...
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Transcript of 5.4 Fundamental Theorem of Calculus Objectives SWBAT: 1)apply the fundamental theorem of calculus...
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5.4 Fundamental Theorem of Calculus
ObjectivesSWBAT:
1) apply the fundamental theorem of calculus2) understand the relationship between the derivative
and definite integral as expressed in both parts of the fundamental theorem of calculus
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The Fundamental Theorem of Calculus (FTOC)
• The FTOC has two parts. The two parts tie together the concept of integration and differentiation.
• Some people regard the FTOC as the most important computational discovery in the history of mathematics!!!
• How excited are you to learn this?!?!?!
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The total distance traveled was the area under the curve (green region)
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To find the total distance traveled, you can find the antiderivative of velocity (s), substitute in 10, substitute in 0, and subtract.
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The Fundamental Theorem of Calculus (the evaluation part aka part 2)
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Integral Rules• These rules are not formally stated until
Chapter 6, but knowing about derivatives, you should be able to make the following connections.
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Example 3: Evaluate each expression.
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The Fundamental Theorem of Calculus (part 1…the simple version)
where a is a constant.
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Wouldn’t it be great if all of Calculus was this simple?
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The Fundamental Theorem of Calculus (part 1…the extended version)
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Let’s look more closely at what it means to define a function using an integral.
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• Let’s now tie everything together (derivatives and integrals).
• You are almost guaranteed to see something like this on the AP exam.
• (distribute handout of example 12)