Focus Area: FTOC Created by Mr. Lajoie and Mr. Herron.

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Focus Area: FTOC Created by Mr. Lajoie and Mr. Herron

The Second Fundamental Theorem of Calculus 2  *Notice: The lower limit must be a constant; the upper limit is the variable. For a more in-depth explanation of this formula, look at the next slides.

Things to Remember About this Formula: 1 3  The function HAS to be continuous on the interval [a, b].  “a” and “b” are two arbitrary (random) x-values. So the interval [a, b] means “from a to b on the x-axis.”  Would the following two integrals meet this condition? Example 1: Example 2: For the answers, look at the next slide.

Things to Remember about this Formula: Part 1 4 VS.

Things to Remember: Part 2 5  What is the deal with all of these symbols? For a detailed explanation of this part, look at the next slide.

The Formula: A “Basic” Explanation 6 Remember, F(x) is the “antiderivative” of f(x). This part is saying that if we go back and take the derivative of F(x), we will end up with our original function. So, the derivative and the antiderivative are inverse operations similar to multiplication and division, and when you both of them to the same term, they somewhat “cancel out”. Derivative Antiderivative The Answer is the same as the original function, just with “x” instead of “t”!

The Formula: A “Basic” Explanation 7 Except there is one change. Before the function was “f(t)”, but now it is f(x). And notice that “x” was the upper limit of integration.

Example Problem 8  Compute dy/dx if… For the answer, look at the next slide.

Example Problem 9

Example Problem 2-4: 10 For the answers, look at the next slide.

Example Problem 2-4: 11 For the answers, look at the next slide.

Next Steps 12  To see more complicated examples, look at the “Text” resource, which has an example that also includes the Chain Rule at the end and will help prepare you well for the Content Assessment.  Also, try out the Check For Understanding Practice problems.