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TI-89 Titanium Graphing Calculator Lesson

Module 16: The Fundamental Theorem of Calculus

Introduction | Lesson 1 | Lesson 2 | Lesson 3 | Self-Test

## Lesson 16.3: The Fundamental Theorem of Calculus

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### A restatement of the Fundamental Theorem of Calculus is presented in this lesson along with a corollary that is used to find the value of a definite integral analytically.

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Restating the Fundamental Theorem
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You have discovered the Fundamental Theorem in the context of finding areas under a curve but a more general version of this theorem can be proved without an appeal to area. The following is a restatement of the Fundamental Theorem.

If *f* is continuous on [*a*, *b*], then the function

has a derivative at every point in [*a*, *b*], and the derivative is

That is, the derivative of a definite integral of *f* whose upper limit is the variable *x* and whose lower limit is the constant *a* equals the function *f* evaluated at *x*. This is true regardless of the value of the lower limit *a*. The function named *F* is the same as the area function that was previously explored.

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Using the Restated Fundamental Theorem
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- Set Angle mode to Radian
- Execute NewProb from the Clean Up menu

16.3.1 Use the restatement of the Fundament theorem to evaluate the following derivatives, then check your predictions with the TI-89.

Click here for the answer.

16.3.2 Predict the following derivative. Check your answer with the TI-89. Hint: You will have to use the chain rule.

Click here for the answer.

16.3.3 Evaluate the following derivative and check with your TI-89.

Click here for the answer.

16.3.4 Find a more general version of the Fundamental theorem by predicting the following derivative. Check your work with the TI-89.

The function *F* is called an antiderivative of the function *f*.

16.3.5 Use the corollary to predict the value of , then check your work with the TI-89.

Click here for the answer.

16.3.6 Use the corollary of the Fundamental theorem to evaluate then check your work with your calculator.

Click here for the answer.

*derivative*of the definite integral. The value of the definite integral is found using an antiderivative of the function being integrated.