Fundamental Theorem of Calculus

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Presentation transcript:

Fundamental Theorem of Calculus AP CALCULUS AB Chapter 5: The Definite Integral Section 5.4: Fundamental Theorem of Calculus

What you’ll learn about Fundamental Theorem, Part 1 Graphing the Function Fundamental Theorem, Part 2 Area Connection Analyzing Antiderivatives Graphically … and why The Fundamental Theorem of Calculus is a Triumph of Mathematical Discovery and the key to solving many problems.

Section 5.4 – Fundamental Theorem of Calculus Fundamental Theorem of Calculus, Part 1 If f is continuous on [a, b], then the function has a derivative at every point x in [a, b], and

The Fundamental Theorem of Calculus

Example Applying the Fundamental Theorem

Example The Fundamental Theorem with the Chain Rule

Example Variable Lower Limits of Integration

Section 5.4 – Fundamental Theorem of Calculus Fundamental Theorem of Calculus, Part 2 If f is continuous at every point of [a, b], and if F is any antiderivative of f on [a, b], then This is also called the Integral Evaluation Theorem.

The Fundamental Theorem of Calculus, Part 2

Section 5.4 – Fundamental Theorem of Calculus Example:

Example Evaluating an Integral

Section 5.4 – Fundamental Theorem of Calculus To find the Total Area Analytically To find the area between the graph of y=f(x) and x-axis over the interval [a, b] analytically: Partition [a, b] with the zeros of f. Integrate f over each subdivision. Add the absolute value of the integrals.

Section 5.4 – Fundamental Theorem of Calculus Ex: Find the area between the curve and the x-axis over the interval [-4, 4].

Section 5.4 – Fundamental Theorem of Calculus To find Total Area Numerically (on the calculator) To find the area between the graph of y=f(x) and the x-axis over the interval [a, b] numerically, evaluate: On the TI-89: nInt (|f(x)|, x, a, b, 50) On the TI-83 or 84: fnInt (|f(x)|, x, a, b, 50)

Section 5.4 – Fundamental Theorem of Calculus Application: Average Daily Inventory: If I(x) is the number of items on hand on day x, the average daily inventory of the items for the period is If h is the dollar cost of holding one item per day, the average daily holding cost for the period is