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The Fundamental Theorems of Calculus Lesson 5.4
First Fundamental Theorem of Calculus Given f is continuous on interval [a, b] F is any function that satisfies F’(x) = f(x) Then
First Fundamental Theorem of Calculus The definite integral can be computed by finding an antiderivative F on interval [a,b] evaluating at limits a and b and subtracting Try
Area Under a Curve Consider Area =
Area Under a Curve Find the area under the following function on the interval [1, 4]
Second Fundamental Theorem of Calculus Often useful to think of the following form We can consider this to be a function in terms of x View QuickTime Movie View QuickTime Movie
Second Fundamental Theorem of Calculus Suppose we are given G(x) What is G’(x)?
Second Fundamental Theorem of Calculus Note that Then What about ? Since this is a constant …
Second Fundamental Theorem of Calculus Try this
Assignment Lesson 5.4 Page 327 Exercises 1 – 49 odd
6 Integration Antiderivatives and the Rules of Integration
Integrals and the Fundamental Theorem (1/25/06) Today we review the concepts of definite integral, antiderivative, and the Fundamental Theorem of Calculus.
Areas and Definite Integrals. Objectives Students will be able to Calculate a definite integral. Calculate the area between a curve and the x-axis over.
The Area Between Two Curves
1 Fundamental Theorem of Calculus Section The Fundamental Theorem of Calculus If a function f is continuous on the closed interval [a, b] and F.
The Area Between Two Curves Lesson When f(x) < 0 Consider taking the definite integral for the function shown below. The integral gives a negative.
Definite Integrals Finding areas using the Fundamental Theorem of Calculus.
Miss Battaglia AP Calculus. Let u be a differentiable function of x. 1.2.
Clicker Question 1 What is the derivative of f(x) = 7x 4 + e x sin(x)? – A. 28x 3 + e x cos(x) – B. 28x 3 – e x cos(x) – C. 28x 3 + e x (cos(x) + sin(x))
The Fundamental Theorem of Calculus Inverse Operations.
5.4 The Fundamental Theorem. The Fundamental Theorem of Calculus, Part 1 If f is continuous on, then the function has a derivative at every point in,
Why is it the second most important theorem in calculus?
4-3 DEFINITE INTEGRALS MS. BATTAGLIA – AP CALCULUS.
Section 5.3 – The Definite Integral
Section 5.3: Evaluating Definite Integrals Practice HW from Stewart Textbook (not to hand in) p. 374 # 1-27 odd, odd.
The Fundamental Theorem of Calculus Lesson Definite Integral Recall that the definite integral was defined as But … finding the limit is not often.
5.c – The Fundamental Theorem of Calculus and Definite Integrals.
Section 5.4a FUNDAMENTAL THEOREM OF CALCULUS. Deriving the Theorem Let Apply the definition of the derivative: Rule for Integrals!
4.4c 2nd Fundamental Theorem of Calculus. Second Fundamental Theorem: 1. Derivative of an integral.
7.4: The Fundamental Theorem of Calculus Objectives: To use the FTC to evaluate definite integrals To calculate total area under a curve using FTC and.
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