Quote of the Day:
"It is clear that Economics, if it is to be a science at
all, must be a mathematical science ... simply because
it deals with quantities...
As the complete theory of almost every other science
involves the use of calculus, so we cannot have a true
theory of Economics without its aid."
-- W. S. Jevons
Objectives:
The student will learn the properties of the definite
integral and apply them when solving integrals.
The student will learn the Fundamental Theorem of Calculus and apply it.
1. Bellringer -- Numerical Word Search (in groups)
2. The Fundamental Theorem of Calculus
Each branch of mathematics has a fundamental theorem
associated with it.
The Fundamental Theorem of Arithmetic:
Any positive integer can be represented in exactly
one way as a product of primes.
The Fundamental Theorem of Algebra:
Every polynomial of degree n has exactly n zeros.
The Fundamental Theorem of Geometry:
No theorem wears this title, but perhaps the
Pythagorean Theorem deserves it.
The Fundamental Theorem of Calculus — there are
actually two parts to this theorem:
The First Fundamental Theorem of Calculus:
The derivative of the integral of a function
is equal to the function.
The Second Fundamental Theorem of Calculus:
The integral of the derivative of a function is
is equal to the function evaluated at its
endpoints.
The F.T.C. tells us that we can evaluate a definite
integral by taking an indefinite integral and
substituting in the endpoints and taking the
difference:
Remember that you can't spell FUNDAMENTAL without FUN (and MENTAL, and DA).
So, think of The Fundamental Theorem of Calculus as DA MENTAL FUN.
3. Examples:
4. Properties of the Definite Integral
5. Examples of the Properties of the Definite Integral
6. Assignment
p. 394 (15, 16, 19, 20, 21)
p. 406 (3, 6, 9, 11, 13)
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