Transcript mmstc-dewey.weebly.com

Fundamental Theorem of
Calculus
by
Nicholas F., Christopher H., Kathleen H., and Lindsey R.
FTC Part I - Definition
If a function g is continuous on the closed
interval [a,b] and f is the indefinite integral
of g on [a,b] then;
FTC Part I - Definition
This means that a definite integral can be
found by using the indefinite integral of the
function and the bounds of the definite
integral
FTC Part I - Visual
Part I Sample Problem
Using the fundamental Theorem of Calculus
evaluate
Part I Solution
Part II - Definition
If
where a stands for a constant, and f is
continuous in the neighborhood of a, then
g’(x)=f(x)
Part II - Definition
What this does is confirm that integrating a
function will return its antiderivative.
Therefore, if a function is integrated, then the
derivative of the integrand should return a
result equivalent to that of the original
function if the same number is substituted for
x into both equations.
Part II - Visual Explanation
FTC Part II: Sample Problem
Here we must find the derivative of g(x)
Part II Solution
To put it simply:
OR
Apply Chain Rule to g’(x):
=
Part II Answer
The derivative of g(x) =
For proof that
g’(x) = f(x), the function
f is continuous around
a = 8, as shown:
8
Sources
Foerster, Paul A. "6-2 Antiderivative of the Reciprocal Function and Another
Form of the Fundamental Theorem." Calculus: Concepts and Applications.
Berkeley, CA: Key Curriculum, 2005. 271. Print.
Antuna, Pablo. "The Fundamental Theorem of Calculus Made Clear: Intuition."
Http://www.intuitive-calculus.com/. N.p., n.d. Web. 17 May 2015.