5.4 The Fundamental Theorem of Calculus

I. The Fundamental Theorem of Calculus Part I. A.) If f is a continuous function on [a, b], then the function has a derivative at each point in [a, b] and F’(x) = f(x).

B.) Ex- x cannot vary between a and itself, therefore we use t as a dummy variable.

C.) IMPORTANCE: 1.) Continuous functions on an interval have antiderivatives F(x) in that interval. 2.) The process of integration and differentiation are inverse processes.

II. Functions Defined in Terms of Integrals A.) Lets look at the graph: Y10=NINT(2x,x,2,x)

B.) Using Calculus:

2.)3.)

III. Derivatives of Integrals A.) Ex.- Find the derivative of

B.) Ex.- Find a function whose derivative is tan x and whose value at x = 3 is 5. Where f (x) is on [2, 4]. NOTE: The interval must have finite discontinuities and contain (3, tan (3)).

C.) Ex.- Find

D.) Ex.- Find

E.) In general by the Fundamental Theorem of Calculus where u is a function of x…

F.) Ex. Find for each of the following.

1.)2.)