MAT.126 4.4

Honelako gehiago

Change Management

by Firman Setiawan

Increments and Differentials

by Rebeca Bazua

chapter 6 & 7 notes

by LaShundra Bell

IndividualChange

by Anne Adrian

MAT.126 4.4 The Fundamental Theorem of Calculus

Understand and use the Second Fundamental Theorem of Calculus

Theorem 4.11 The Second Fundamental Theorem of Calculus

If f is continuous on an open interval I containing a, then, for every x in the interval,

d/dx [ S_a^x f(t) dt ] = f(x).

Accumulation Function

The Definite Integral as a Function of x

The definite integral as a number

S_a^b f(x) dx

The definite integral as a function of x

F(x) = S_a^b f(t) dt

Understand and use the Net Change Theorem

Theorem 4.12 The Net Change Theorem

The definite integral of the rate of change of a quantity F'(x) gives the total change, or net change, in that quantity on the interval [a,b].

S_a^b F'(x) dx = F(b) - F(a) Net change of F

Find the average value of a function over a closed interval

Definition of the average value of a function on an interval

If f is integrable on the closed interval [a,b], then the average value of f on the interval is

1/(b-a) S_a^b f(x) dx.

Understand and use the Mean Value Theorem for Integrals

Theorem 4.10 Mean Value Theorem for Integrals

If f is continuous on the closed interval [a,b], then there exists a number c in the closed interval [a,b] such that

S_a^b f(x) dx = f(c)(b-a).

Evaluate a definite integral using the Fundamental Theorem of Calculus

Guidelines for using the Fundamental Theorem of Calculus

See page 283 in your text.

Note especially item 3: You do not need a constant of integration when working with definite integrals.

Theorem 4.9 The Fundamental Theorem of Calculus

If a function f is continuous on the closed interval [a,b] and F is an antiderivative of f on the interval [a,b], then

S_a^b f(x) dx = F(b) - F(a).

Informally

Differentiation and (definite) integration are inverse operations.

Slope is a quotient. The limit of the slope of secant lines gives the slope of a tangent line.

Area is a product. The limit of the area of rectangles gives the area of a region under a curve.

Quotients and products are inverse operations (division and multiplication).