lecture13 - Lecture 13 - Antiderivatives and the Definite...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Lecture 13 - Antiderivatives and the Definite Integral 13.1 Antiderivatives and Indefinite Integrals Definition 13.1 If f and g are two functions defined on an interval I , then we say that f is an antiderivative of g if f ( x ) = g ( x ) for all x in I . Example: Let f ( x ) = 6 e 3 x , and let g 1 ( x ) = 2 e 3 x , g 2 ( x ) = 2 e 3 x + 37, and g 3 ( x ) = 2 e 3 x- 12. Then g 1 ,g 2 , and g 3 are all antiderivatives of f : g 1 ( x ) = 2 e 3 x · 3 = 6 e 3 x g 2 ( x ) = 2 e 3 x · 3 = 6 e 3 x g 3 ( x ) = 2 e 3 x · 3 = 6 e 3 x Note that we can always change the constant and we still have an antiderivative. The notation we use for this is the following: Z f ( x ) d x = g ( x ) + C if and only if g ( x ) = f ( x ) The expression R f ( x ) d x is called an indefinite integral , and C is an arbitrary constant called the constant of integration . So the preceding example could be written as Z 6 e 3 x d x = 2 e 3 x + C Example: A firm knows that the marginal cost function at a production level of x units is M ( x ) = x 2- 2 x + 10. Suppose that the fixed initial cost is $35,000. Find the cost function. solution: Remember, marginal cost is the derivative of the cost function. So M ( x ) = C ( x ). Hence to find C ( x ) we should find the antiderivative of M ( x ). So we calculate C ( x ) = Z M ( x ) d x = Z ( x 2- 2 x + 10) d x For simple functions such as this one can “hunt around” for the antiderivative. In this case,For simple functions such as this one can “hunt around” for the antiderivative....
View Full Document

This note was uploaded on 01/07/2010 for the course MAT mat1300 taught by Professor Pieterhofstra during the Fall '09 term at University of Ottawa.

Page1 / 5

lecture13 - Lecture 13 - Antiderivatives and the Definite...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online