1 / 16

13.1 Antiderivatives and Indefinite Integrals

13.1 Antiderivatives and Indefinite Integrals. The Antiderivative. The reverse operation of finding a derivative is called the antiderivative. A function F is an antiderivative of a function f if F ’( x ) = f ( x ). Find the antiderivative of f(x) = 5

afi
Télécharger la présentation

13.1 Antiderivatives and Indefinite Integrals

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 13.1 Antiderivatives and Indefinite Integrals

  2. The Antiderivative • The reverse operation of finding a derivative is called the • antiderivative. A function F is an antiderivative of a function f if • F ’(x) = f (x). • Find the antiderivative of f(x) = 5 • Find several functions that have the derivative of 5 • Answer: 5x; 5x+ 1; 5x -3; • 2) Find the antiderivative of f(x) = x2 • Find several functions that have the derivative of x2 • Answer: Theorem 1: If a function has more than one antiderivative, then the antiderivatives differ by at most a constant.

  3. The graphs of antiderivatives are vertical translations of each other. • For example: f(x) = 2x Find several functions that are the antiderivatives for f(x) Answer: x2, x2 + 1, x2 + 3, x2 - 2, x2 + c (c is any real number)

  4. Indefinite Integrals Let f (x) be a function. The family of all functions that are antiderivatives of f (x) is called the indefinite integral and has the symbol The symbol  is called an integral sign, and the function f (x) is called the integrand. The symbol dx indicates that anti-differentiation is performed with respect to the variable x. By the previous theorem, if F(x) is any antiderivative of f, then The arbitrary constant C is called the constant of integration.

  5. Indefinite Integral Formulas and Properties (power rule) It is important to note that property 4 states that a constant factor can be moved across an integral sign. A variable factorcannotbe moved across an integral sign.

  6. Example 1: A) B) C)

  7. Example 1 (continue) D)

  8. Example 1 (continue) E)

  9. Example 2 A)

  10. Example 2 (continue) B)

  11. Example 2 (continue) C)

  12. Example 2 (continue) D)

  13. Example 2 (continue) E)

  14. Example 3 Find the equation of the curve that passes through (2,6) if its slope is given by dy/dx = 3x2 at any point x. The curve that has the derivative of 3x2 is Since we know that the curve passes through (2, 6), we can find out C Therefore, the equation is y = x3 - 2

  15. Example 4 Find the revenue function R(x) when the marginal revenue is R’(x) = 400 - .4x and no revenue results at a 0 production level. What is the revenue at a production of 1000 units? The marginal revenue is the derivative of the function so to find the revenue function, we need to find the antiderivative of that function So R(x) = 400x -.2x2, we know need to find R(1000) Therefore, the revenue at a production level of 1000 units is $200,000

  16. Example 5 The current monthly circulation of the magazine is 640,000 copies. Due to the competition from a new magazine, the monthly circulation is expected to decrease at a rate of C’(t)= -6000t1/3 copies per month, t is the # of months. How long will it take the circulation of the magazine to decrease to 460,000 copies per month? We must solve this equation: C(t) = 460,000 with C(0) = 640,000 To find the function C(t), take the antiderivative So, it takes about 16 months

More Related