1 / 16

ANTIDERIVATIVES AND INDEFINITE INTEGRATION

When you are done with your homework, you should be able to. Write the general solution of a differential equationUse indefinite integral notation for antiderivativesUse basic integration rules to find antiderivativesFind a particular solution of a differential equation. Thales lived in 600 BC.

mead
Download Presentation

ANTIDERIVATIVES AND INDEFINITE INTEGRATION

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. ANTIDERIVATIVES AND INDEFINITE INTEGRATION Section 4.1

    2. When you are done with your homework, you should be able to… Write the general solution of a differential equation Use indefinite integral notation for antiderivatives Use basic integration rules to find antiderivatives Find a particular solution of a differential equation

    3. Thales lived in 600 BC. He is famous for being the first person to… …use deduction in mathematics. …measure the size of the earth. …characterize the conic sections. All of the above.

    4. ANTIDERIVATIVES A function F is an antiderivative of f on an interval I if for all x in I. Why does the definition use “an antiderivative” instead of “the antiderivative”?

    5. Theorem: Representation of Antiderivatives If F is an antiderivative of f on an interval I, then G is an antiderivative of f on the interval I if and only if G is of the form , for all x in I where C is a constant. How is this theorem different from the last definition?

    6. Some terms to be familiar with… The constant C is called the constant of integration.  The family of functions represented by G is the general antiderivative of f. is the general solution of the differential equation

    7. DIFFERENTIAL EQUATION A differential equation in x and y is an equation that involves x, y and derivatives of y. Examples: and

    8. Solving a Differential Equation Find the general solution of the differential equation . Solution: We need to find a function whose derivative is 6. The function has a derivative of 6. Using the previous theorem, we write the general solution as .

    9. Solve the differential equation Both A and C

    10. Solve the differential equation

    11. NOTATION FOR ANTIDERIVATIVES When solving a differential equation of the form , we solve for , giving us the equivalent differential form . This means you isolate dy by multiplying both sides of the equation by dx. It is easier to see if you write the left side as instead of

    12. Notation continued… The operation of finding all solutions of this equation is called antidifferentiation or indefinite integration and is denoted by an integral sign . The general solution is denoted by

    13. Solve the differential equation

    14. SOLVING A VERTICAL MOTION PROBLEM The Grand Canyon is 1800 meters deep at its deepest point. A rock is dropped from the rim above this point. Write the height of the rock as a function of the time t in seconds. How long will it take for the rock to reach the canyon floor?

    15. Vertical motion continued… Use as the acceleration due to gravity. Neglect air resistance. Recall that represents initial velocity, represents initial position. So . How did we get from the acceleration function to the position function?

    16. Continued…

More Related