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ANTIDERIVATIVES AND INDEFINITE INTEGRATION

ANTIDERIVATIVES AND INDEFINITE INTEGRATION. Section 4.1. 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

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ANTIDERIVATIVES AND INDEFINITE INTEGRATION

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  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 dybymultiplying 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…

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