1. Warm-Up Explain the difference between propagated error and relative error.

2. 4.1 : Anti-derivatives Greg Kelly, Hanford High School, Richland, Washington

3. First, a little review: Consider: then: or It doesn’t matter whether the constant was 3 or -5, since when we take the derivative the constant disappears. However, when we try to reverse the operation: Given:findWe don’t know what the constant is, so we put “C” in the answer to remind us that there might have been a constant.

4. If we have some more information we can find C. Given: and when, find the equation for. This is called an initial value problem. We need the initial values to find the constant. An equation containing a derivative is called a differential equation. It becomes an initial value problem when you are given the initial condition and asked to find the original equation. Given a derivative function, the anti-derivative is the function whose derivative was given. The process of finding an anti- derivative is called indefinite integration (or anti-differentiation).

5. 4.1 Anti-derivatives and Indefinite Integration

6. Definition of an Anti-derivative A function F is an antiderivative of f on an interval if for all x in the interval.

7. 4.1 Anti-derivatives and Indefinite Integration

9. Is there a general rule for ? Discuss with your group.

10. 4.1 Anti-derivatives and Indefinite Integration

12. 4.1 Antiderivatives and Indefinite Integration

16. 4.1 Anti-derivatives and Indefinite Integration

19. A differential equation is an equation involving a derivative. The solution to a differential equation is the function that satisfies the equation. You can find a particular solution if you are given an initial condition.

20. 4.1 Anti-derivatives and Indefinite Integration

22. You Try. Solve the differential equation with the given initial condition:

23. You Try

24. 4.1 Antiderivatives and Indefinite Integration

25. Practice Exercises Original Rewrite Integrate Simplify

27. 4.1 BC Homework Day 1: pg. 255, 5-14 all, 15-47 odd, 55-61 odd 77,79 Day 2: MMM pg. 124-125

28. 4.1 AB Homework Tues: pg. 255, 5-14 all, 15-47 odd, 55-59 odd, 77,79 Wed: MMM pg. 124-125 Thurs: Anti-Differentiation WS (4.1) Fri: 4.1 WS

29. 4.1 Anti-derivatives and Indefinite Integration

33. HWQ 11/5 Find the indefinite integral and check the result by differentiation:

34. HWQ 11/6 The acceleration of a particle moving along a straight line is given by a=6t. If at t=0, the particle’s velocity is 1 and its position is 3, find an equation for the position of the particle at any time.