1. Winter wk 7 – Tues.15.Feb.05 Calc. Ch.6.1: Constructing Antiderivatives Antiderivatives = integrals Finding antiderivatives graphically Finding antiderivatives numerically 6.2: Finding antiderivatives analytically Next week: 6.3: Introduction to differential equations Energy Systems, EJZ

2. Ch.6.1: Antiderivatives graphically Antiderivatives = integrals If f=x 2 then the derivative of f is F=df/dx=____ If F=2x, then the antiderivative of F is f=__ In other words, if F=f’=df/dx, then

3. Visualizing Antiderivatives Given a graph of the slope F=df/dx, sketch f. Practice with conceptests. Notice that the intercept is undetermined. Practice with Ch.6.1 Exercises 1-4, 11-13 (p.265)

4. Conceptest 1

5. Conceptest 1 answer

6. Conceptest 2

7. Conceptest 2 answer

8. Conceptest 3

9. Conceptest 3 answer Now practice with 6.1 Exercises 1-4

10. Fundamental theorem of calculus Antiderivatives = integrals Given the graph of the slope F=df/dx, sketch f, showing all critical points (f’=0) and inflection points (where f”=0 or f” changes sign: concavity of f changes)

11. Find function f from slope f’ Practice with Ex. 9-13 p.265

12. Antiderivatives - numerically How to find f from values of F=df/dx? Recall that  f =  x df/dx. Start at f 0, and increase f step by step: Calculate  f for each  x Add f 0 +  f, etc. for each  x Tabulate f(x)

13. Conceptest 4

14. Conceptest 4 answer Now practice with 6.1 Ex. 5,6 p.265

15. 6.2 Antiderivatives - analytically What is an antiderivative of f’(x) = 0? What function f does not change? What is an antiderivative of g’(x) = constant? How does g compare to f above? Write f and f’ for these plots of f(x)

16. Finding f from df/dx If df/dx=constant (f’=k) then Recall polynomial rule:

17. Finding f from df/dx

18. Plot inverse functions If y=log x, thenIf y=ln x, then x=_________ x=__________ Sketch e x

19. Trigonometric functions Practice 6.2 odd numbered problems p.271-272