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Welcome to Calculus!

Welcome to Calculus!. Get out your ASGN. What other questions can I answer. Before our quiz?. 2.7 Notes . Write the 2 definitions of the slope of the tangent to graph of f at (a, f (a)). f(a+h). f(a). a. a+h. The formal name of this is. The derivative!. Alternate definition:.

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Welcome to Calculus!

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  1. Welcome to Calculus! Get out your ASGN

  2. What other questions can I answer Before our quiz?

  3. 2.7 Notes Write the 2 definitions of the slope of the tangent to graph of f at (a,f(a))

  4. f(a+h) f(a) a a+h

  5. The formal name of this is . . .

  6. The derivative! Alternate definition: Watch this video! Calculus-help

  7. The derivative! Alternate definition: Watch this video! Calculus-help

  8. Example • Let f(x) = 3x2 – 4x +7. Find the derivative of f(x) at (1, 6) • The notation for this is f’(1) • f’(1)=2

  9. Example Find f’(a) if f(x)=x2-5x+2, leave your answer in terms of a.

  10. Velocity and Speed • “Velocity is the derivative of the position function.” • Also the speed of the particle is the absolute value of the velocity.

  11. Example • A given position function: • s = f(t) = 1/(1 + t) • t is seconds and s is meters. • Find the velocity and speed after 2 seconds. • Hint: The derivative of f when t = 2

  12. Solution (cont’d)

  13. Solution (cont’d) • Thus, the • velocity of the particle after 2 seconds is f ´(2) = – (1/9) m/s , • and the • speed of the particle is |f ´(2)| = 1/9 m/s .

  14. Estimate the derivative at the indicated points C D A B E F

  15. Graph example Sketch the graph of a function f for which f(0)=2 f ’(0)=0 f ’(-1)=-1 f ’(1)=3 f ’(2)=1

  16. Let’s use a calculator! • Let f(x) = 2x . Estimate the value of the slope of the tangent at x=0. • The definition of slope of a tangent gives • We can use a calculator to approximate the values of (2h – 1)/h . Use Table, then Y1(small number).

  17. Solution to (a) (cont’d)

  18. Solution (b) from Drawing a tangent on the calculator

  19. Review • Calculator based slope of a tangent • Definition of derivative of a function • Interpretation of derivative as • the slope of a tangent • a rate of change

  20. ASGN 15 p. 153 3-7odd 11,13,25

  21. Derivitive: Slope of tangent Calculus-help

  22. Quiz review: Look over the following topics, your notes, and assignments and find a question you would like to clarify. Topics so far in Chapter 2 2.1 slope of a secant average velocity average rate of change from table 2.2 limits from a graph limits numerically limits piece wise limit absolute value 2.3 limits algebraically 2.4 continuity discontinuity: removable, infinite, jump Intermediate value theorem 2.5 limit as x goes to infinity limit as x goes to infinity asymptotes: vertical and horizontal 2.6 Slope of a tangent line equation of a tangent line Instantaneous rate of change

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