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Understanding the Derivative: A Comprehensive Guide to Calculus Concepts

Dive into the essential concepts of calculus focusing on derivatives and the slope of tangents. This guide covers definitions, examples, and important applications like velocity and speed in motion. You'll find illustrative examples including the derivative of a polynomial function and practical skills like using a calculator to estimate slopes. Review critical topics, including limits, continuity, and rates of change, to solidify your understanding of calculus. Prepare for your quiz with confidence as you explore these principles, ensuring a solid foundation in calculus.

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Understanding the Derivative: A Comprehensive Guide to Calculus Concepts

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Presentation Transcript

  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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