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Miss Battaglia AB/BC Calculus

1.1-1.2 Introduction to Calculus and Limits Objectives: Understand what calculus is, tangent line problem and area problem; Find limits graphically and numerically . Miss Battaglia AB/BC Calculus. What is calculus?. Very advanced algebra and geometry

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Miss Battaglia AB/BC Calculus

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  1. 1.1-1.2 Introduction to Calculus and LimitsObjectives: Understand what calculus is, tangent line problem and area problem; Find limits graphically and numerically Miss Battaglia AB/BC Calculus

  2. What is calculus? • Very advanced algebra and geometry • Look at the two pictures, the problem in both cases is to determine the amount of energy required to push the crate to the top. calculus problem regular math problem things are constantly changing unchanging force/unchanging speed Limit Process Precalculus Mathematics Calculus

  3. The Tangent Line Problem • Find the slope of the tangent line at P msec=

  4. The Area Problem • Approximate the area of the region • As you increase the number of rectangles, the approximation becomes better and better.

  5. An Introduction to Limits Suppose you are asked to find x approaches 0 from the left x approaches 0 from the right f(x) approaches e f(x) approaches e

  6. Estimating a limit numerically • Evaluate the function at several points near x=0 and use the results to estimate the limit Use the table on the calculator

  7. Finding a Limit • Find the limit of f(x) as x approaches 2, where f is defined as

  8. Behavior that Differs from the Right and from the Left • Show that the limit does not exist f(x)=1 f(x)=-1

  9. Unbounded behavior • Discuss the existence of the limit

  10. Oscillating Behavior • Discuss the existence of the limit.

  11. Common Types of Behavior Associated with Nonexistence of a Limit • f(x) approaches a different number from the right side of c than it approaches from the left side. • f(x) increases or decreases without bound as x approaches c. • f(x) oscillates between two fixed values as x approaches c.

  12. Definition of a Limit ε represents a small positive number “f(x) becomes arbitrarily close to L” means f(x) lies in the interval (L- ε,L+ ε) | f(x) - L | < ε Let f be a function defined on an open interval containing c (except possibly at c) and let L be a real number. The statement means that for each ε>0 there exists a δ>0 such that if 0<|x-c|<δ, then |f(x)-L|<ε

  13. Finding a δ for a Givenε Given the limit find δ such that |(2x-5)-1|<0.01 whenever 0<|x-3|<δ if , then

  14. Using the ε-δDefinition of a Limit Use the ε-δdefinition of a limit to prove that

  15. Using the ε-δDefinition of a Limit Use the ε-δ definition of limit to prove that

  16. Classwork/Homework • Read 1.1 and 1.2 • Page 54 #1, 3, 11, 13 (use table on calculator), 15-33 all, 35, 46

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