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Functions, Limits, and Continuity

Functions, Limits, and Continuity. Section 1.3. Functions, Limits, and Continuity. Limit Fundamental concept of calculus Separates calculus from algebra Two types Toward infinity Toward a point. Functions, Limits, and Continuity. Limit toward infinity

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Functions, Limits, and Continuity

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  1. Functions, Limits, and Continuity Section 1.3

  2. Functions, Limits, and Continuity • Limit • Fundamental concept of calculus • Separates calculus from algebra • Two types • Toward infinity • Toward a point

  3. Functions, Limits, and Continuity • Limit toward infinity • As the value of the input variable grows large (or small) without bound, what happens to the value of the output variable • “End behavior of the function”

  4. Functions, Limits, and Continuity Y= Y1 = 6X+2 GRAPH

  5. Functions, Limits, and Continuity “Limit as z goes to infinity of g(z) is infinity” “Limit as z goes to negative infinity of g(z) is negative infinity”

  6. Functions, Limits, and Continuity

  7. Functions, Limits, and Continuity Y= Y1 = 9/(1+e^(-x)) GRAPH

  8. Functions, Limits, and Continuity

  9. Functions, Limits, and Continuity • Numerically evaluating limits toward infinity • Calculate value of output variable for progressively larger (smaller) values of the input variable TBLSET Indpnt: Ask Depend: Auto TABLE

  10. Functions, Limits, and Continuity 8.9996 8.9777 8.5732 6.5795 “Limit as x goes to infinity of r(x) is 9” 1 3 5 10

  11. Functions, Limits, and Continuity 8.9996 8.9777 8.5732 6.5795 1 3 5 10

  12. Functions, Limits, and Continuity

  13. Functions, Limits, and Continuity

  14. Functions, Limits, and Continuity

  15. Functions, Limits, and Continuity

  16. Functions, Limits, and Continuity

  17. Functions, Limits, and Continuity • Toward a point • As the input variable gets arbitrarily close to a certain value, what happens to the value of the output variable? • Limit must be the same from both sides

  18. Functions, Limits, and Continuity “Limit as z goes to 4 from the right of g(z) is 26” 26 4 “Limit as z goes to 4 from the left of g(z) is 26”

  19. Functions, Limits, and Continuity 26

  20. Functions, Limits, and Continuity

  21. Functions, Limits, and Continuity • Continuity • A function is continuous if it is defined at every input with no breaks or sudden jumps in the output value • “Function can be drawn without lifting your pen”

  22. Functions, Limits, and Continuity Yes

  23. Functions, Limits, and Continuity No

  24. Functions, Limits, and Continuity Yes

  25. Functions, Limits, and Continuity 1993 Does the limit exist at 1993? Is the function continuous? Is the function continuous from 1993 on?

  26. Functions, Limits, and Continuity • Application: Continuously compounded interest P = principal r = interest rate of a particular term t = number of terms principal is invested A = accumulated value

  27. Functions, Limits, and Continuity • Application: Continuously compounded interest P = principal r = annual interest rate t = number of years principal is invested A = accumulated value

  28. Functions, Limits, and Continuity • Term compounding • Compounding matches the term of the rate • e.g., annual interest rate with annual compounding

  29. Functions, Limits, and Continuity • General compounding • Compounding happens n times per term

  30. Functions, Limits, and Continuity • Example: Semi-annual compounding • Happens 2 times per year

  31. Functions, Limits, and Continuity • Continuous compounding • Infinite number of periods per term

  32. Functions, Limits, and Continuity t = 15 years P = $1,000 r = 8% per year Annual Monthly Continuous

  33. Functions, Limits, and Continuity • In-Class

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