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Calculus

Calculus. Weijiu Liu Department of Mathematics University of Central Arkansas. Overview. What is calculus? . Calculus is a subject about the study of limiting processes and it consist of three basic concepts: limit, differentiation , and integration . Who founded calculus?.

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Calculus

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  1. Calculus Weijiu Liu Department of Mathematics University of Central Arkansas

  2. Overview

  3. What is calculus? Calculus is a subject about the study of limiting processes and it consist of three basic concepts: limit, differentiation, and integration.

  4. Who founded calculus? Isaac Newton (1643 – 1727) was the greatest English mathematician of his generation. He laid the foundation for differential and integral calculus. His work on optics and gravitation make him one of the greatest scientists the world has known. http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Newton.html

  5. Gottfried Wilhelm von Leibniz(1646 – 1716) was a German mathematician who developed the present day notation for the differential and integral calculus though he never thought of the derivative as a limit. His philosophy is also important and he invented an early calculating machine. http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Leibniz.html For a history of the calculus, see: http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html

  6. Why is calculus needed? • Geometrical problems. Slope of a sceant line: y=f(x) Tangent line Q(x,f(x)) Q(x,f(x)) Q(x,f(x)) Slope of a tangent line:

  7. The area A of a range under the graph of a function y y y = f(x) y = f(x) x x a b a b

  8. Physical problems. s Average velocity Instantaneous velocity:

  9. m a x b Work: m a x b Work:

  10. More advanced problems. Description of particle motion:

  11. Description of evolution of chemical concentration – partial differential equation: Convection Diffusion Equation: Convection Diffusoin

  12. Description of the string vibration – the wave equation

  13. How to study calculus? • Read your textbook • Understand concepts clearly • Do your homework on time • Ask your instructor and classmates around you whenever you have a question • Learn from your classmate • Form a group to discuss

  14. Tentative schedule • Chapter 1 – Limits, 3 weeks • Chapter 2 – Differentiation, 3 weeks • Chapter 3 – Application of differentiation, 3 weeks • Chapter 4 – Integration, 3 weeks • Chapter 5 – Application of definite integrals, 2 weeks

  15. Chapter 1-- Limits

  16. Problem of Limit Tangent line problem Slope of a sceant line: y=f(x) Tangent line Q(x,f(x)) Q(x,f(x)) Q(x,f(x)) Slope of a tangent line: Problem of Limit: As x gets closer and close to , to what number is a function g(x) like getting closer and closer to even though g(x) is not well defined at ?

  17. Find limits of a function graphically Consider the function As x>0 gets closer and closer to 0 f(x) is getting closer and closer to 1. As x<0 gets closer and closer to 0 f(x) is getting closer and closer to 1. We say the limit of f(x) as x approaches 0 from the right is 1, written We say the limit of f(x) as x approaches 0 from the left is 1, written One-sided limits We say the limit of f(x) as x approaches 0 is 1, written

  18. Find limits of a function numerically x sin x / x -0.10000000000000 0.99833416646828 -0.01000000000000 0.99998333341667 -0.00100000000000 0.99999983333334 -0.00010000000000 0.99999999833333 -0.00001000000000 0.99999999998333 -0.00000100000000 0.99999999999983 -0.00000010000000 1.00000000000000 -0.00000001000000 1.00000000000000 -0.00000000100000 1.00000000000000 -0.00000000010000 1.00000000000000 1 0

  19. x sin x / x 0.10000000000000 0.99833416646828 0.01000000000000 0.99998333341667 0.00100000000000 0.99999983333334 0.00010000000000 0.99999999833333 0.00001000000000 0.99999999998333 0.00000100000000 0.99999999999983 0.00000010000000 1.00000000000000 0.00000001000000 1.00000000000000 0.00000000100000 1.00000000000000 0.00000000010000 1.00000000000000 0 1

  20. In general, if f(x) is getting closer and closer to L as x gets closer and closer to a, we say that the limit of f(x) as x approaches a is L, written Theorem.

  21. Example. So does not exist!

  22. Example. So does not exist!

  23. Graph of problem 4 of Exercises 1.2

  24. Intermediate Value Theorem a b f(a) f(b) Midp f(midp) 2.0000 3.0000 -2.0000 13.0000 2.5000 3.6250 2.0000 2.5000 -2.0000 3.6250 2.2500 0.3906 2.0000 2.2500 -2.0000 0.3906 2.1200 -0.9519 2.1200 2.2500 -0.9519 0.3906 2.1800 -0.3598 2.1800 2.2500 -0.3598 0.3906 2.2100 -0.0461 2.2100 2.2500 -0.0461 0.3906 2.2300 0.1696 2.2100 2.2300 -0.0461 0.1696 2.2200 0.0610 2.2100 2.2200 -0.0461 0.0610 2.2100 -0.0461 2.2100 2.2200 -0.0461 0.0610 2.2100 -0.0461 2.2100 2.2200 -0.0461 0.0610 2.2100 -0.0461

  25. a b f(a) f(b) Midp f(midp) -1.0000 0 1.0000 -2.0000 -0.5000 -0.1250 -1.0000 -0.5000 1.0000 -0.1250 -0.7500 0.5781 -0.7500 -0.5000 0.5781 -0.1250 -0.6300 0.2700 -0.6300 -0.5000 0.2700 -0.1250 -0.5700 0.0948 -0.5700 -0.5000 0.0948 -0.1250 -0.5400 0.0025 -0.5400 -0.5000 0.0025 -0.1250 -0.5200 -0.0606 -0.5400 -0.5200 0.0025 -0.0606 -0.5300 -0.0289 -0.5400 -0.5300 0.0025 -0.0289 -0.5400 0.0025 -0.5400 -0.5300 0.0025 -0.0289 -0.5400 0.0025 -0.5400 -0.5300 0.0025 -0.0289 -0.5400 0.0025

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