Course Objectives

1. Define the term "Function".
2. Introduce Function Notation.
3. Classify functions (i.e. constant, liner, quadratic, cubic, polynomial, rational, algebraic, and transcendental).
4. Review concepts of function operations, composition in particular.
5. Define even and odd functions.
6. Review the definition of the 6 trigonometric functions, the Pythagorean identities, and the sum or difference of two angles.
7. Review the graphs of the 6 trigonometric functions.
8. Provide an intuitive feeling for the concept of a limit.
9. Show some limits that do not exist.
10. Define a limit.
11. Provide a strategy for finding limits using direct substitution and limits using direct substitution and limit properties.
12. Discuss the limits of the trigonometric functions and the two special limits:
    
    
    
    
13. Define the term "Continuity", at a point and on an open interval.
14. Introduce one-sided limits.
15. Discuss the properties of continuity.
16. Explain the Intermediate Value Theorem.
17. Provide a common sense approach to infinite limits.
18. Use infinite limits to find vertical asymptotes.
19. Introduce the derivative using the tangent line problem.
20. Set up the 4-step process to the definition of derivative.
21. Realize that differentiability implies continuity.
22. Discuss the derivative in terms of velocity and acceleration.
23. Establish the differentiation rules for constants, Powers, constant multiples, sums and differences, and the sine and cosine functions.
24. Prove and demonstrate the differentiation rules of products, quotients, secants, tangents, cosecants, and cotangents.
25. Establish the Chain and Power rules for the composition of functions.
26. Apply the Chain Rule to trigonometric functions.
27. Summarize the ruled for differentiation and use in combination with one another.
28. Distinguish between explicit and implicit forms of an equation.
29. Apply the technique of implicit differentiation.
30. Find the 2nd derivative implicitly.
31. Use implicit differentiation to do related rate problem.
32. Define extrema and relative extrema of function.
33. Establish guidelines for finding extrema on a closed interval.
34. Explain Rolle's Theorem and the Mean Value Theorem.
35. Use the concepts of increasing and decreasing functions to establish the 1st Derivative Test for Extrema.
36. Define concavity and points of inflection.
37. Use the 2nd derivative test.
38. find limits at infinity and horizontal asymptotes.
39. Provide a summary of curve sketching for rational (Slant asymptotes), radical, polynomial, and trigonometric functions.
40. Apply extrema of functions to the solution of problems from physics, engineering, biology, and business topics of marginals, demand, and elasticity.
41. Review the idea of inverse function and its existence.
42. Introduce exponential functions with their graphs.
43. Find the derivatives for a^u and e^u and show some applications dealing with compound interest and biological growth and decay.
44. Introduce the logarithmic functions as inverse of the exponentials.
45. Find the derivatives of the logarithmic functions and use the logarithmic properties as an aid to differentiation.
46. Use logarithmic differentiation.
47. Review inverse trigonometric functions.
48. Develop the derivatives of the inverse trigonometric functions.
49. Review all of the basic integration rules as antiderivatives.
50. Establish the motion of "antiderivative" and its notation.
51. Develop the basic integration rules as antiderivatives.
52. Develop substitution as a technique of integration (change of variables).
53. Define sigma notation and the limit of a sequence.
54. Use the properties and formulas of summation.
55. Find the area of a region as an approximation, using upper and lower sums.
56. Find the area of a plane region by the limit definition.
57. Define the definite integral using Riemann sums.
58. Corrolate the definite integral and the area of a region in a plane.
59. Use the Fundamental Theorem of Calculus to evaluate definite integrals.

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Syllabus

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Outline

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Examples of Objectives

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