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Part 1: Functions

1. Determine which algebraic formulas or graphs are functions.
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2. Graph a function by plotting points.
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3. Graph a function in an appropriate viewing window using a graphing calculator or computer algebra system, so that its important features are visible.
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4. Summarize the important features of a function by looking at its graph: where the function is increasing and decreasing, where it has maximum and minimum values, vertical and horizontal asymptotes, and its long-term behavior.
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5. Determine whether a problem is complex enough to require the use of technology.
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6. Know when to believe the output of a calculator or computer algebra system and when to question it.
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7. Distinguish between exact and approximate solutions to problems.
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8. Find the roots of a function algebraically.
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9. Find the equation of a line given two points on the line.
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10. Find the distance between two points in the plane using the distance formula.
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11. Factor a quadratic expression into linear factors.
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12. Find the roots of a quadratic function using the quadratic formula or by completing the square.
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13. Simplify an expression involving exponential functions using properties of exponents.
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14. Simplify an expression involving logarithmic functions using properties of logarithms.
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15. Solve an equation like 3=2^x for x.
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16. Work with a function defined by a table.
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17. Work with a piecewise-defined function.
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18. Identify the domain and range of a function from either its algebraic formula, its graph, or a verbal description.
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19. Identify whether a function is polynomial, exponential, trigonometric, logarithmic, or rational by looking at its graph.
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20. Understand the important features of polynomial, exponential, trigonometric, logarithmic, and rational functions.
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21. Understand the applications of polynomial, exponential, and trigonometric functions.
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22. Compare the graphs of exponential and logarithmic functions with different bases.
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23. Compare the graphs of trigonometric functions with different periods and amplitudes.
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24. Interpret the input and output values of trigonometric functions in the context of a right triangle or a unit circle.
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25. Find the sine and cosine of 0, π/2, π, and 3π/2 without the use of technology.
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26. Quickly sketch ln(x), sin(x), cos(x), tan(x), 1/x and e^x without the use of technology.
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27. Understand, recognize, and apply the identity (sin(x))^2+(cos(x))^2=1.
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28. Given the graph of f(x), sketch the graphs of f(x)+2, f(x+2), f(2x), 2f(x), and -f(x), and understand the effects of these transformations.
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29. Identify even and odd functions.
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30. Identify periodic functions. Use the knowledge that a function is periodic to aid in calculations.
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31. Translate back and forth between functions defined in real-world terms and functions defined algebraically.
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Part 2: Derivatives Intuitively

32. Estimate the slope of a function at a particular point.
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33. Sketch the derivative of a function given its graph.
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34. Sketch a possible antiderivative of a function given its graph.
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35. Find the equation of the tangent line to a function at a particular point, given its algebraic formula, or estimate it from a graph.
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36. Translate between statements about the velocity, speed, or acceleration of an object and statements about its first or second derivative.
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37. Translate between statements about rates of change in a real-world example and statements about the first or second derivative of a function.
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38. Zoom in on the graph of a function to estimate its derivative.
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39. Find the intervals where a function is positive, negative, increasing, decreasing, concave up, and concave down.
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40. Find the local and global maximum and minimum points of a function.
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41. Find the inflection points of a function.
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42. Translate between properties of the graphs of f, f', and f''. 1 2 3
    

43. Determine which properties of the graphs of f' and f'' can be obtained from the graph of f and which cannot, and vice-versa.
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44. Understand and apply the first and second derivative tests.
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45. Find an example of a function given information about the intervals over which it's positive, negative, increasing, decreasing, concave up, or concave down, or about its maximum and minimum points, or about its first or second derivative.
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Part 3: Derivatives Formally

46. Find the limit of a function at a particular point, given its algebraic formula, or estimate it from a graph.
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47. Find the limit of a function at a particular point from the right or from the left.
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48. Determine when the limit of a function at a point does not exist.
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49. Understand the difference between f(a) and the limit of f(x) as f approaches a.
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50. Find the intervals over which a function is continuous.
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51. Identify which of the basic types of algebraically defined functions are always continuous.
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52. Find the limit of a function as x goes to infinity, or at an x-value where y goes to infinity. Translate back and forth between limits involving infinity and horizontal and vertical asymptotes.
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53. Find the limit of f(x)/g(x) as x approaches infinity if both f (x) and g(x) go to infinity as x goes to infinity.
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54. At a particular x-value, use information about the limits of f(x) and g(x) to find the limits of f(x)+g(x), f(x)*g(x), f(x)/g(x), and f(g(x)).
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55. Understand the formal definition of a limit, and explain it in your own words.
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56. Understand the formal definition of the derivative, and explain it in your own words. Explain the formal definition of the derivative in terms of the limit of slopes of secant lines.
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57. Use the formal definition of the derivative to find the derivative of simple algebraically defined functions.
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58. Understand and apply the difference between an average rate of change and an instantaneous rate of change.
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Part 4: Calculating Derivatives Algebraically

59. Take the derivative or antiderivative of a polynomial.
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60. Take the derivative or antiderivative of f(x)=b^x$ for some positive number $b$.
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61. Take the derivative or antiderivative of sin(x) and cos(x).
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62. Take the derivative of log_b(x)$ for some positive number b.
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63. Take the derivative of a function using the product rule.
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64. Take the derivative of a function using the quotient rule.
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65. Take the derivative of a function using the chain rule.
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66. Take the derivative of a function using a combination of the various techniques.
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67. Given incomplete information about a function and its derivative, use the product, quotient, and chain rules to find the remaining information.
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68. Use the derivative of a function to find its local and global maximum and minimum points, inflection points, and intervals over which it's increasing, decreasing, concave up, and concave down.
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69. Use the derivative of a function to find the equation of the tangent line to the function at a given point.
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Part 5: Applications of Derivatives

70. Find the general family of solutions of a simple differential equation like y''=a.
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71. Given a differential equation and a function f, check that f is a solution to the differential equation.
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72. Solve an initial value problem given a general solution to a differential equation and some initial conditions.
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73. Interpret a statement about a real-world process as a differential equation, and vice-versa.
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74. Recognize a given real-world process as an object in free-fall, exponential growth or decay, or Newton's law of cooling, and write the general family of functions that model this process.
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75. Know and apply all the steps of solving an optimization problem, including algebraic and graphical techniques.
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76. Find the Taylor polynomial of a function at a given point with a given degree.
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77. Determine whether a given polynomial is a reasonable candidate for the Taylor polynomial of a given function.
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78. Use the Taylor polynomial of a function to estimate its y-values.
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79. Explain the meaning of the Intermediate Value Theorem, explain why all of its hypotheses are necessary, and apply it to a given function.
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80. Explain the meaning of the Extreme Value Theorem, explain why all of its hypotheses are necessary, and apply it to a given function.
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81. Explain the meaning of the Mean Value Theorem, explain why all of its hypotheses are necessary, and apply it to a given function.
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Part 6: Integrals

82. Calculate the definite integral of a function using areas of geometric shapes.
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83. Calculate the indefinite integral of a function using the antiderivative.
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84. Calculate the definite integral of a function using the antiderivative.
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85. Use the algebraic properties of integrals and given definite integrals to find other integrals.
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86. Determine whether a given definite integral is positive or negative.
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87. Use information about the graph of a function to determine information about the graph of its integral.
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88. Understand and apply the Fundamental Theorem of Calculus, both the indefinite and definite integral versions.
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89. Explain the meaning of both statements of the Fundamental Theorem of Calculus.
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