1. AP Calculus
First day!!

2. Calc - DO NOW 8/4/14 On the index card you picked up:
On the lined side of the card, write:
Your full name
Your year in school (9th, 10th, 11th, 12th)
Your address (if you have one)
One the unlined side of the card, write:
What did you do over the summer?
What do you hope to get out of this year? What are your goals?

3. Policies and procedures
Our First Year of Calculus
What is AP Calculus? Calc Standards
Tutoring: Thursday 3:30-4:30
Textbook / Graphing Calculators
Homework!
Website: mathmcdermott.weebly.com

4. Parent-Student Contract
Signed by you and a parent at home.
English or Spanish depending on what your parents speak.
Bring back tomorrow!
Complete Contact Information

5. Calculator Contract Signed by you and a parent at home.
English or Spanish depending on what your parents speak.
Bring back tomorrow!
Complete Contact Information

6. The Mathematical “Rule of 3”
We will analyze problems:
Numerically
Graphically
Analytically (only way to prove something)

7. The AP Calculus Exam
Part A (No Calculator) – 1 hr 55 min 28 Multiple Choice, 4 short answer Part B (With Calculator) – 1 hr 20 min 17 Multiple Choice, 2 short answer May 5, 2015 – 277 Days!

8. A Preview of Calculus

9. What Do You Think?
What things could be considered the greatest achievements of the human mind?

10. It's the Greatest!
Consider that all these things emerged because of technological advances.
Those advances relied on CALCULUS !
Calculus has made it possible to:
Build giant bridges
Travel to the moon
Predict patterns of population change

13. True or False?
False
False
False

14. The Genius of Calculus is Simple
It relies on only two ideas
The Derivative
The Integral
Both come from a common sense analysis of motion
Motion is change in position over time
All you have to do is drop your pencil to see it happen

15. What Is Calculus
It is the mathematics of change
It is the mathematics of
tangent lines
slopes
areas
volumes
It enables us to model real life situations
It is dynamic
In contrast to algebra/precalc which is static

16. What Is Calculus One answer is to say it is a "limit machine"
Involves three stages
Precalculus/algebra mathematics process
Building blocks to produce calculus techniques
Limit process
The stepping stone to calculus
Calculus
Derivatives, integrals

17. Contrasting Algebra & Calculus
Use f(x) to find the height of the curve at x = c
Find the limit of f(x) as x approaches c

18. Contrasting Algebra & Calculus
Find the average rate of change between t = a and t = b
Find the instantaneous rate of change at t = c

19. Contrasting Algebra & Calculus
Area of a rectangle
Area between two curves

20. Tangent Line Problem Approximate slope of tangent to a line
Start with slope of secant line

21. Tangent Line Problem
Now allow the Δx to get smaller

22. The Area Problem We seek the area under a curve, the graph f(x)
We approximate that area with a number of rectangles
Sum = 31.9
Actual = 33.33

23. The Area Problem
The approximation is improved by increasing the number of rectangles
Number of rectangles = 10
Sum = 32.92
Actual = 33.33

24. The Area Problem View Applet Example
The approximation is improved by increasing the number of rectangles
Number of rectangles = 25
Sum = 33.19
Actual = 33.33
View Applet Example

25. Calc DO NOW – 8/4/14 Hand in the following documents:
Signed Home-School Connection
Calculator Contract Form (w/calc #)
Open your textbook to Section 2.1
Take out guided notes and your notebook.

26. Introduction to Limits
2.1.1

27. What is a limit?
Sketch the graph of the function

28. x
0.75
0.9
0.99
0.999 1
1.001
1.01
1.1
1.25
f (x)

29. The informal Definition of a Limit
If f (x) becomes arbitrarily close to a single number L as x approaches c from either side, the limit of f (x), as x approaches c, is L. This limit is written as

30. Find the limit using a table of values.

32. Find the limit graphically.

34. Find the limit using either method.

36. Find the limit

37. Differs from Right and Left

38. Unbounded Behavior

39. Oscillating Behavior

40. One Sided Limits
The limit from the right means that x approaches c from values greater than c. This limit is denoted as
The limit from the left means that x approaches c from values less than c. This limit is denoted as

41. Find the limit of as x approaches – 2 from the right.
One-sided Limits
Find the limit of as x approaches – 2 from the right.

42. The Greatest Integer Function
Find the limit of the greatest integer function as x approaches 0 from the left and from the right.

43. Homework
p. 66/ 29, 31, 37-44, 45-48

44. Exit Ticket Quick Write:
In your own words, describe what a limit is. How can you find the limit of a function at a given x-value?

45. Evaluating Limits Analytically
2.1.2

46. Some Basic Limits
Let b and c be real numbers and let n be a positive integer. 1. 2. 3.

47. Example

48. More Limit Properties
Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits.
Scalar Multiple
Sum or Difference
Product
Quotient
Power

49. Find the limit of the function

50. Functions that Agree at All but One Point
Let c be a real number and let f (x) = g(x) for all x ≠ c in an open interval containing c. If the limit of g(x) as x approaches c exists, then the limit of f (x) also exists and

51. Find the limit of the function
How can we confirm this result?

52. Find the limit of the function
How can we confirm this result?

53. Find the limit of the function
How can we confirm this result?

54. Homework
p. 66/ 5, 6, 7-23odd, 51-57odd

55. Exit Ticket
Evaluate the limit:

56. Special Trig Limits
2.1.3

57. then exists and is equal to L.
The Squeeze Theorem
If h(x) ≤ f (x) ≤ g(x) for all x in an open interval containing c, except possibly at c, itself, and if
then exists and is equal to L.

58. Special Trigonometric Limits

59. Two Special Trigonometric Limits

60. Evaluating Trigonometric Limits

61. Evaluating Trigonometric Limits
How can we confirm this result?

62. Homework
p. 66/ 24-28, 71-74

63. Three Strikes
Evaluate:

64. Three Strikes
Evaluate:

65. Three Strikes
Evaluate:

66. Three Strikes
Evaluate:

67. Three Strikes
Evaluate:

68. Three Strikes
Evaluate:

69. Three Strikes
Evaluate:

70. Three Strikes
Evaluate:

71. Three Strikes
Evaluate:

72. Three Strikes
Evaluate:

73. Three Strikes
Evaluate:

74. Three Strikes
If , then find:

75. Infinite Limits
2.2

76. Graph the function

77. Infinite Limits
A limit in which f (x) increases or decreases without bound as x approaches c is called an infinite limit.

78. Determine the limit
Find the limit of f (x) as x approaches 1 from the left and from the right.

79. Vertical Asymptote
If f (x) approaches infinity (or negative infinity) as x approaches c from the right or the left, then the line x = c is a vertical asymptote of the graph f.

80. has a vertical asymptote at x = c.
Vertical Asymptotes
Let f and g be continuous on an open interval containing c. If and there exists an open interval containing c such that for all in the interval, then the graph of the function given by
has a vertical asymptote at x = c.

81. Vertical Asymptotes
Find the equations of the vertical asymptotes of the following functions.

82. Properties of infinite limits
Let b and c be real numbers, let n be a positive integer, and let f and functions with the following limits:
Sum or Difference
Product
Quotient

83. Graph the following on your TI

84. What if we don’t have our calculators?

85. Limits at Infinity
If r is a positive rational number and c is any real number, then Furthermore, if xr is defined when x < 0, then

86. Find the limit at infinity

87. Find the limits at infinity:

88. Guidelines for Finding Limits at Infinity of Rational Functions
If the degree of the numerator is less than the degree of the denominator, then the limit of the rational function is 0.
If the degree of the numerator is equal to the degree of the denominator, then the limit of the rational function is the ratio of the leading coefficients.
If the degree of the numerator is greater than the degree of the denominator, then the limit of the rational function does not exist.

89. Order of magnitude of a function
(aka – how quickly a function rises)
Slowest rise: logarithmic function
Medium rise: power function
Fastest rise: exponential function
Show that the order of function is strict.
Another example: which one is bigger n^1.001 or n*log n?

90. Order-of-Magnitude Analysis and Big O Notation
A comparison of growth-rate functions: b) in graphical form

91. Infinite Limits
Evaluate the limit:

92. pg. 76 ~ 9-33(O), 35-38, 39-55(O), 56 Homework
Typo on HW Calendar – sorry!

93. 1d47ffed On Socrative, enter Room:
Reflection
On Socrative, enter Room:
1d47ffed
And answer the question in complete sentences.

94. Continuity
2.3

95. Continuity at a Point
A function is continuous at c if the following three conditions are met:
f (c) is defined.
exists.

96. The existence of a limit
Let f be a function and let c and L be real numbers. The limit of f (x) as x approaches c is L if and only if

97. Continuity on an open interval
Continuity on an Open Interval: A function is continuous on an open interval (a, b) if it is continuous at each point in the interval.
A function that is continuous on the entire real line is everywhere continuous.

98. Two Types of Discontinuity
Removable - (Holes) We can easily redefine the function at c to make the function continuous.
Non-Removable - Infinite or finite gaps occur.

99. Discuss the Continuity

100. Discuss the Continuity

101. Discuss the Continuity

102. Discuss the Continuity

103. Continuity on a closed interval
A function f is continuous on the closed interval [a, b] if it is continuous on the open interval (a, b) and
The function f is continuous from the right at a and continuous from the left at b.
and

104. Continuity on a Closed interval
Discuss the continuity of

105. Intermediate Value Theorem
If f is continuous on the closed interval [a, b] and k is any number between f (a) and f (b), then there is at least one number c in [a, b] such that

106. Intermediate Value Theorem
Use the Intermediate Value Theorem to show that the polynomial function has a zero in the interval [0, 1].

107. Intermediate Value Theorem
Use the Intermediate Value Theorem to show that for all spheres with radii in the interval [1, 5], there is one with a volume of 275 cubic centimeters.

108. Homework
Pg. 84 ~ 1-9(O), 11-16, 19, 21, 23, 41-50