Calculus (Math 1A) Lecture 5

Transcription

1 Calculus (Math 1A) Lecture 5 Vivek Shende September 5, 2017

2 Hello and welcome to class!

3 Hello and welcome to class! Last time

4 Hello and welcome to class! Last time We discussed composition, inverses, exponentials, and logarithms.

5 Hello and welcome to class! Last time We discussed composition, inverses, exponentials, and logarithms. Today

6 Hello and welcome to class! Last time We discussed composition, inverses, exponentials, and logarithms. Today We talk about motion, Zeno s paradox, tangent lines, and limits.

7 Zeno of Elea (450 b.c.)

8 Zeno s paradox of the arrow

9 Zeno s paradox of the arrow

10 Zeno s paradox of the arrow At each instant, the arrow occupies a single position in space.

11 Zeno s paradox of the arrow At each instant, the arrow occupies a single position in space. Motion means changing position; as the arrow does not change position in any instant, it is not moving at any instant.

12 Zeno s paradox of the arrow At each instant, the arrow occupies a single position in space. Motion means changing position; as the arrow does not change position in any instant, it is not moving at any instant. If it is not moving at any instant, when is it moving?

13 Lesson from Zeno

14 Lesson from Zeno We should be careful discussing anything at an instant.

15 Lesson from Zeno We should be careful discussing anything at an instant. Answer to Zeno (?):

16 Lesson from Zeno We should be careful discussing anything at an instant. Answer to Zeno (?): While the amount of motion at an instant is zero,

17 Lesson from Zeno We should be careful discussing anything at an instant. Answer to Zeno (?): While the amount of motion at an instant is zero, it makes sense to discuss the rate of motion at an instant, and this is not zero.

18 Lesson from Zeno We should be careful discussing anything at an instant. Answer to Zeno (?): While the amount of motion at an instant is zero, it makes sense to discuss the rate of motion at an instant, and this is not zero. The correct notion of being in motion at an instant is having a nonzero rate of motion.

19 Rate

20 Rate rate of change = amount of change time it took

21 Rate rate of change = amount of change time it took If what is changing is described by a number a varying with time as a(t),

22 Rate rate of change = amount of change time it took If what is changing is described by a number a varying with time as a(t), then the rate of change over some interval [t 0, t 1 ] is

23 Rate rate of change = amount of change time it took If what is changing is described by a number a varying with time as a(t), then the rate of change over some interval [t 0, t 1 ] is a(t 1 ) a(t 0 ) t 1 t 0

24 Rate rate of change = amount of change time it took If what is changing is described by a number a varying with time as a(t), then the rate of change over some interval [t 0, t 1 ] is a(t 1 ) a(t 0 ) t 1 t 0 I.e., the slope of the line through (t 0, a(t 0 )) and (t 1, a(t 1 )).

25 Secant line

26 Secant line A line connecting two points on the graph of a function is called a secant line. It has slope f / x.

27 Tangent line

28 Tangent line The tangent line to through a point on a curve is the (?) line through that point meeting no other nearby points of the curve.

29 Tangent line The tangent line to through a point on a curve is the (?) line through that point meeting no other nearby points of the curve.

30 Tangent line

31 Tangent line

32 Tangent from secants Nearby secant lines get closer and closer to the tangent line.

33 Tangent from secants Nearby secant lines get closer and closer to the tangent line.

34 Tangent from secants Nearby secant lines get closer and closer to the tangent line.

35 Tangent from secants Nearby secant lines get closer and closer to the tangent line.

36 Tangent from secants Nearby secant lines get closer and closer to the tangent line.

37 Tangent from secants

38 Tangent from secants Let s try with formulas.

39 Tangent from secants Let s try with formulas. Consider f (x) = x 3, and let s find the tangent line through (2, 8).

40 Tangent from secants Let s try with formulas. Consider f (x) = x 3, and let s find the tangent line through (2, 8). We ll do this by studying the secant lines passing through (2, 8) and (x, x 3 ), when x is close to 2.

41 Tangent from secants Let s try with formulas. Consider f (x) = x 3, and let s find the tangent line through (2, 8). We ll do this by studying the secant lines passing through (2, 8) and (x, x 3 ), when x is close to 2. Since x is close to 2, we write it as 2 + h.

42 Tangent from secants Let s try with formulas. Consider f (x) = x 3, and let s find the tangent line through (2, 8). We ll do this by studying the secant lines passing through (2, 8) and (x, x 3 ), when x is close to 2. Since x is close to 2, we write it as 2 + h. 2 + h getting close to 2 is the same as h getting close to 0.

43 Tangent from secants The slope of the secant through (2, f (2) = 8) and (2 + h, f (2 + h)):

44 Tangent from secants The slope of the secant through (2, f (2) = 8) and (2 + h, f (2 + h)): f x = f (2 + h) f (2) (2 + h) 2 = (2 + h)3 2 h =

45 Tangent from secants The slope of the secant through (2, f (2) = 8) and (2 + h, f (2 + h)): f x = f (2 + h) f (2) (2 + h) 2 = (2 + h)3 2 h = (h 3 + 6h h + 8) 8 h = h 2 + 6h + 12

46 Tangent from secants As h gets close to zero, h 2 + 6h + 12 gets close to 12.

47 Tangent from secants As h gets close to zero, h 2 + 6h + 12 gets close to 12.

48 Tangents from secants The slope of the tangent to the graph of f (x) = x 3 at (2, 8) is 12.

49 Tangents from secants The slope of the tangent to the graph of f (x) = x 3 at (2, 8) is 12. The tangent line itself is given by (y 8) = 12(x 2)

50 Tangent from secants Here we have f (x) = x 3 and g(x) = 12(x 2) + 8.

51 Tangent from secants Here we have f (x) = x 3 and g(x) = 12(x 2) + 8.

52 Try it yourself!

53 Try it yourself! Using only these ideas (finding tangents from secants), find the tangent line to x 2 + 2x + 3 at (1, 6).

54 Limits

55 Limits In order to compute the slope of the tangent line, we ended up trying to understand how the quantities behave as x approaches 2. f (x) f (2) x 2

56 Limits In order to compute the slope of the tangent line, we ended up trying to understand how the quantities behave as x approaches 2. f (x) f (2) x 2 To do this kind of thing in an organized way, it helps to introduce the notion of limit.

57 Limits

58 Limits For a function f (x), the question lim f (x) =? x x 0 asks about the behavior of f (x) near but not at x 0.

59 Limits For a function f (x), the question lim f (x) =? x x 0 asks about the behavior of f (x) near but not at x 0. In terms of the previous discussion, the near but not at distinction is important because the secant through (x 0, f (x 0 )) and (x, f (x)) is not defined when x = x 0.

60 Limits For a function f (x), the question lim f (x) =? x x 0 asks about the behavior of f (x) near but not at x 0. In terms of the previous discussion, the near but not at distinction is important because the secant through (x 0, f (x 0 )) and (x, f (x)) is not defined when x = x 0. In terms of the formula, x = x 0 is not in the domain of (f (x) f (x 0 ))/(x x 0 ).

61 Limits More precisely, we say lim x x 0 f (x) = y 0

62 Limits More precisely, we say lim x x 0 f (x) = y 0 if, by restricting our attention to x very close to x 0

63 Limits More precisely, we say lim x x 0 f (x) = y 0 if, by restricting our attention to x very close to x 0 we can guarantee that f (x) is very close to y 0.

64 Limits Even more precisely, we say lim x x 0 f (x) = y 0

65 Limits Even more precisely, we say lim x x 0 f (x) = y 0 if, for every ɛ > 0, there is some δ > 0, such that in order to guarantee that f (x) is within ɛ of y 0, it suffices to take x 0 within δ over x.

66 Limits

67 Limits We ll return to these more precise definitions in a couple days.

68 Limits We ll return to these more precise definitions in a couple days. For now, let s look at examples to gain some intuition.

69 Limits lim x 2 =? x 1

70 Limits lim x 2 =? x 1 We are asking: as x gets close to 1, what happens to x 2?

71 Limits As x gets close to 1, what happens to x 2?

72 Limits As x gets close to 1, what happens to x 2?

73 Limits As x gets close to 1, what happens to x 2? Looks like it gets close to 1.

74 Limits More quantitatively:

75 Limits More quantitatively: writing x = 1 + h, we have x 2 = (1 + h) 2 = 1 + 2h + h 2

76 Limits More quantitatively: writing x = 1 + h, we have x 2 = (1 + h) 2 = 1 + 2h + h 2 x 2 1 = (1 + 2h + h 2 ) 1 = 2h + h 2

77 Limits More quantitatively: writing x = 1 + h, we have x 2 = (1 + h) 2 = 1 + 2h + h 2 x 2 1 = (1 + 2h + h 2 ) 1 = 2h + h 2 That is, the difference between x 2 and 1 becomes small as h 0, i.e., as x 1.

78 Limits More quantitatively: writing x = 1 + h, we have x 2 = (1 + h) 2 = 1 + 2h + h 2 x 2 1 = (1 + 2h + h 2 ) 1 = 2h + h 2 That is, the difference between x 2 and 1 becomes small as h 0, i.e., as x 1. The above formula in fact tells us how small, which we will use when we return to the precise ɛ δ definition.

79 Try it yourself lim (x 2 + 3x + 2) =? x 4

80 Limits

81 Limits We will see (next week) that for any polynomial, rational, algebraic, trigonometric, exponential function, f, or for functions that are combinations of these, and for any point x 0 in the domain of f, one has lim f (x) = f (x 0 ) x x 0

82 Limits We will see (next week) that for any polynomial, rational, algebraic, trigonometric, exponential function, f, or for functions that are combinations of these, and for any point x 0 in the domain of f, one has lim f (x) = f (x 0 ) x x 0 But this is not true for all functions!

83 Limits Consider the function with the following graph:

84 Limits In symbols we might write 1 x > 0 h(x) = 1/2 x = 0 0 x < 0

85 Limits In symbols we might write 1 x > 0 h(x) = 1/2 x = 0 0 x < 0 Sometimes this is called the Heaviside function

86 Heaviside

87 Heaviside

88 Limits What s the limit of this function as x 0?

89 Limits There is no limit as x 0.

90 Limits There is no limit as x 0. Indeed, a limit would have to be equal to both 0

91 Limits There is no limit as x 0. Indeed, a limit would have to be equal to both 0 because numbers arbitrarily close to zero but smaller have h(x) = 0,

92 Limits There is no limit as x 0. Indeed, a limit would have to be equal to both 0 because numbers arbitrarily close to zero but smaller have h(x) = 0, and 1

93 Limits There is no limit as x 0. Indeed, a limit would have to be equal to both 0 because numbers arbitrarily close to zero but smaller have h(x) = 0, and 1 because numbers arbitrarily close to zero but larger have h(x) = 1.

94 Limits At which x 0 does lim x x0 f (x) exist? What is it?

95 Limits At which x 0 does lim x x0 f (x) exist? What is it? The limit exists except when x = 1, 1, 2. When defined, the limit is given by the value of the function.

96 Limits At which x 0 does lim x x0 f (x) exist? What is it?

97 Limits At which x 0 does lim x x0 f (x) exist? What is it? The limit always exists, and is given by the value of the function.

98 Limits At which x 0 does lim x x0 f (x) exist? What is it?

99 Limits At which x 0 does lim x x0 f (x) exist? What is it? The limit exists except at x = 2, and is given by the value of the function.

100 Limits At which x 0 does lim x x0 f (x) exist? What is it?

101 Limits At which x 0 does lim x x0 f (x) exist? What is it? The limit always. It is given by the value of the function except at x = 1, where it is 2.

102 Limits at infinity

103 Limits at infinity We say lim f (x) = x x 0 when, (roughly), as x gets close to x 0, the quantity f (x) becomes increasingly large and positive.

104 Limits at infinity We say lim f (x) = x x 0 when, (roughly), as x gets close to x 0, the quantity f (x) becomes increasingly large and positive. Similarly, we say lim x x 0 f (x) = when f (x) becomes very negative near x 0.

105 Limits at infinity Example:

106 Limits at infinity Example: For small x near 0, the quantity 1 x 2 is large and positive:

107 Limits at infinity Example: For small x near 0, the quantity 1 x 2 is large and positive: lim x 0 1 x 2 =

108 Limits at infinity Example:

109 Limits at infinity Example: For small x near 0, the quantity 1 x has large absolute value, but can be either positive or negative.

110 Limits at infinity Example: For small x near 0, the quantity 1 x has large absolute value, but can be either positive or negative. 1 lim x 0 x is undefined

111 One-sided limits

112 One-sided limits We say lim x x + 0 f (x) = y when, roughly, as x gets close to x 0 while remaining larger than it, the quantity f (x) approaches y.

113 One-sided limits We say lim x x + 0 f (x) = y when, roughly, as x gets close to x 0 while remaining larger than it, the quantity f (x) approaches y. We say lim x x 0 f (x) = y when, roughly, as x gets close to x 0 while remaining smaller than it, the quantity f (x) approaches y.

114 One-sided limits If lim x x0 f (x) exists,

115 One-sided limits If lim x x0 f (x) exists, then

116 One-sided limits If lim x x0 f (x) exists, then Conversely, if lim x x 0 f (x) and lim x x + 0 f (x) exist

117 One-sided limits If lim x x0 f (x) exists, then Conversely, if lim x x 0 f (x) and lim x x + 0 f (x) exist and are equal,

118 One-sided limits If lim x x0 f (x) exists, then Conversely, if lim x x f (x) and lim 0 x x + f (x) exist and are equal, 0 then also lim x x0 f (x) exists and is equal to both.

119 One-sided limits If lim x x0 f (x) exists, then Conversely, if lim x x f (x) and lim 0 x x + f (x) exist and are equal, 0 then also lim x x0 f (x) exists and is equal to both. However, it is possible that lim x x 0 but are not equal. f (x) and lim x x + 0 f (x) exist

120 One-sided limits If lim x x0 f (x) exists, then Conversely, if lim x x f (x) and lim 0 x x + f (x) exist and are equal, 0 then also lim x x0 f (x) exists and is equal to both. However, it is possible that lim x x f (x) and lim 0 x x + f (x) exist 0 but are not equal. In this case lim x x0 f (x) does not exist.

121 One-sided limits Which one-sided limits lim x x + f (x) and lim 0 x x 0 What are they? When are they equal? f (x) exist?

122 One-sided limits Which one-sided limits lim x x + f (x) and lim 0 x x 0 What are they? When are they equal? f (x) exist? Except at x = 0, the usual limit lim x x0 f (x) exists and is equal to f (x 0 ); hence the same for both one-sided limits.

123 One-sided limits Which one-sided limits lim x x + f (x) and lim 0 x x 0 What are they? When are they equal? f (x) exist? Except at x = 0, the usual limit lim x x0 f (x) exists and is equal to f (x 0 ); hence the same for both one-sided limits. We also have lim x 0 + = 1 and lim x 0 = 0.

124 One-sided limits At which x 0 does lim x x + 0 f (x) or lim x x 0 f (x) exist? What is it?

125 One-sided limits At which x 0 does lim x x + 0 f (x) or lim x x 0 f (x) exist? What is it? Already the ordinary limit existed except at x = 1, 1, 2. At these values, the one-sided limits exist, but are different.

126 One-sided limits At which x 0 does lim x x + 0 f (x) or lim x x 0 f (x) exist? What is it? Already the ordinary limit existed except at x = 1, 1, 2. At these values, the one-sided limits exist, but are different. E.g.: lim f (x) = 1 x 1 lim f (x) = 2 x 1 +

127 One-sided limits at infinity

128 One-sided limits at infinity 1 lim x 0 + x = lim x 0 1 x =

129 sin(1/x) The function sin(1/x) can be made to assume any value between 1 and 1 for x arbitrarily close to zero (on either side).

130 sin(1/x) The function sin(1/x) can be made to assume any value between 1 and 1 for x arbitrarily close to zero (on either side).

131 sin(1/x) The function sin(1/x) can be made to assume any value between 1 and 1 for x arbitrarily close to zero (on either side). Thus it has no limit as x 0 (or one-sided limits).

132 (1/x) sin(1/x)