Calculus (Math 1A) Lecture 4

Transcription

1 Calculus (Math 1A) Lecture 4 Vivek Shende August 30, 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 shifting, stretching, and composition.

5 Hello and welcome to class! Last time We discussed shifting, stretching, and composition. Today

6 Hello and welcome to class! Last time We discussed shifting, stretching, and composition. Today We finish discussing composition, then discuss inverses and the exponential function.

7 Picking up from last time Let s write mult a (x) = ax add b (x) = x + b

8 Picking up from last time Let s write mult a (x) = ax add b (x) = x + b these functions compose as follows: mult a mult a = mult aa add b add b = add b add b

9 Picking up from last time Let s write mult a (x) = ax add b (x) = x + b these functions compose as follows: mult a mult a = mult aa add b add b = add b add b Last time we called these functions str a and sh b because we were thinking about how composition with them affects the graph.

10 Composing stretches and shifts Exercise: Find some c, d such that add b mult a = mult c add d

11 Composing stretches and shifts Exercise: Find some c, d such that add b mult a = mult c add d Let s just evaluate both sides. (add b mult a )(x) = add b (ax) = ax + b (mult c add d )(x) = mult c (x + d) = cx + cd

12 Composing stretches and shifts Exercise: Find some c, d such that add b mult a = mult c add d Let s just evaluate both sides. (add b mult a )(x) = add b (ax) = ax + b (mult c add d )(x) = mult c (x + d) = cx + cd So we should take c = a and d = b/a.

13 Composing stretches and shifts Exercise: Find some c, d such that add b mult a = mult c add d Let s just evaluate both sides. (add b mult a )(x) = add b (ax) = ax + b (mult c add d )(x) = mult c (x + d) = cx + cd So we should take c = a and d = b/a. add b mult a = mult a add b/a

14 Associativity A priori, f (g h) and (f g) h have different meanings:

15 Associativity A priori, f (g h) and (f g) h have different meanings: the first means first do the function g h, then do the function f to the result

16 Associativity A priori, f (g h) and (f g) h have different meanings: the first means first do the function g h, then do the function f to the result the second means first do the function h, then do the function g h to the result

17 Associativity A priori, f (g h) and (f g) h have different meanings: the first means first do the function g h, then do the function f to the result the second means first do the function h, then do the function g h to the result However, unpacking further, both amount to: first do the function h, then do the function g to the result of that, then do the function f to the result of that.

18 Associativity A priori, f (g h) and (f g) h have different meanings: the first means first do the function g h, then do the function f to the result the second means first do the function h, then do the function g h to the result However, unpacking further, both amount to: first do the function h, then do the function g to the result of that, then do the function f to the result of that. We write f g h to mean either of these.

19 Associativity example

20 Associativity example We saw doing a vertical shift then a vertical stretch is not the same as doing these in the reverse order.

21 Associativity example We saw doing a vertical shift then a vertical stretch is not the same as doing these in the reverse order. What about a vertical shift up by b, then a horizontal stretch by a factor of 1/a, versus the same in the opposite order?

22 Associativity example We saw doing a vertical shift then a vertical stretch is not the same as doing these in the reverse order. What about a vertical shift up by b, then a horizontal stretch by a factor of 1/a, versus the same in the opposite order? Given our starting function f,

23 Associativity example We saw doing a vertical shift then a vertical stretch is not the same as doing these in the reverse order. What about a vertical shift up by b, then a horizontal stretch by a factor of 1/a, versus the same in the opposite order? Given our starting function f, the first of these would be (add b f ) mult a,

24 Associativity example We saw doing a vertical shift then a vertical stretch is not the same as doing these in the reverse order. What about a vertical shift up by b, then a horizontal stretch by a factor of 1/a, versus the same in the opposite order? Given our starting function f, the first of these would be (add b f ) mult a, and the second add b (f mult a ).

25 Associativity example We saw doing a vertical shift then a vertical stretch is not the same as doing these in the reverse order. What about a vertical shift up by b, then a horizontal stretch by a factor of 1/a, versus the same in the opposite order? Given our starting function f, the first of these would be (add b f ) mult a, and the second add b (f mult a ). Associativity tells us that these are the same.

26 The identity function

27 The identity function Consider the function I (x) = x.

28 The identity function Consider the function I (x) = x. Composing with it does not affect other functions: (f I )(x) = f (I (x)) = f (x) (I f )(x) = I (f (x)) = f (x)

29 The identity function Consider the function I (x) = x. Composing with it does not affect other functions: (f I )(x) = f (I (x)) = f (x) (I f )(x) = I (f (x)) = f (x) I (x) is to composition

30 The identity function Consider the function I (x) = x. Composing with it does not affect other functions: (f I )(x) = f (I (x)) = f (x) (I f )(x) = I (f (x)) = f (x) I (x) is to composition as 1 is to multiplication,

31 The identity function Consider the function I (x) = x. Composing with it does not affect other functions: (f I )(x) = f (I (x)) = f (x) (I f )(x) = I (f (x)) = f (x) I (x) is to composition as 1 is to multiplication, as 0 is to addition.

32 The identity function Consider the function I (x) = x. Composing with it does not affect other functions: (f I )(x) = f (I (x)) = f (x) (I f )(x) = I (f (x)) = f (x) I (x) is to composition as 1 is to multiplication, as 0 is to addition. For this reason it is sometimes called the identity function.

33 The identity function Consider the function I (x) = x. Composing with it does not affect other functions: (f I )(x) = f (I (x)) = f (x) (I f )(x) = I (f (x)) = f (x) I (x) is to composition as 1 is to multiplication, as 0 is to addition. For this reason it is sometimes called the identity function. (You do not need to remember this name.)

34 The inverse of a function We say g is the inverse of f if: f (g(x)) = x Sometimes we write g = f 1. g(f (x)) = x

35 The inverse of a function We say g is the inverse of f if: f (g(x)) = x Sometimes we write g = f 1. g(f (x)) = x DO NOT GET CONFUSED: f 1 DOES NOT MEAN 1/f.

36 The inverse of a function We say g is the inverse of f if: f (g(x)) = x Sometimes we write g = f 1. g(f (x)) = x DO NOT GET CONFUSED: f 1 DOES NOT MEAN 1/f. What is being inverted is not the value of the function, but instead the operation that the function is doing.

37 The inverse of a function We say g is the inverse of f if: f (g(x)) = x Sometimes we write g = f 1. g(f (x)) = x DO NOT GET CONFUSED: f 1 DOES NOT MEAN 1/f. What is being inverted is not the value of the function, but instead the operation that the function is doing. Another perspective: f g = I = g f. I.e., the inverse function is what one must compose with to get the identity function.

38 The inverse of a linear function What s the inverse of f (x) = ax + b?

39 The inverse of a linear function What s the inverse of f (x) = ax + b? That is, what function g(x) has the properties that g(f (x)) = g(ax + b) = x f (g(x)) = ag(x) + b = x

40 The inverse of a linear function What s the inverse of f (x) = ax + b? That is, what function g(x) has the properties that g(f (x)) = g(ax + b) = x f (g(x)) = ag(x) + b = x Solving the second equation for g gives g(x) = x b a.

41 The inverse of a linear function What s the inverse of f (x) = ax + b? That is, what function g(x) has the properties that g(f (x)) = g(ax + b) = x f (g(x)) = ag(x) + b = x Solving the second equation for g gives g(x) = x b a. This also satisfies the first equation, so is the inverse function.

42 The inverse of a linear function What s the inverse of f (x) = ax + b? That is, what function g(x) has the properties that g(f (x)) = g(ax + b) = x f (g(x)) = ag(x) + b = x Solving the second equation for g gives g(x) = x b a. This also satisfies the first equation, so is the inverse function. In particular, (add b ) 1 = add b and (mult a ) 1 = mult 1/a.

43 The inverse of f (x) = x 2

44 The inverse of f (x) = x 2 We want a function g such that g(x 2 ) = x and g(x) 2 = x.

45 The inverse of f (x) = x 2 We want a function g such that g(x 2 ) = x and g(x) 2 = x. Of course we want to take g(x) = x.

46 The inverse of f (x) = x 2 We want a function g such that g(x 2 ) = x and g(x) 2 = x. Of course we want to take g(x) = x. Note however that x has domain only [0, ) and moreover x 2 = x, which is equal to x only if x is non-negative.

47 The inverse of f (x) = x 2 We want a function g such that g(x 2 ) = x and g(x) 2 = x. Of course we want to take g(x) = x. Note however that x has domain only [0, ) and moreover x 2 = x, which is equal to x only if x is non-negative. Thus is the inverse, not of the function f (x) = x 2 with domain (, ),

48 The inverse of f (x) = x 2 We want a function g such that g(x 2 ) = x and g(x) 2 = x. Of course we want to take g(x) = x. Note however that x has domain only [0, ) and moreover x 2 = x, which is equal to x only if x is non-negative. Thus is the inverse, not of the function f (x) = x 2 with domain (, ), but instead of the function f (x) = x 2 with domain and range [0, ).

49 Graph of the inverse:

50 Graph of the inverse: For g(x) is the inverse of f (x), consider a point on the graph of g: (x, g(x)) = (f (g(x)), g(x))

51 Graph of the inverse: For g(x) is the inverse of f (x), consider a point on the graph of g: (x, g(x)) = (f (g(x)), g(x)) This looks like a point on the graph of f (x) with its coordinates reversed.

52 Graph of the inverse: For g(x) is the inverse of f (x), consider a point on the graph of g: (x, g(x)) = (f (g(x)), g(x)) This looks like a point on the graph of f (x) with its coordinates reversed. The graph of the inverse is the reflection of the original graph across the line y = x.

53 When is there an inverse?

54 When is there an inverse? The function f (x) has an inverse g(x), if and only if:

55 When is there an inverse? The function f (x) has an inverse g(x), if and only if: for every y in the range of f, there is only one x in the domain of f such that f (x) = y.

56 When is there an inverse? The function f (x) has an inverse g(x), if and only if: for every y in the range of f, there is only one x in the domain of f such that f (x) = y. This is a necessary condition since, if f (x) = f (x ) and g is an inverse to f,

57 When is there an inverse? The function f (x) has an inverse g(x), if and only if: for every y in the range of f, there is only one x in the domain of f such that f (x) = y. This is a necessary condition since, if f (x) = f (x ) and g is an inverse to f, then x = g(f (x)) = g(f (x )) = x

58 When is there an inverse? The function f (x) has an inverse g(x), if and only if: for every y in the range of f, there is only one x in the domain of f such that f (x) = y. This is a necessary condition since, if f (x) = f (x ) and g is an inverse to f, then x = g(f (x)) = g(f (x )) = x Moreover, under this condition, we can simply define g(y) to be the unique x such that f (x) = y.

59 When is there an inverse? The function f (x) has an inverse g(x), if and only if: for every y in the range of f, there is only one x in the domain of f such that f (x) = y. This is a necessary condition since, if f (x) = f (x ) and g is an inverse to f, then x = g(f (x)) = g(f (x )) = x Moreover, under this condition, we can simply define g(y) to be the unique x such that f (x) = y.

60 When is there an inverse? In terms of the graph: if f has an inverse g,

61 When is there an inverse? In terms of the graph: if f has an inverse g, its graph, being the graph of a function, must satisfy the vertical line test

62 When is there an inverse? In terms of the graph: if f has an inverse g, its graph, being the graph of a function, must satisfy the vertical line test or equivalently, the graph of f must satisfy a horizontal line test

63 When is there an inverse? In terms of the graph: if f has an inverse g, its graph, being the graph of a function, must satisfy the vertical line test or equivalently, the graph of f must satisfy a horizontal line test (i.e. every horizontal line meets its graph at most once)

64 When is there an inverse? In terms of the graph: if f has an inverse g, its graph, being the graph of a function, must satisfy the vertical line test or equivalently, the graph of f must satisfy a horizontal line test (i.e. every horizontal line meets its graph at most once) If the graph of f satisfies the horizontal line test,

65 When is there an inverse? In terms of the graph: if f has an inverse g, its graph, being the graph of a function, must satisfy the vertical line test or equivalently, the graph of f must satisfy a horizontal line test (i.e. every horizontal line meets its graph at most once) If the graph of f satisfies the horizontal line test, intersecting with this line can be used to define (and calculate) the inverse.

66 When is there an inverse? These functions, on their original domains do not have inverses:

67 When is there an inverse? These functions, on their original domains do not have inverses: However, an inverse can be define after restricting the domain, e.g. to [0, ) for f (x) = x 2 and to ( π, π] for sin(x).

68 Inverse functions you know

69 Inverse functions you know Often the inverse of a function is not a function whose name you already know.

70 Inverse functions you know Often the inverse of a function is not a function whose name you already know. Many years ago, you learned addition.

71 Inverse functions you know Often the inverse of a function is not a function whose name you already know. Many years ago, you learned addition. Then subtraction, as follows:

72 Inverse functions you know Often the inverse of a function is not a function whose name you already know. Many years ago, you learned addition. Then subtraction, as follows: if you want to subtract 4 from 5, you ask for the number so that, if you add 4 to it, you get 5.

73 Inverse functions you know Often the inverse of a function is not a function whose name you already know. Many years ago, you learned addition. Then subtraction, as follows: if you want to subtract 4 from 5, you ask for the number so that, if you add 4 to it, you get 5. Said differently, if a 4 (x) = x + 4 is the function which adds 4, then subtracting 4 from y is (a 4 ) 1 (y).

74 Inverse functions you know Often the inverse of a function is not a function whose name you already know. Many years ago, you learned addition. Then subtraction, as follows: if you want to subtract 4 from 5, you ask for the number so that, if you add 4 to it, you get 5. Said differently, if a 4 (x) = x + 4 is the function which adds 4, then subtracting 4 from y is (a 4 ) 1 (y). Similarly, you learned division as the inverse to multiplication

75 Inverse functions you know Likewise, the meaning of the expression n x is: the quantity whose n th power is x.

76 Inverse functions you know Likewise, the meaning of the expression n x is: the quantity whose n th power is x. In other words, the function f (x) = n x is by definition the inverse function to g(x) = x n.

77 Inverse functions you know Likewise, the meaning of the expression n x is: the quantity whose n th power is x. In other words, the function f (x) = n x is by definition the inverse function to g(x) = x n. Note for n even, we have to restrict the domain of g(x) to [0, ) before discussing its inverse.

78 Exponentials For n an integer, a n means: multiply a by itself n times.

79 Exponentials For n an integer, a n means: multiply a by itself n times. We just discussed the meaning of a 1/n in this case.

80 Exponentials For n an integer, a n means: multiply a by itself n times. We just discussed the meaning of a 1/n in this case. Putting these together, you can make sense of when m, n are integers. a m/n = (a m ) 1/n = (a 1/n ) m

81 Exponentials For n an integer, a n means: multiply a by itself n times. We just discussed the meaning of a 1/n in this case. Putting these together, you can make sense of when m, n are integers. a m/n = (a m ) 1/n = (a 1/n ) m That is, a r makes sense when r is a rational number (fraction).

82 Exponentials Note that when a > 1, then for integers N < M one has a N < a M.

83 Exponentials Note that when a > 1, then for integers N < M one has a N < a M. The same holds for rational numbers: if r < s, then a r < a s.

84 Exponentials Note that when a > 1, then for integers N < M one has a N < a M. The same holds for rational numbers: if r < s, then a r < a s. You can see this by putting r, s over a common denominator as r = m/n and s = l/n. Since then m < l, we have: a r = a m/n = (a 1/n ) m < (a 1/n ) l = a l/n = a s

85 Exponentials Thus we see that for any real number a > 1, the expression a x is an increasing function defined on rational numbers x.

86 Exponentials Thus we see that for any real number a > 1, the expression a x is an increasing function defined on rational numbers x. Fact: there is an increasing function defined on the real numbers, also denoted a x, which agrees with the function we have defined so far on the rational numbers.

87 Exponentials Thus we see that for any real number a > 1, the expression a x is an increasing function defined on rational numbers x. Fact: there is an increasing function defined on the real numbers, also denoted a x, which agrees with the function we have defined so far on the rational numbers. To compute this for some arbitrary real number x, you could take better and better rational approximations of x.

88 Exponentials Thus we see that for any real number a > 1, the expression a x is an increasing function defined on rational numbers x. Fact: there is an increasing function defined on the real numbers, also denoted a x, which agrees with the function we have defined so far on the rational numbers. To compute this for some arbitrary real number x, you could take better and better rational approximations of x. Saying this precisely requires a discussion of limits.

89 Exponentials Thus we see that for any real number a > 1, the expression a x is an increasing function defined on rational numbers x. Fact: there is an increasing function defined on the real numbers, also denoted a x, which agrees with the function we have defined so far on the rational numbers. To compute this for some arbitrary real number x, you could take better and better rational approximations of x. Saying this precisely requires a discussion of limits. For a < 1, the same holds, except the function is decreasing.

90 Exponentials Here are f (x) = 2 x, g(x) = 3 x, (1/2) x, and e x.

91 Exponentials Here are f (x) = 2 x, g(x) = 3 x, (1/2) x, and e x.

92 Exponentials Here are f (x) = 2 x, g(x) = 3 x, (1/2) x, and e x.

93 Exponentials Here are f (x) = 2 x, g(x) = 3 x, (1/2) x, and e x.

94 Exponentials Here are f (x) = 2 x, g(x) = 3 x, (1/2) x, and e x.

95 Exponentials Here are f (x) = 2 x, g(x) = 3 x, (1/2) x, and e x.

96 Exponentials Here are f (x) = 2 x, g(x) = 3 x, (1/2) x, and e x.

97 Exponential versus polynomial Here are f (x) = x 2, g(x) = x 4, x 6, and e x.

98 Exponential versus polynomial Here are f (x) = x 2, g(x) = x 4, x 6, and e x.

99 Exponential versus polynomial Here are f (x) = x 2, g(x) = x 4, x 6, and e x.

100 Exponential versus polynomial Here are f (x) = x 2, g(x) = x 4, x 6, and e x.

101 Exponential versus polynomial Here are f (x) = x 2, g(x) = x 4, x 6, and e x.

102 Exponential versus polynomial Here are f (x) = x 2, g(x) = x 4, x 6, and e x.

103 Exponential versus polynomial Here are f (x) = x 2, g(x) = x 4, x 6, and e x.

104 Some uses of exponential functions Radioactive decay

105 Some uses of exponential functions Interest

106 Some uses of exponential functions Population growth (when not limited by resources)

107 Some uses of exponential functions A sample problem: You invest $100 at 3% interest, compounded annually. How many years will it take to grow to $10, 000?

108 Some uses of exponential functions A sample problem: You invest $100 at 3% interest, compounded annually. How many years will it take to grow to $10, 000? Here we need to solve: 100 (1.05) x = or simplifying (1.05) x = 100

109 Some uses of exponential functions A sample problem: You invest $100 at 3% interest, compounded annually. How many years will it take to grow to $10, 000? Here we need to solve: or simplifying 100 (1.05) x = (1.05) x = 100 To find x, we need an inverse to the exponential function.

110 Logarithms The logarithm is by definition the inverse of the exponential.

111 Logarithms The logarithm is by definition the inverse of the exponential. More precisely, the function y log b (y) is the inverse of the function x b x.

112 Logarithms The logarithm is by definition the inverse of the exponential. More precisely, the function y log b (y) is the inverse of the function x b x. One reads log b (y) as the logarithm in base b of y.

113 Logarithms The logarithm is by definition the inverse of the exponential. More precisely, the function y log b (y) is the inverse of the function x b x. One reads log b (y) as the logarithm in base b of y. So if (1.05) x = 100 then x = log

114 Logarithm and exponential facts The log and exp have the following complementary behaviors: a b a c = a b+c log a (BC) = log a (B) + log a (C) a b /a c = a b c log a (B/C) = log a (B) log a (C) (a b ) c = a bc log a (B C ) = C log a B

115 Logarithm and exponential facts The log and exp have the following complementary behaviors: a b a c = a b+c log a (BC) = log a (B) + log a (C) a b /a c = a b c log a (B/C) = log a (B) log a (C) (a b ) c = a bc log a (B C ) = C log a B The last property can be rewritten: log m n = log a n log a m

116 Logarithm and exponential facts The log and exp have the following complementary behaviors: a b a c = a b+c log a (BC) = log a (B) + log a (C) a b /a c = a b c log a (B/C) = log a (B) log a (C) (a b ) c = a bc log a (B C ) = C log a B The last property can be rewritten: log m n = log a n log a m Note this means that to compute logarithm in any base, it is enough to know how to compute logarithm in one particular base.

117 e When doing calculus, it is particularly convenient to use the base e.

118 e When doing calculus, it is particularly convenient to use the base e. The number e is a certain irrational number.

119 e When doing calculus, it is particularly convenient to use the base e. The number e is a certain irrational number. Approximately it is e

120 What is e? Being irrational means that any description of e must involve a limit

121 What is e? Being irrational means that any description of e must involve a limit since we only know how to write rational numbers explicitly. So the following won t really make sense until we discuss limits.

122 What is e? Being irrational means that any description of e must involve a limit since we only know how to write rational numbers explicitly. So the following won t really make sense until we discuss limits. e = n=0 1 n!

123 What is e? Being irrational means that any description of e must involve a limit since we only know how to write rational numbers explicitly. So the following won t really make sense until we discuss limits. e = n=0 1 n! ( e = lim ) N N N

124 What is e? Being irrational means that any description of e must involve a limit since we only know how to write rational numbers explicitly. So the following won t really make sense until we discuss limits. e = n=0 1 n! ( e = lim ) N N N Also: the tangent line at (0, 1) to the graph of e x has slope 1.