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1. Functions and Models Flashcards

1
Q

What is a function?

A

A function f is a rule that assigns each element x in a set D exactly one element, called f(x), in a set E.

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2
Q

When do functions arise?

A

Whenever one quantity depends on another.

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3
Q

In defining a function, what is the set D called?

A

The domain of the function.

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4
Q

What is f(x)?

A

The value of f at x, and is read f of x.

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5
Q

For what kind of numbers do we usually consider functions for which the sets D and E to be a part of?

A

Real numbers

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6
Q

What is the range of f?

A

The set of all possible values of f(x) as x varies throughout the domain.

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7
Q

What is an independent variable in math?

A

A symbol that represents a number in the domain of f.

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8
Q

What is a dependent variable in math?

A

A symbol that represents a number in the range of f.

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9
Q

What is a helpful way to think of a function?

A

It’s helpful to think of a function as a machine. If x is in the domain of the function f, then when x enters the machine, it’s accepted as an input and the machine produces an output f(x) according to the rule of the function.

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10
Q

What is, apparently, another useful way to picture the function other than the machine analogy?

A

By an arrow diagram. Each arrow connects an element of D to an element of E.

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11
Q

What is the most common method of visualizing a function?

A

By its graph.

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12
Q

How can we read the value of f(x) from a graph?

A

Since the y-coordinate of any point (x,y) on the graph is y=f(x), we can read the value of f(x) from the graph as being the height of the graph above the point x. The graph of f also allows us to picture the domain of f on the x-axis and its range on the y-axis.

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13
Q

What are the four ways of representing a function?

A
  • verbally - by a description in words
  • numerically - by a table of values
  • visually - by a graph
  • algebraically - by an explicit function
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14
Q

Describe the vertical line test.

A

A curve int he xy plane is the graph of a function of x if and only if no vertical line intersects the curve more than once.

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15
Q

Why does a failed vertical line test discredit the curve’s “functionness”?

A

If a line intersects the curve twice, then the curve can’t represent a function because a function can’t assign two different values to an x.

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16
Q

What are piecewise defined functions?

A

Functions that are defined by different formulas in different parts of their domains.

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17
Q

What is a subtle example of a piecewise function?

A

The absolute value function.
lal = a if a≥0
lal = -a if a<0

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18
Q

What is an even function?

A

A function which satisfies f(-x) = f(x) for every number x in its domain.
ex: f(x) = x^2

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19
Q

What is the geometric significance of a even function?

A

Its graph is symmetric with respect to the y-axis.

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20
Q

What is an odd function?

A

A function which satisfies f(-x) = -f(x) for every number x in its domain.
ex: f(x) = x^3

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21
Q

What is the geometric significance of an odd function?

A

Its graph is symmetric about the origin. If we already have the graph of f for x≥0, we can obtain the entire graph by rotating this portion through 180deg about the origin.

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22
Q

When is a function called increasing on an interval I?

A

When

f(x1) < f(x2) whenever x1 < x2 in I.

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23
Q

When is a function called decreasing on an interval I?

A

When

f(x1) > f(x2) whenever x1 < x2 in I.

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24
Q

What is a mathematical model and what is its purpose?

A

A mathematical description of a real-world phenomenon whose purpose is to understand the phenomenon and perhaps to make predictions about future behavior.

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25
Q

What is the first stage of mathematical modeling?

A

Given a real-world problem, our first task is to formulate a mathematical model by identifying and naming the independent and dependent variables and making assumptions that simplify the phenomenon enough to make it mathematically tractable. We use our knowledge of the physical situation and our mathematical skills to obtain equations that relate the variables.

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26
Q

How can we accomplish the first stage of mathematical modeling in situations where there are no physical laws to guide us?

A

We may need to collect data and examine the data in the form of a table in order to discern patterns. From this numerical representation of a function, we may wish to obtain a graphical representation by plotting the data. The graph might even suggest a suitable algebraic formula in some cases.

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27
Q

What is the second stage of mathematical modeling?

A

The second stage is to apply the mathematics that we know to the mathematical model that we have formulated in order to derive mathematical conclusions.

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28
Q

What is the third stage of mathematical modeling?

A

In the third stage, we take the mathematical conclusions derived from stage 2 and interpret them as information about the original real-world phenomenon by way of offering explanations and making predictions.

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29
Q

What is the final stage of mathematical modeling?

A

The final stage is to test our predictions by checking against new real data. If the predictions don’t compare well with reality, we need to refine our model or to formulate a new model and start the cycle again.

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30
Q

How are mathematical models idealization?

A

A mathematical model is never a completely accurate representation of a physical situation. A good model simplifies reality enough to permit mathematical calculations but is accurate enough to provide valuable conclusions.

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31
Q

What do we mean when we say that y is a linear function of x?

A

That the graph of the function is a line, so we can use the slope-intercept form of the equation of a line to write a formula for the function as:
y = f(x) = mx + b
where m is the slope of the line and b is the y-intercept.

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32
Q

Describe a characteristic feature of linear functions.

A

A characteristic feature of linear functions is that they grow at a constant rate. Thus, the slope of the graph can be interpreted as the rate of change of y with respect to x.

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33
Q

What is an empirical model?

A

A mathematical model based entirely on collected data. We eek the curve that “fits” the data in the sense that it captures the basic trend of the data points.

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34
Q

What i linear regression?

A

A statistical procedure for obtaining linear models from lines of best fit.

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35
Q

What is interpolation?

A

Estimating a value between observed values.

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36
Q

What is extrapolation?

A

Estimating a value outside the regions of observations.

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37
Q

When is a function P called a polynomial?

A

When:
P(x) = [a_n x^n] + [a_(n-1) x^(n-1)] + … + [a_2 x^2] + [a_1 x] + a_0
where n is a non-negative integer and the numbers a_0, a_1, a_2, …,a_n are constants called the coefficients of the polynomial.

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38
Q

What is the domain of a polynomial?

A

ℝ = (-∞,∞)

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39
Q

What is the degree of a polynomial?

A

The degree of the polynomial is n if the leading coefficient a_n ≠ 0

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40
Q

In what form is a polynomial of degree 1?

A

A polynomial of degree 1 is of the form P(x) = mx + b and so it is a linear function.

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41
Q

In what form is a polynomial of degree 2?

A

A polynomial of degree 2 is of the form
P(x)= ax^2 + bx + c
and is called a quadratic function. Its graph is always a parabola obtained by shifting the parabola
y = ax^2. The parabola opens upward if a > 0 and downward if a < 0.

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42
Q

In what form is a polynomial of degree 3?

A

A polynomial of degree 3 is of the form
P(x) = ax^3 + bx^2 + cx + d a≠0
and is called a cubic function.

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43
Q

For which fields of studies are polynomials commonly used to model various quantities?

A

The natural and social sciences.

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44
Q

What is a power function?

A

A function in the form f(x) = x^a, where a is a constant.

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45
Q

What is the general shape of the graph of f(x) = x^n power function?

A

If n is even, then f(x) = x^n is an even function and its graph is similar to the parabola y = x^2.
If n is odd, then f(x) = x^n is an odd functions and its graph is similar to that of y = x^3.

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46
Q

What is a root function?

A

A power function where f(x) = x^(1/n) where n is a positive number.

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47
Q

What is the general shape of the graph of root functions?

A

For even values of n, the graph is similar to that of
y = x^(1/2) whose domain is positive
For odd values of n, the graph is similar to that of
y = x^(1/3) whose domain includes all real numbers.

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48
Q

What is a reciprocal function?

A

A power function where

f(x) = x^(-1) = 1/x

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49
Q

What is the general shape of the graph of a reciprocal function?

A

Its graph has the equation y = 1/x or xy = 1,

and its graph is a hyperbola with the coordinate axes as its asymptotes.

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50
Q

What is a rational function and what is its domain?

A

A rational function f is a ratio of two polynomials:
f(x) = P(x)/Q(x)
where P and Q are polynomials.
The domain consists of all values of x such that Q(x) ≠ 0.

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51
Q

When is a function called an algebraic function?

A

A function f is called an algebraic function if it can be constructed using algebraic operations (such as addition, subtraction, multiplication, division, and taking roots) starting with polynomials. Any rational function is automatically an algebraic function.

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52
Q

How are angles conventionally measured in calculus?

A

In radians.

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53
Q

What are the domain and range of sine and cosine functions?

A 
domain = (-∞,∞)
range is the closed interval [-1,1]
Thus, for all values of x:
-1 ≤ sin(x) ≤ 1
-1 ≤ cos(x) ≤ 1
54
Q

When do the zeros of the sine function occur?

A

At the integer multiple of π.

55
Q

What is an important property of the sine and cosine functions?

A

They are periodic functions and have period 2π. This means that for all values of x:
sin(x+2π) = sin(x)
cos(x+2π) = cos(x)

56
Q

What does the periodic nature of sin and cosine functions make them suitable for?

A

Modeling repetitive phenomena.

57
Q

How is the tangent function related to the sine and cosine functions?

A

tan(x) = sin(x)/cos(x)

58
Q

What are the domain and range of the tangent function?

A

The tangent function is undefined whenever cos(x) = 0, that is, when x = ±π/2, ±3π/2, …
Its range is (-∞,∞)

59
Q

What are the reciprocals of sine, cosine, and tangent respectively?

A

Cosecant, secant, and cotangent

60
Q

What are exponential functions?

A

The exponential functions are the functions of the form
f(x) = a^x
where the base “a” is a positive constant.

61
Q

What are logarithmic functions?

A

The logarithmic functions f(x) = log_a(x), where the base “a” is a positive constant, are the inverse functions of the exponential functions.

62
Q

How does y = f(x) + c affect the graph of f(x)?

A

By shifting the graph of y=f(x) a distance of c units upward.

63
Q

How does y = f(x) - c affect the graph of f(x)?

A

By shifting the graph of y=f(x) a distance of c units downward.

64
Q

How does y = f(x-c) affect the graph of f(x)?

A

By shifting the graph of y=f(x) a distance of c units to the right.

65
Q

How does y = f(x+c) affect the graph of f(x)?

A

By shifting the graph of y=f(x) a distance of c units to the left.

66
Q

How does y = f(x) change in response to y = cf(x) if c>1?

A

By stretching the graph of y=f(x) vertically by a factor of c.

67
Q

How does y = f(x) change in response to y = (1/c)f(x) if c>1?

A

By shrinking the graph of y=f(x) vertically by a factor of c.

68
Q

How does y = f(x) change in response to y = f(cx) if c>1?

A

By shrinking the graph of y=f(x) horizontally by a factor of c.

69
Q

How does y = f(x) change in response to y = f(x/c) if c>1?

A

By stretching the graph of y=f(x) horizontally by a factor of c.

70
Q

How does y = f(x) change in response to y = -f(x) if c>1?

A

By reflecting the graph of y=f(x) about the x-axis.

71
Q

How does y = f(x) change in response to y = f(-x) if c>1?

A

By reflecting the graph of y=f(x) about the y-axis.

72
Q

How can two functions, f and g, be combined to form the new functions f+g, f-g, fg, f/g?

A

They can be combined in a similar manner to the way we add, subtract, multiply, and divide, respectively.

73
Q

What is the domain of added, subtracted, or multiplied functions?

A

If the domains of f is A and the domain of g is B, then the domain of f+g,…, is the intersections A∩B because both f(x) and g(x) have to be defined.

74
Q

What is the domain of divided functions?

A

If the domains of f is A and the domain of g is B, then the domain of f/g is the intersections A∩B whenever g≠0, because both f(x) and g(x) have to be defined.

75
Q

What is a composite function?

A

Given two functions f and g, the composite function f ∘ g (also called the composition of f and g) is defined by
(f ∘ g)(x) = f(g(x))

76
Q

How do you compute composite functions?

A

By substitution.

77
Q

What is the domain of a composite function?

A

(f ∘ g) is defined whenever both f(x) and g(x) are defined.

78
Q

What is an important basic thing to remember about composite functions?

A

f ∘ g ≠ g ∘ f

79
Q

Is it possible to take the composition of three or more functions, and if so, how is it?

A

Yes it’s possible.
The composite function f ∘ g ∘ h is found by first applying h, then g, then f as follows:
( f ∘ g ∘ h)(x) = f(g(h(x)))

80
Q

Describe the difference between a power function and an exponential function.

A

The function f(x) = 2^x is called an exponential function because the variable, x, is in the exponent. It should not be confused with the power function g(x) = x^2, in which the variable is the base.

81
Q

What is the value of an exponential function in which the variable x = 0?

A

1

82
Q

What is the value of an exponential function in which the variable x = -n?

A

a^(-n) = 1/(a^n)

83
Q

What is the value of an exponential function in which the variable x is a rational number?

A

a^(p/q) = Rootbase_q(a^p).

84
Q

Describe the graphs of exponential functions.

A

All graphs pass through the point (0,1) because a^0 = 1. As the base “a” gets larger, the exponential function grows more rapidly.

85
Q

Describe the graphs of the three basic kinds of exponential functions.

A

y = a^x
If 0 < a < 1, the exponential function decreases.
If a = 1, it is constant.
If a > 1, it increases

86
Q

According to the law of exponents, what is another way of writing a^(x+y)?

A

a^(x+y) = (a^x)(a^y)

87
Q

According to the law of exponents, what is another way of writing a^(x-y)?

A

a^(x-y) = (a^x)/(a^y)

88
Q

According to the law of exponents, what is another way of writing (a^x)^y?

A

(a^x)^y = a^(xy)

89
Q

According to the law of exponents, what is another way of writing (ab)^x?

A

(ab)^x = (a^x)(b^x)

90
Q

Which number is the most convenient base for an exponential function in all of calculus?

A

The number e = 2.71828

91
Q

Why is the number e so important for calculus?

A

Because the slope of the tangent line to y = e^x at (0,1) is exactly 1.

92
Q

What do we call the function f(x) = e^x?

A

The natural exponential function.

93
Q

What is an inverse function?

A

A function that “reverses” another function, denoted by f^(-1)(x), read f inverse of x.

94
Q

Which functions possess inverses?

A

One-to-one functions.

95
Q

When is a function one-to-one?

A

A function f is called one-to-one if it never takes on the same value twice.
f(x1) ≠ f(x2) whenever x1 ≠ x2

96
Q

How can you tell graphically if a function is one-to-one?

A

With a horizontal line test.

A function is one-to-one if and only if no horizontal line intersects its graph more than once.

97
Q

Why are one-to-one functions important?

A

Because they are precisely the functions which possess inverse functions.

98
Q

Define a one-to-one function.

A

Let f be a one-to-one function with domain A and range B. Then its inverse functions f^(-1) has a domain B and a range A and is defined by
f^(-1)(y) = x ⇔ f(x) = y
for any y in B.

99
Q

What is the domain of f^(-1)?

A

The range of f

100
Q

What is the range of f^(-1)?

A

The domain of f

101
Q

What is it that’s important to remember about the -1 in the inverse function notation?

A

Do not mistake the -1 in the f^(-1) for an exponent.

102
Q

How do you find the inverse function function to a one-to-one function?

A
  1. Write y = f(x)
  2. Solve this equation for x in terms of y (if possible)
  3. To express f^(-1) as a function of x, interchange x and y. The resulting equation is y = f^(-1)(x).
103
Q

How do obtain a graph of a f^(-1) from f?

A

By reflecting the graph of f along the line y = x

104
Q

Why are exponential functions one-to-one?

A

If a>0 and a≠1, the exponential function f(x) = a^x is either increasing or decreasing and passes a horizontal line test. it

105
Q

What is the inverse function of the exponential function f(x) = a^x?

A

The logarithmic function with base a:

f^(-1)(x) = log_a

106
Q

What is the equation to transform an exponential function to a logarithmic one and back?

A

log_a(x) = y ⇔ a^y = x

Thus, if x>0, then log_a(x) is the exponent to which base “a” must be raised to give x.

107
Q

Apply the cancellation equations to the logarithmic and exponential functions.

A 
log_a(a^x) = x 
a^(log_a(x)) = x for every x>0
108
Q

What is the domain and range of an exponential function?

A

Domain ℝ

Range (0 - ∞)

109
Q

What is the domain and range of a logarithmic function?

A

Domain (0 - ∞)

Range ℝ

110
Q

How do you obtain the graph of the logarithmic function y = log_a(x) from the graph of the exponential function y = a^x?

A

By reflecting the graph of y = a^x about the line y=x.

111
Q

Describe the graphs of logarithmic functions.

A

All logarithmic functions pass through the point (1,0) since log_a(1) = 0.
The fact that y = a^x is a very rapidly increasing function for x>0 is reflected in the fact that y = log_a(x) is a very slowly increasing function for x>1.

112
Q

What are the laws of logarithms?

A
  1. log_a(xy) = log_a(x) + log_a(y)
  2. log_a(x/y) = log_a(x) - log_a(y)
  3. log_a(x^r) = rlog_a(x) (where r is any real number)
113
Q

What number is the most convenient base for logarithms?

A

The number e.

114
Q

What is the logarithm with base e called?

A

The natural logarithm. ln

115
Q

What is the (logarithmic) change of base formula?

A

For any positive number a (a≠1), we have:

log_a(x) = lnx/lna

116
Q

(about logarithms) Because the curve y = e^x crosses the y-axis with a slope of 1, it follows that…?

A

It follows that the reflected curve y = lnx crosses the x-axis with a slope of 1.

117
Q

What is an interesting feature about growth rate of natural logarithms?

A

Although lnx is an increasing function, it grows very slowly when x>1. In fact, lnx grows more slowly than any positive power of x.

118
Q

Why do we have difficulty finding the inverse of trigonometric functions?

A

Because the trigonometric functions aren’t one-to-one.

119
Q

How do we find the inverse of trigonometric functions if they aren’t one-to-one?

A

By restricting their domains so that they become one-to-one.

120
Q

What is the inverse functions of the restricted sine function called ?

A

The inverse sine function (sin^(-1)) or the arcsine function (arcsin).

121
Q

Define arcsin.

A

sin^(-1)(x) = y ⇔ sin(y) = x and (-π/2) ≤ y ≤ (π/2)

122
Q

Define arccos

A

cos^(-1)(x) = y ⇔ cos(y) = x and 0 ≤ y ≤ π

123
Q

Define arctan

A

tan^(-1)(x) = y ⇔ tan(y) = x and (-π/2) ≤ y ≤ (π/2)

124
Q

What is the domain and range of the arcsin function?

A

domain: [-1,1]
range: [ (-π/2), (π/2)]

125
Q

How do you obtain the graph of the arcsin function from the restricted sine function?

A

By reflecting the restricted sine function about the line y = x

126
Q

What is the domain and range of the arccos function?

A

domain: [-1,1]
range: [ 0, π]

127
Q

How do you obtain the graph of the arccos function from the restricted cosine function?

A

By reflecting the restricted cosine function about the line y = x

128
Q

What are the cancellation equations for the inverse sine function?

A 
sin^(-1)(sin^(x)) = x    for (-π/2) ≤ x ≤ (π/2)
sin(sin^(-1)(x)) = x     for  -1  ≤ x ≤ 1
129
Q

What are the cancellation equations for the inverse cosine function?

A 
cos^(-1)(cos^(x)) = x    for 0 ≤ x ≤ π
cos(cos^(-1)(x)) = x     for  -1  ≤ x ≤ 1
130
Q

What is the domain and range of the arctan function?

A

domain: ℝ
range: [ (-π/2), (π/2)]

131
Q

How do you obtain the graph of the arctan function from the restricted tangent function?

A

We know that the lines x = ±π/2 are vertical asymptotes of the graph of tan. Since the graph of arctan is obtained by reflecting the graph of the restricted tangent function about the line y=x, it follows that the lines y = π/2 and y = -π/2 are horizontal asymptotes of the graph of arctan.

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