MAC1140 Exam 1 Vocabulary Terms

Which of the following is equivalent to: {x∈R:x<−4}?

( -∞, -4)

Which of the following is equivalent to: {x∈R:x<−4}?

(-∞, -1)

Which of the following is equivalent to: {x ∈ R : x < −4}?

(-∞, 10)

The virtue of studying universal properties in general is...

- By learning a few universal properties, it cuts down significantly on how many specific properties you must learn. - Universal properties tend to be very important and useful properties that apply widely.

The goal of generalizing numeric models is to...

- Expend more time and effort up front, to save considerable time and effort later. - Set up a standard by which to calculate values. That is to say, to create 'form' where certain known quantities can be "plugged in" and the answer is immediately calculated.

What is the point of Contextual Based Learning?

- To put content in context to improve retention. - To aid learning, rather than memorizing information. - To help build intuition and understanding of content, allowing one to connect new content to already understood content.

Which of the following are valid excuses for graphs?

-To build intuition about how two or more variables are related to one another. -To get approximate values for things like extrema and intercepts. -To get an idea of how functions behave at various approximate values. -Getting global behavior/information about your function at a glance.

What is the primary goal/purpose of this course?

-To get everyone to the same mathematical level to progress to future math courses. -To teach students how to problem solve and think logically an deductively. -To introduce the necessary fundamentals of notation and mechanics for calculus (except trig).

Model

A (collection of) mathematical expressions used to represent a specific situation.

Information

A collection of knowledge or observations used to problem solve.

A common error is to...

A common error is to overgeneralize. Just because you may be able to generalize a piece of information, doesn't mean you will want to. The first step is to identify which elements we are able to generalize, but that doesn't mean we will generalize every piece of information we can.

Deductive Reasoning

A form of inference that uses a set of given information, and extracts new information that must be true as a result. For example (eg): Given the information: It has rained There is no cover for the ground Then we can deduce that the ground is wet. This would be a logical deduction, or "deductive reasoning".

Inverse Function

A function g(y) is an inverse to another function f(x) if the following two compositions are true.

Mathematical Relationship

A link between two or more pieces of information or data. Specifically a 'relation' (aka relationship) need not involve variables, although often the pieces of information or data in a relation are eventually generalized into variables.

Mathematical Symbol: Actual Definition

A mathematical symbol (usually referred to simply as 'symbol' when context is clear) is a non-numeric piece of writing (typically a letter) that holds the place of some (typically numeric) value, to be decided (or deduced) later.

Cartesian Coordinates

A method of graphing a function where the domain and range meet at a right angle (ie the so-called "x-y plane".)

Parent Function

A parent functions is the 'prototypical' form of the given function type. That is to say, the 'parent function' of a function type is the base (ie most basic) version of that function without any manipulations, shifts, or changes to it's form. For example: The parent function of the quadratic function would be f(x)=x2. This is the base type without anything added to it. This is most commonly referenced by asking a question. For example: 'What is the parent function type of the function f(x)=x2+2x−3?' In this case the answer would be f(x)=x2 since the given function was a quadratic, and x2 is the parent function for a quadratic.

Generalize

A process of going from a (somewhat) specific problem (eg a numeric model) to a methodology or (Generalized) Model to attain answers to (less specific) variations of the original problem.

Function

A specific type of mathematical relationship that relates independent and dependent variables, and yields precisely one value for each dependent variable, for any fixed combination of specific values for the independent variable(s). That is: An equation that has one "output value" for a given set of "input values". Note: A function must have a Domain and a Codomain as part of it's definition. The key thing to remember is that a function is a relationship between variables within some context.

Mathematical Expression

A statement that involves variables and/or constants and some relationship between them. A mathematical expression does not contain the symbol ' = ' Mathematical expressions are statements that link a number of variables together, but specifically do not contain an equals sign.

Mathematical Symbol: Conceptual Definition

A symbol is a letter, in some alphabet, that is a temporary stand-in for a piece of information. It is typically either a variable or replaced by a concrete value when one forms a model.

Arbitrary Constant

A symbol that is unaffected by independent variables, and whose value is determined by the model and initial conditions of the system. This is typically a result of some initial information used in your model and is a byproduct of choices in your model, but they are unaffected by independent variables.

Variable

A symbol used to represent an unknown piece of data.

Mathematical Reasoning

A technique of problem solving that relies on deductive reasoning and symbolic manipulation.

Transformation

A technique to change the shape or size of a function in a predictable (and reversible) way. Often used to "rescale" the function's graph. For Example: Scaling the graph to make it bigger, smaller, or flipping the graph across some line are examples of 'transformations'.

Rigid Translation

A technique to move the function about on a graph without changing it's (relative) size. For example: Movements of the graph up, down, left, or right would count as 'rigid transformations'.

Dependent Variable

A variable whose value is unknown and is generated by the Model. In most cases dependent variables are determined (solved for) by the Model.

Independent Variable

A variable whose value is unspecified, but will be provided by the statement of the problem.

Graph of a Function

A visual representation of the relationship between domain and range, ie the "x-y coordinate picture" of a function.

Solution

An answer that depends on the question asked. That is: There is no such thing as a "universal solution to a function".

This is typically the hard part of the "clarifying" steps and then the future "quantifying" steps, but it is necessary since otherwise your solution may

Be wholly inaccurate due to faulty assumptions.

What is the difference between the codomain and range?

Codomain: Type of thing that the output is Range: The actual achievable output

Types of Information: Data

Data is Quantifiable Information that has been quantified. This is a gray line; data is rarely distinguished from quantifiable information. An example might be that the "quantifiable information" would be that a brick can be bought with money (ie there is some number that corresponds to it's cost) whereas the "data" would be the specific cost of the brick (eg the brick costs $2.50). In one case we have information that tells us there exists a piece of data (a number), but we don't have it yet, whereas in the other case we have the actual data (number) already.

Mathematics is the language of _____ reasoning.

Deductive

Discontinuities

Discontinuities are domain values (x-values) where a function fails to be continuous. By convention we only count points where the function is still defined on either side of the discontinuities, thus we wouldn't say x−−√ is 'discontinuous' for x<0 because it's domain simply ends at 0, there is no 'disruption' in the domain because the domain is only on one side of the value 0. Discontinuities can be found in a number of forms; holes, infinite (or asymptotic) discontinuities, and jumps. Classifying these discontinuities analytically is beyond this scope of this course, but we will give geometric examples in this topic.

What is the importance of discontinuities?

Discontinuities represent the places where your function (and thus your model) is behaving in a weird way; which suggests further studying may be worthwhile.

Extrema

Extrema of a function are the maximum or minimum values that the function attains. These can be broken up into local or relative extrema, and absolute or global extrema. Local/Relative Extrema: are points that are maximums or minimums within some 'small enough' section of x values near the x value of the extrema. By 'close enough' we mean that for some specific x value (let's say x0, f(x0) is bigger (or smaller if it's a local minimum) than f(x) for any x within some distance you can specify (like 'within 12') of x0. Absolute/Global Extrema: are points that attain the absolute highest (or lowest) y values that a function can attain. Remember: Point of Inflection

What is a good explanation of what an extrema of a function is?

Extrema represent the maximum and minimum values of a function within some defined interval of it's domain.

Accuracy

How close to correct a value is. For example, 3.14 is a more accurate value of π than 3.151592, even though 3.151592 is a more precise number than 3.14.

Precision

How exact (aka how specific) a value is. For example, 2.1343435 is more precisely determined than 2.134 since it has considerably more digits given.

Types of Information: Quantifiable Information

Information that can be numerically or algebraically represented (but may not be yet, see Data below); eg the design of the patio or the fact that bricks are available to buy at the local hardware store.

Types of Information: Extraneous Information

Information that is not relevant to the problem solving process/method you are using. This is often (somewhat paradoxically) the most important type of information, in that recognizing that some piece of knowledge can be ignored is often the key to seeing how a solution can be achieved. In our example; if you keep spending all your time trying to quantify the relative costs of paint colors and you don't stop to ask if the patio is going to be painted, then you are wasting a lot of time if it turns out that paint isn't necessary.

An item is in the range if it's something that...

Is a possible output of the function for a given input that is the domain.

Memorizing

Is the antithesis of learning.

An element is in the domain if...

It is an element that satisfies all/any conditions that were provided in the problem to describe the domain.

We will start with a geometric view of rigid translations because

It is hard to manipulate functions via rigid translations without understanding what they are, and it is hard to understand what they are without literally seeing what's happening.

Mathematics is the language of

Language of deduction! It is a very carefully developed language built with the intent of bringing precision and structure to what we casually refer to as "thinking".

Equation

Mathematical expressions or relations that involve two or more variables and an equality. That is to say, an equation is when two mathematical expressions are 'equal' to one another. (Mathematical) Equations are statements that link a number of variables together using an equals sign

In other words, modeling is...

More of an artform than a science.

Range

Possible output values of the dependent variable of a given function and domain pair.

Which of the following is true?

Precision and Accuracy are both important, but separate things. Which is more important, or what it means when you get one and not the other, is specific to the context of the problem.

Data

Quantified versions of information.

What kind of problems are most likely to involve/require 'mathematical reasoning'?

Synthesizing or interpret (primarily) numeric information.

Vertical Line Test

The Vertical Line test is the process of checking the graph of a relation to see if it is a function by checking every vertical line to see if it intersects the function in more than one place. The easiest way to do this is to imagine (or use a straight edge) a straight vertical line sweeping from left to right across the entire plot, and see if there is any point at which the line intersects the graph of the relation in more than one place.

In a residential neighborhood most families have multiple cars; at least one for each parent and maybe one for the kids over 16. You have learned to recognize every car in your neighborhood and which house it belongs to. What is the domain of this association (recognizing the car and then recalling which house it belongs to)?

The cars in the neighborhood.

Functional Argument

The content that a function is being applied to. For Example: The "x" in "f(x)" or the "2x+1" in the "g(2x+1)" are both examples of 'functional arguments'.

Function is really relation between...

The domain and codomain.

What is the codomain?

The houses in the neighborhood.

Functional Output or Value

The point in the codomain that a function returns or 'output's. For Example: If f(x)=3x+1 and we compute f(5)=3⋅5+1=16, then the '16' is an example of the 'functional output' (also referred to as function value)

X or Y Intercepts

The points at which a function intersects either the x or y axis (respectively). These are points and must always be written as points. For example: One would say "The x-intercept is (5,0)". It is incorrect to say "The x-intercept is 5."

The relationship...

The relationship goes from (takes elements within) the Domain Name 'to' (relates the element to) the Codomain Name.

Codomain

The set (or type) of values a dependent variable can possibly have. Note that the dependent variable may not actually attain all the values of the codomain. For example; a dependent variable may belong to the codomain of "all real numbers", but if it is a distance, then it would have to be a positive number. See the lecture notes below for an explanation of "codomain" versus "range"

Which of the following is the English-translation of the following: {x:x∈Z,−30<x<45}?

The set of integers (strictly) between negative 30 and 45.

Which of the following might be a reason to care about the zeros of a function?

The zero is the output, so it may represent something like the break-even point on a profit curve, or when an object hits the ground on a height function.

Zeros of a function

The zeros of a function are the domain values that yield zero as the output. Put another way, the zeros of a function are the x-values only of the x-intercepts. These are not points, but they may be written either as points or as values. For example: One could say "The zero of the function is (5,0)". It is slightly more conventional to say "The zero of the function is 5."

The primary purpose of recitation is which of the following?

To do as many examples and demonstrate mechanics and individual steps of new techniques .

Which of the following is the best description of "Geometric Perspective"?

To get a 'wide lense view' of data; to have a visual summation of a large-scale trends and important features of data.

The primary purpose of lectures is to...

To introduce new techniques and give broader context for how and why the technique should be used.

Which of the following might be a good reason to care about the intercepts of a function?

Using zero as the input to get the y-intercept tells you something about the initial condition of the problem.

Domain

Valid input values for the independent variables of an equation.

Y values are the...

Vertical distance between the x-axis and the point. So if we had multiple y-values for the same x-value, there would be two points on the same vertical line.

Is this association a function?

Yes, each car belongs to one house.

Which of the following would be the best set-builder notation to describe "the set of all positive real numbers"?

{ x : x > 0, x ∈ R}

Which of the following is equivalent to: (−4,∞)?

{ x ∈ R : x > -4 }

Which of the following is equivalent to: (10,∞)?

{ x ∈ R : x > 10}