# Tag Info

### Why do we teach even and odd functions?

One of the major themes of precalculus is what I call “connecting geometry to algebra”. Being able to translate between an algebraic statement like $f(x)= f(-x)$, and the geometric statement that the ...
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### f(x+h) in the difference quotient

You have already applied some good diagnostic tests. I recommend the following additional diagnostics What happens if you ask them to evaluate each of the following: $f(3y)$: Passing this test ...
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### How to help new students accept function notation

You might remind them that $y$ is just a name for a number. When they draw a plot, they draw a bunch of points: maybe $y=3$ here, $y=5$ there, and $y=-2$ over there. But at some point (no pun ...
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### How to help new students accept function notation

Start by talking about functions in general, not only about functions that can be expressed by a simple formula in x and y. Examples: The function that maps every non-empty list to its first element. ...
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### What are some examples of great functions that are not too elementary (easy)?

I'm personally a fan of simple examples: $x e^x$ (has nice critical point, point of inflection) $e^{-x^2}$ (with appropriate rescaling, the normal distribution from statistics) $\frac{x}{x^2+1}$ (a ...
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### Why do we teach even and odd functions?

Here you can see that knowing if the function is even or odd can help you when you are integrating over the interval $[-a, a]$. You can reduce really-hard-to-look-at integrals to zero just by knowing ...
• 178
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### A more rigorous approach to Precalculus

My experience in teaching Precalculus is that you will be surprised about what you actually need to teach compared to what you thought you needed to teach. For example, if you ask students to compute ...
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### Why do we teach even and odd functions?

Besides applicability in topics like integration and Fourier analysis, it also connects algebra to calculus at least in the way that multiplication of even/odd functions behaves like addition even/odd ...
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### How to help new students accept function notation

The crucial thing the students need to realise is that the (e.g.) $x$ that turns up in the function definition is a bound variable. That's what allows it to be freely renamed or indeed omitted without ...

### Why do we teach linear algebra in precalculus classes?

Vector algebra is a standard 3rd-semester calculus topic (e.g., see OpenStax Calculus 3, Ch. 2-3). This includes calculations of the dot product, cross product, and related values. Standard ...
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### For purposes of teaching, should constant functions be considered "linear functions"?

A linear function is not necessarily a first degree polynomial function: zero function is also linear. In France the terminology is more appropriate than the traditional English one: a linear ...
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### Are there examples of central symmetry, without axial symmetry, in nature?

Does this example of a flower with rotational, but not reflective, symmetry hit what you are looking for? (Name: Pinwheel Flower or Tabernaemontana divaricata)
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### Why do we teach even and odd functions?

Learning to think about functions abstractly should be one goal in precalculus, and function symmetry helps. Also suppose we carefully protected a student from knowing anything about function symmetry....
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### How to help new students accept function notation

Because x and y are just variable names It happens that sometimes y=f(x), but other times z=f(x,y), w=f(x,y,z), or x=f(y) for that matter. All of these variable names are syntactically equivalent, ...
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### Enlighten younger students about the concept of "procedural justice" in mathematics?

On the contrary, many seem surprisingly impatient when being asked to prove 1+1=4/2, whose proof (with properly delimited deepness) involves nothing beyond and possibly well below most people's ...
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### How to help new students accept function notation

TL;DR: A function is a verb. It's an action. Variables are nouns, objects. Verbs (functions) connect nouns semantically, i.e. how A (or x) relates to B (or y), ...
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