42
votes
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 ...
- 23.5k
40
votes
Accepted
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 ...
- 23.5k
25
votes
Why do we teach that every line is a linear function?
The usage you object to is, in fact, the original meaning of "linear". "Linear" means "having to do with lines". The notion of "linear" in the sense of "linear transformation" is a more modern, ...
- 17.2k
24
votes
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 ...
- 241
23
votes
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.
...
- 580
23
votes
Accepted
How do you explain concavity of a polynomial without any calculus?
Here is a proposed definition:
A function $f$ is said to be concave up on an interval $[a,b]$ if for all $x,y \in [a,b]$ with $x<y$, the line $L$ connecting $(x,f(x))$ and $(y,f(y))$ satisfies $L(t)...
- 23.5k
22
votes
How to help new students accept function notation
You should tell them these two main benefits:
(1) Function notation is concise! For example, instead of writing "Find $y$ when $x=5$" one can simply write "Find $f(5)$" This becomes very appreciable ...
- 496
20
votes
Accepted
Math topics that reward going beyond cookbook methods
One answer would be: any real-world word/application problems; insofar as one avoids making them in a repetitive template by "just changing the numbers". I find that most any textbook has an "...
- 23.6k
20
votes
f(x+h) in the difference quotient
I know a teacher who (at least in the past) would require students to write underlined blanks in place of the input whenever they were evaluating a function from its formula:
[Examples with $f(x)=5x^2-...
- 8,856
19
votes
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 ...
- 3,882
18
votes
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
18
votes
Accepted
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 ...
- 20.8k
16
votes
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 ...
- 2,972
16
votes
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 ...
- 693
16
votes
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 ...
- 23.6k
15
votes
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 ...
- 310
15
votes
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)
- 3,947
14
votes
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....
- 9,473
14
votes
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, ...
- 243
13
votes
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 ...
- 11.5k
13
votes
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), ...
- 131
12
votes
Accepted
Is there a more intuitive way to solve combined rates of work problems?
By throwing in all of the $r$, $t$, and $d$, not to mention $x$, you're overly complicating things, especially if your brother is algebra-averse. There's really only one unknown, and you can call it $...
- 817
12
votes
For purposes of teaching, should constant functions be considered "linear functions"?
While I haven't done a systematic survey, my impression is that the overwhelming majority of pre-calculus and calculus texts define a linear function to be one of the form $f(x) = mx + b$ with no ...
- 1,506
12
votes
Why do we teach even and odd functions?
Even and odd parity are probably the simplest examples of function symmetries.
In applied mathematics, the general observation of function symmetries allows to simplify calculations (as stated by ...
- 498
12
votes
Examples (for beginners) of real functions which are not given by elementary formulae
How about $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = y$ where $y$ is the unique solution to $y^5+x^2y+5=0$? This does not have an elementary formula, but students can understand that for any ...
- 23.5k
12
votes
What are some examples of great functions that are not too elementary (easy)?
Don't forget purely graphical problems. Give a graph of $f$ or $f'$, ask for the student to sketch the other one.
- 23.5k
12
votes
Accepted
Are there examples of central symmetry, without axial symmetry, in nature?
It is easy to have axial symmetry, without an inversion center (e.g. the picture Opal showed).
It is more difficult to have the converse. Many molecules with inversion centers also have rotational ...
- 136
11
votes
The Order in Pre-Calculus Textbooks
Mitchell,
Great question. I think that precalculus is usually a wasteland for some random classical techniques that used to be required to study calculus.
I think that you can take a completely ...
- 984
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