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
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
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 ...
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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
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 ...
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10
votes
f(x+h) in the difference quotient
The issue here seems to be with substitution, a fundamental operation in mathematics.
Specifically, substituting $x$ with an expression that itself contains $x$.
The issue of how to perform ...
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9
votes
Why do we teach linear algebra in precalculus classes?
The College Board made curriculum decisions for their new AP Precalculus course that align with sentiments you express. The course is divided into four units, where unit four is titled Functions ...
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8
votes
f(x+h) in the difference quotient
My solution is something like Nick's. I think it has helped my students, but I don't have hard data.
I scribble out the x's. So $f(x)=5x^2-3x+1$ becomes $f(scribble)=5scribble^2-3scribble+1$. And I ...
- 19.4k
8
votes
Sources on inequity in precalculus sequence
Here's a hair-raising article I sometimes use as a touchstone:
Kenschaft, Patricia Clark. "Racial equity requires teaching elementary school teachers more mathematics." Notices of the AMS 52....
- 23.6k
8
votes
3
votes
Differentiation in integer solutions
I think the scope of your question is too narrow. If you ask more broadly about the use of calculus in number theory (not just differentiation for solving Diophantine equations) then books on analytic ...
- 3,048
3
votes
f(x+h) in the difference quotient
You start with $f(x)=x^2+x+1$. How about having students determine $f(y)$ where $y=z +1$?
First, substitute $y$ for $x$ to obtain $f(y)=y^2+ y +1$.
Then, substitute $z+1$ for $y$ to obtain $f(z+1)=(z+...
- 924
3
votes
f(x+h) in the difference quotient
I think the problem originates from conflicting mathematical notation. In elementary school multiplication is explicitly written out $a \times b$. By middle school it becomes just $ab$, so $a(b+c)$ ...
- 1,036
3
votes
Are there examples of central symmetry, without axial symmetry, in nature?
Just to clarify:
There are zero two dimensional "examples of central symmetry [(x,y) -> (-x,-y)], without axial symmetry[Cn], in nature". This is because it is impossible mathematically. ...
- 39
3
votes
Write $y=\sqrt{3+x}$ as the composite of two functions
IMHO, students often struggle with concepts if they don't see their usefulness. So, embedding the question into a context like "to help finding the first derivative using the chain rule" ...
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