How can I overcome the lack of support for my passion for mathematics?

[nycmathguy]

Homework Statement

Given the slope of the tangent line, find Slope of the secant line.

Relevant Equations

f(x) = x^2 - 1

Slope of a Tangent Line For f (x) = x^2 − 1:

Find the slope of the secant line containing the points P = (−1, f (−1)) and
Q = (−1 + h, f (−1 + h)).

Solution:

I think I need to find f(-1).

f(-1) = (-1)^2 - 1

f(-1) = 1 - 1

f(-1) = 0

Point P becomes (-1, 0).

I now must find f(-1 + h).

f(-1 + h) = (-1 + h)^2 - 1

f(-1 + h) = 1 + - 2h + h^2 - 1

f(-1 + h) = h^2 - 2h

Point Q is now (-1 + h, h^2 - 2h).

Let m = slope of the secant line.

m = (0 - h^2 - 2h)/(-1 + h - (-1))

m = (-h^2 - 2h)/h

m = -h -2

Factor out -1.

m = -(h + 2)

I hope this is right.

 

[Ssnow]

Hi, I think there is an error of sign in the denominator of $m$, it is $-1-h-(-1)$ ... (also an error of sign in the numerator, find it ...)
The procedure seems correct!
Ssnow

 

[nycmathguy]

Hi, I think there is an error of sign in the denominator of $m$, it is $-1-h-(-1)$ ...
The procedure seems correct!
Ssnow

Denominator:

-1 - h - (-1)

-1 - h + 1

Both -1 & 1 cancel out. This leaves me with -h

Yes?

 

[Ssnow]

yes now you must to see the numerator ...

 

[Ssnow]

It is $0 -(h^2-2h)$, so ...
Ssnow

 

[nycmathguy]

It is $0 -(h^2-2h)$, so ...
Ssnow

This leads to -h^2 + 2h.

Numerator: -h^2 + 2h.

Denominator: -h

m = (-h^2 + 2h)/-h

m = h - 2

Yes?

 

[Ssnow]

This leads to -h^2 + 2h.

Numerator: -h^2 + 2h.

Denominator: -h

m = (-h^2 + 2h)/-h

m = h - 2

Yes?

Yes! In fact when $h\rightarrow 0$ you find the derivative $f'(-1)=-2$, in general for the secant you have that the slope is $h-2$.
Ssnow

 

[nycmathguy]

Yes! In fact when $h\rightarrow 0$ you find the derivative $f'(-1)=-2$, in general for the secant you have that the slope is $h-2$.
Ssnow

Very good. One problem at a time is better than posting lots of questions at the same time.

 

[Mark44]

A couple of tips to save yourself some typing. Instead of writing this:

I think I need to find f(-1).
f(-1) = (-1)^2 - 1
f(-1) = 1 - 1
f(-1) = 0

... you can write it this way:
$f(-1) = (-1)^2 - 1 = 1 - 1 = 0$
You don't need to keep writing f(-1) at the start of each line.
And instead of writing this:

I now must find f(-1 + h).
f(-1 + h) = (-1 + h)^2 - 1
f(-1 + h) = 1 + - 2h + h^2 - 1
f(-1 + h) = h^2 - 2h

... you can write it like so:
$f(-1 + h) = (-1 + h)^2 - 1 = 1 + - 2h + h^2 - 1 = h^2 - 2h$

 

[nycmathguy]

A couple of tips to save yourself some typing. Instead of writing this:

... you can write it this way:
$f(-1) = (-1)^2 - 1 = 1 - 1 = 0$
You don't need to keep writing f(-1) at the start of each line.
And instead of writing this:

... you can write it like so:
$f(-1 + h) = (-1 + h)^2 - 1 = 1 + - 2h + h^2 - 1 = h^2 - 2h$

You are right but I like typing my work vertically like this:

f(x) = this
f(x) = that
f(x) = where
f(x) = who
f(x) = etc...

 

[Mark44]

Aside from the extra typing, doing what you're doing might make it harder to keep your focus on what you're trying to do.

Here is some of the work on finding the slope of the secant line between P(-1, f(-1)) and Q(-1 + h, f(-1 + h)).
The slope of this line segment is $\frac{\Delta y}{\Delta x}$, the change in function values divided by the change in x values.

$\frac{f(-1 + h) - f(-1)}{-1 + h - (-1)} = \frac{(-1 + h)^2 - 1 -[(-1)^2 - 1]}{-1 + h + 1}$
$= \frac{1 - 2h + h^2 - 1 - 0}h = \frac{-2h + h^2}h = \dots$

The way you're doing it, by calculating each expression in the numerator separately, increases the likelihood of losing track of what you're trying to accomplish. It's also a signal to readers that you are a noob at this. Does your textbook do things the way you're doing them?

 

[nycmathguy]

Aside from the extra typing, doing what you're doing might make it harder to keep your focus on what you're trying to do.

Here is some of the work on finding the slope of the secant line between P(-1, f(-1)) and Q(-1 + h, f(-1 + h)).
The slope of this line segment is $\frac{\Delta y}{\Delta x}$, the change in function values divided by the change in x values.

$\frac{f(-1 + h) - f(-1)}{-1 + h - (-1)} = \frac{(-1 + h)^2 - 1 -[(-1)^2 - 1]}{-1 + h + 1}$
$= \frac{1 - 2h + h^2 - 1 - 0}h = \frac{-2h + h^2}h = \dots$

The way you're doing it, by calculating each expression in the numerator separately, increases the likelihood of losing track of what you're trying to accomplish. It's also a signal to readers that you are a noob at this. Does your textbook do things the way you're doing them?

What is a noob? I like to break down my work as much as possible for students who may be visiting the site, thinking about joining and who might appreciate the step by step approach. It takes a long time finally reach the final answer but a few people might appreciate everything I do.

 

[Mark44]

What is a noob?

Slang for someone new to something. The term has been around for ages.

I like to break down my work as much as possible for students who may be visiting the site, thinking about joining and who might appreciate the step by step approach.

My comment is not about skipping steps, but rather the extra cruft that readers have to wade through to grasp what you're doing.

IMO, writing for potential visitors to the site is not something that should concern you. Of much more importance is the people who are actually reading your posts.

If you really want to get better at math, you ought to take some of the comments to heart. Most of the people helping you are much further along in their mathematical knowledge, and some, like myself, are much older than you.

 

[nycmathguy]

Slang for someone new to something. The term has been around for ages.My comment is not about skipping steps, but rather the extra cruft that readers have to wade through to grasp what you're doing.

IMO, writing for potential visitors to the site is not something that should concern you. Of much more importance is the people who are actually reading your posts.

If you really want to get better at math, you ought to take some of the comments to heart. Most of the people helping you are much further along in their mathematical knowledge, and some, like myself, are much older than you.

I am not good at computer abbreviations. What is IMO? Some members are older than me at 56. What do you say to people who are curious about your passion for math at your age? My family thinks I need mental help for answering textbook questions for the sake of learning.

They seen zero practicality in what I am doing. My friends and family think about learning but only to make more money. They YEARN to know why a 56 year old security guard with three college degrees cares about mathematics.

 

[Mark44]

I am not good at computer abbreviations. What is IMO?

IMO == In my opinion

Some members are older than me at 56. What do you say to people who are curious about your passion for math at your age?

It doesn't usually come up. I've been passionate about mathematics since about age 9.

They YEARN to know why a 56 year old security guard with three college degrees cares about mathematics.

Sorry to hear that your family and friends have no interest in learning and improving one's mind.

 

[nycmathguy]

IMO == In my opinion

It doesn't usually come up. I've been passionate about mathematics since about age 9.

Sorry to hear that your family and friends have no interest in learning and improving one's mind.

This is the story of my life. My friends and family only care about the practicality of education. If learning A and B does not lead to C, where C is money, then it's useless to them.