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#1 2021-04-11 06:22:15

mathland
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Registered: 2021-03-25
Posts: 438

Rate of Change of Quadratic Function

Show that the rate of change of a quadratic
function f (x) = ax^2 + bx + c is a linear function of x.

Can someone get me started here?

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#2 2021-04-11 19:22:20

Bob
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Registered: 2010-06-20
Posts: 9,128

Re: Rate of Change of Quadratic Function

hi

Have you met differential calculus yet?

It's here: https://www.mathsisfun.com/calculus/der … ction.html

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#3 2021-04-12 10:02:23

mathland
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Registered: 2021-03-25
Posts: 438

Re: Rate of Change of Quadratic Function

Bob wrote:

hi

Have you met differential calculus yet?

It's here: https://www.mathsisfun.com/calculus/der … ction.html

Bob

I just started differential calculus. Can you set this one up for me?

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#4 2021-04-12 20:35:14

Bob
Administrator
Registered: 2010-06-20
Posts: 9,128

Re: Rate of Change of Quadratic Function

Ok let's have a go.  The function is  f (x) = ax^2 + bx + c .  If you take the general point on the curve it has coordinates

(x, ax^2 + bx + c)  I'll call that point A.

We then pick a new point B that is close to A.  Newton was one of two mathematicians who invented calculus. The other was Leibniz.  They both came up with the idea at about the same time and a controversy raged between them about who first had the idea and whether the other 'stole' it.  I'll used Newton's notation.  He invented a new symbol  Δx to mean 'a little bit' in the x direction.  It's a single algebraic symbol not two multiplied together.

So the coordinates of B are [x +  Δx, a(x +  Δx)^2 + b(x+ Δx) + c]  I've only used square brackets for the outer to avoid confusion with the round brackets .

So the gradient of the chord AB is

( a(x +  Δx)^2 + b(x+ Δx) + c - ax^2 - bx - c) / (x + Δx - x)

Have a go at simplifying that.  It will probably help if you expand the bracket (x + Δx)^2 first.

You'll find there's lots that cancels out and finally you can divide Δx into the numerator.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#5 2021-04-14 12:21:20

mathland
Banned
Registered: 2021-03-25
Posts: 438

Re: Rate of Change of Quadratic Function

Bob wrote:

Ok let's have a go.  The function is  f (x) = ax^2 + bx + c .  If you take the general point on the curve it has coordinates

(x, ax^2 + bx + c)  I'll call that point A.

We then pick a new point B that is close to A.  Newton was one of two mathematicians who invented calculus. The other was Leibniz.  They both came up with the idea at about the same time and a controversy raged between them about who first had the idea and whether the other 'stole' it.  I'll used Newton's notation.  He invented a new symbol  Δx to mean 'a little bit' in the x direction.  It's a single algebraic symbol not two multiplied together.

So the coordinates of B are [x +  Δx, a(x +  Δx)^2 + b(x+ Δx) + c]  I've only used square brackets for the outer to avoid confusion with the round brackets .

So the gradient of the chord AB is

( a(x +  Δx)^2 + b(x+ Δx) + c - ax^2 - bx - c) / (x + Δx - x)

Have a go at simplifying that.  It will probably help if you expand the bracket (x + Δx)^2 first.

You'll find there's lots that cancels out and finally you can divide Δx into the numerator.

Bob


Cool reply. I thank you for taking the time to explain what is happening here. I will try to simplify this algebraic fraction when time allows.

Questions:

1. Are you a math professor?

2. If you are a math professor, do you teach math beyond Calculus 3 aka Vector Calculus?

3. There is a huge difference between what is called the REGULAR CALCULUS SEQUENCE (CALCULUS l, ll, and lll) and Advanced Calculus. Do you also teach Advanced Calculus? I think the course also goes by the name Tensor Calculus. 

4. Are you on FB?

5. Do you have your own math website?

6. I, like most students, struggle with word problems, especially SAT, GRE and GMAT word problems. Do I plan to take the SAT, GRE AND GMAT? No. I am 56 years old but the word problems found in test prep books for the SAT, GRE AND GMAT make good practice problems. What is the best way to improve word problems skills? My greatest struggle is converting WORDS to algebraic expressions or equations.

7. Do you think it is normal for a man my age to have a passion for learning math, even though I am never going to become a math teacher?

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#6 2021-04-14 20:47:07

Bob
Administrator
Registered: 2010-06-20
Posts: 9,128

Re: Rate of Change of Quadratic Function

1. Are you a math professor?

No.  after getting a maths degree I became a teacher (11-18).

2. If you are a math professor, do you teach math beyond Calculus 3 aka Vector Calculus?

No.  I'd have to look up what that is.

3. There is a huge difference between what is called the REGULAR CALCULUS SEQUENCE (CALCULUS l, ll, and lll) and Advanced Calculus. Do you also teach Advanced Calculus? I think the course also goes by the name Tensor Calculus.

Never done tensors.

4. Are you on FB?

What is FB?

5. Do you have your own math website?

Used to when I taught.  It was part of my broadband package.  When my A level ICT students constructed a website I would publish it live for them on my site, so they could see if it really did as expected and get another student to test it for them.  When I switched broadband provider, I had to give up the site.

6. I, like most students, struggle with word problems, especially SAT, GRE and GMAT word problems. Do I plan to take the SAT, GRE AND GMAT? No. I am 56 years old but the word problems found in test prep books for the SAT, GRE AND GMAT make good practice problems. What is the best way to improve word problems skills? My greatest struggle is converting WORDS to algebraic expressions or equations.

I don't know what any of those acronyms stand for. 

MIF has a page on this https://www.mathsisfun.com/algebra/word … lving.html

Here's one of the problems from that page.  I'll take you through how I would solve it.

The denominator of a fraction is 3 more than the numerator. If both the numerator and the denominator are increased by 4, the fraction becomes 4/5
What was the original fraction?

Let one of the unknowns be x.  Then 'translate' the information into expressions with x in.

Let the original numerator be x.

The denominator is 3 more, so it is x+3

Both are increased by another 4, so the new numerator becomes x + 4, and the new denominator becomes x + 7

So now the fraction is (x+4)/(x+7) and we're told that is 4/5.  So now I can make and solve an equation.

Multiply by 5(x+7)

Check this works.

Original was 8/11.  New is 12/15 = 4/5 Tick.

7. Do you think it is normal for a man my age to have a passion for learning math, even though I am never going to become a math teacher?

Not many would want to do this but I think it's great that you want to.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#7 2021-04-15 15:04:27

mathland
Banned
Registered: 2021-03-25
Posts: 438

Re: Rate of Change of Quadratic Function

Bob wrote:

1. Are you a math professor?

No.  after getting a maths degree I became a teacher (11-18).

2. If you are a math professor, do you teach math beyond Calculus 3 aka Vector Calculus?

No.  I'd have to look up what that is.

3. There is a huge difference between what is called the REGULAR CALCULUS SEQUENCE (CALCULUS l, ll, and lll) and Advanced Calculus. Do you also teach Advanced Calculus? I think the course also goes by the name Tensor Calculus.

Never done tensors.

4. Are you on FB?

What is FB?

5. Do you have your own math website?

Used to when I taught.  It was part of my broadband package.  When my A level ICT students constructed a website I would publish it live for them on my site, so they could see if it really did as expected and get another student to test it for them.  When I switched broadband provider, I had to give up the site.

6. I, like most students, struggle with word problems, especially SAT, GRE and GMAT word problems. Do I plan to take the SAT, GRE AND GMAT? No. I am 56 years old but the word problems found in test prep books for the SAT, GRE AND GMAT make good practice problems. What is the best way to improve word problems skills? My greatest struggle is converting WORDS to algebraic expressions or equations.

I don't know what any of those acronyms stand for. 

MIF has a page on this https://www.mathsisfun.com/algebra/word … lving.html

Here's one of the problems from that page.  I'll take you through how I would solve it.

The denominator of a fraction is 3 more than the numerator. If both the numerator and the denominator are increased by 4, the fraction becomes 4/5
What was the original fraction?

Let one of the unknowns be x.  Then 'translate' the information into expressions with x in.

Let the original numerator be x.

The denominator is 3 more, so it is x+3

Both are increased by another 4, so the new numerator becomes x + 4, and the new denominator becomes x + 7

So now the fraction is (x+4)/(x+7) and we're told that is 4/5.  So now I can make and solve an equation.

Multiply by 5(x+7)

Check this works.

Original was 8/11.  New is 12/15 = 4/5 Tick.

7. Do you think it is normal for a man my age to have a passion for learning math, even though I am never going to become a math teacher?

Not many would want to do this but I think it's great that you want to.

Bob

Hello Bob. See below.

1. By 11-18 you mean ages 11 to 18, right? This is your teaching age category. Yes?

2. Vector Calculus is known as Calculus 3 in America. It is the Calculus that deals with vectors, points in space, 3D graphs, double and triple integrals, etc.

3. FB = FACEBOOK. Are you on FB?

4. If you are on FB, I can create a math group for us to further discuss mathematics.

5. SAT, GRE AND GMAT are prerequisite exams.
High school students take the SAT to get placed into advanced classes or to earn college credits. The GRE stands for GRADUATE RECORD EXAMINATION. The GRE is taken by students wishing to enter Graduate School. However, most Graduate Schools no longer require the GRE as the demand for more students has become a great need for colleges. The GMAT is taken by people wanting to enter certain business schools. I like the SAT, GRE AND GMAT practice word problems. Very challenging but fun.

6. The practice word problem you posted is easy. The SAT, GRE AND GMAT problems are more involved. I will post a few questions in the coming days, if time allows.

7. I get criticized by friends and family for studying math. Why? What's the big crime? I don't get it. I am not doing anything wrong. I am not hurting anyone.

I am simply learning advanced material that fascinates me. Besides, I am a 56, lonely, family not too close, my adult son living his own life, etc. What else am I to do at 56? Math keeps my memory power in tact, at least I hope it does.

People do not like to see others happy. I had the same problem when I decided to enroll in the Family Radio School of the Bible many years ago. Friends and family criticizing my study time with Scripture. Why?

Bob, I am a loner. Most loners are misunderstood people. I am most happy in my alone time. Sociable people envy loners. Sociable people don't understand why loners are happy when alone. This is a puzzle for the gregarious bunch. Anyway, let me know about FB.

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#8 2021-04-15 20:16:52

Bob
Administrator
Registered: 2010-06-20
Posts: 9,128

Re: Rate of Change of Quadratic Function

No Facebook.  I am horrified by how intrusive membership of Facebook is.  Too much personal information shared widely.  The scandal involving Cambridge Analytica and Facebook trying to fix election results just shows how dangerous it is.

Shortly after the "Leave the EU" referendum result the BBC had a Question Time programme and the topic of the vote came up.  A member of the audience was not embarrassed to announce that she was a supporter of 'remain' right up to the day of the vote.  Then she thought about those stories that the EU has a regulation forbidding bent bananas and decided to switch to 'leave'.  That fake news came from social media.

For a special limited reason I did briefly join under a fake name.  Despite entering false information about myself, and keeping my data private, I still received Friends Requests from people I know.  I cancelled my membership.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#9 2021-04-16 07:21:24

mathland
Banned
Registered: 2021-03-25
Posts: 438

Re: Rate of Change of Quadratic Function

Bob wrote:

No Facebook.  I am horrified by how intrusive membership of Facebook is.  Too much personal information shared widely.  The scandal involving Cambridge Analytica and Facebook trying to fix election results just shows how dangerous it is.

Shortly after the "Leave the EU" referendum result the BBC had a Question Time programme and the topic of the vote came up.  A member of the audience was not embarrassed to announce that she was a supporter of 'remain' right up to the day of the vote.  Then she thought about those stories that the EU has a regulation forbidding bent bananas and decided to switch to 'leave'.  That fake news came from social media.

For a special limited reason I did briefly join under a fake name.  Despite entering false information about myself, and keeping my data private, I still received Friends Requests from people I know.  I cancelled my membership.

Bob

I myself may leave FB soon. I am a Trump guy. FB is a far left social media that has connected me with friends of long ago. Still, FB greatly dislikes Trump. I fear what America will be like after 4 years under the Biden administration.  SAT, GRE, AND GMAT word problems this entire weekend. I will show my work. Ready to have fun with math this weekend?

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