(Calculus) Derivative Thinking QuestionAP Calculus DerivativeTextual explanation of a derivativeFor what values of $a$ will $y=ax$ be a tangent to $x^2+y^2+20x-10y+100=0$How to find the slope of curves at origin if the derivative becomes indeterminateTangent line at point PProof of tangent lines to a curveHow can I find m using the discriminant?Find equation of tangent to the circleIf first derivative of a point on curve gives the slope of tangent,what does the second derivative give,slope of tangent,i.e zero?Calculus and Vectors Question
Define, (actually define) the "stability" and "energy" of a compound
Property of summation
How difficult is it to simply disable/disengage the MCAS on Boeing 737 Max 8 & 9 Aircraft?
How to deal with taxi scam when on vacation?
What's the meaning of “spike” in the context of “adrenaline spike”?
Does Mathematica reuse previous computations?
Python if-else code style for reduced code for rounding floats
Declaring defaulted assignment operator as constexpr: which compiler is right?
What do Xenomorphs eat in the Alien series?
Use void Apex method in Lightning Web Component
Is it possible to upcast ritual spells?
How to make healing in an exploration game interesting
Is it normal that my co-workers at a fitness company criticize my food choices?
Is it true that good novels will automatically sell themselves on Amazon (and so on) and there is no need for one to waste time promoting?
Did Ender ever learn that he killed Stilson and/or Bonzo?
Are there other languages, besides English, where the indefinite (or definite) article varies based on sound?
Look at your watch and tell me what time is it. vs Look at your watch and tell me what time it is
Is a party consisting of only a bard, a cleric, and a warlock functional long-term?
A limit with limit zero everywhere must be zero somewhere
Why does Bach not break the rules here?
How to read the value of this capacitor?
Should we release the security issues we found in our product as CVE or we can just update those on weekly release notes?
What did Alexander Pope mean by "Expletives their feeble Aid do join"?
Do I need to be arrogant to get ahead?
(Calculus) Derivative Thinking Question
AP Calculus DerivativeTextual explanation of a derivativeFor what values of $a$ will $y=ax$ be a tangent to $x^2+y^2+20x-10y+100=0$How to find the slope of curves at origin if the derivative becomes indeterminateTangent line at point PProof of tangent lines to a curveHow can I find m using the discriminant?Find equation of tangent to the circleIf first derivative of a point on curve gives the slope of tangent,what does the second derivative give,slope of tangent,i.e zero?Calculus and Vectors Question
$begingroup$
Recently, my Calculus and Vectors (Grade 12) teacher gave our class a thinking question/assignment to work on over the march break, and after working on for some time, I've become stuck on it.
The Question:
Consider f(x), a general quadratic function in standard form, and g(x) its reciprocal. For which values of x are the slopes of their respective tangent lines equal?
Consider two cases: one where it is true for exactly one value of x, and the other where it is true for exactly two values of x. In the latter case, you can assume that the steepness a does not equal 0 of f(x) is equal to both it’s y-intercept and also to the slope of its tangent at x = 1.
Find the required conditions on the parameters a,b,c in terms of a.
My Progress So Far:
So I know that $f(x) = ax^2 +bx + c$ and $g(x) = 1/f(x)$. After I solved for the derivative of each function, I set them both equal to each other and started solving for it. But then I came to a equation of $(ax^2 + bx + c)^2 = -1$ and that doesn't work.
Next I tried something else. Since I know from the 2nd case that $f(x) = c$ and $f(x) = f'(1)$ when a cannot equal 0, I set $c = f'(1)$ and got $c = 2a + b$. But after that, I don't know where to go.
I'm not expecting a full solution, but if anyone could give me a hint for solving this question, I would appreciate it. I know that it says to write everything in terms of a, but I'm not sure how to approach that method.
calculus derivatives quadratics tangent-line slope
edited 50 mins ago
Jon due
948
asked 1 hour ago
Rasheed AmanzaiRasheed Amanzai
213
New contributor
Rasheed Amanzai is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
Recently, my Calculus and Vectors (Grade 12) teacher gave our class a thinking question/assignment to work on over the march break, and after working on for some time, I've become stuck on it.
The Question:
Consider f(x), a general quadratic function in standard form, and g(x) its reciprocal. For which values of x are the slopes of their respective tangent lines equal?
Consider two cases: one where it is true for exactly one value of x, and the other where it is true for exactly two values of x. In the latter case, you can assume that the steepness a does not equal 0 of f(x) is equal to both it’s y-intercept and also to the slope of its tangent at x = 1.
Find the required conditions on the parameters a,b,c in terms of a.
My Progress So Far:
So I know that $f(x) = ax^2 +bx + c$ and $g(x) = 1/f(x)$. After I solved for the derivative of each function, I set them both equal to each other and started solving for it. But then I came to a equation of $(ax^2 + bx + c)^2 = -1$ and that doesn't work.
Next I tried something else. Since I know from the 2nd case that $f(x) = c$ and $f(x) = f'(1)$ when a cannot equal 0, I set $c = f'(1)$ and got $c = 2a + b$. But after that, I don't know where to go.
I'm not expecting a full solution, but if anyone could give me a hint for solving this question, I would appreciate it. I know that it says to write everything in terms of a, but I'm not sure how to approach that method.
calculus derivatives quadratics tangent-line slope
edited 50 mins ago
Jon due
948
asked 1 hour ago
Rasheed AmanzaiRasheed Amanzai
213
New contributor
Rasheed Amanzai is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
I made a typo in solving and so my prior comment did not make sense. To get your equation $f(x)^2=-1$ you likely made a cancellation on both sides, but what happens if the term you canceled is zero?
$endgroup$
– Michael
1 hour ago
$begingroup$
The term that I cancelled was 2ax+b (which is f'(x)), I don't know if the term cancelled can be zero since I do not know the unknowns. (I apologize if I didn't answer your question).
$endgroup$
– Rasheed Amanzai
1 hour ago
1
$begingroup$
So you got $-f'(x)=f'(x)f(x)^2$ and want to find an $x$ where this is possible. It is not possible if $f'(x) neq 0$ as you already showed and so the only way possible is if $f'(x)=0$. So then for what values of $a,b$ is it possible to have $2ax+b=0$?
$endgroup$
– Michael
55 mins ago
$begingroup$
Now that made a lot more sense to me. So then the values of a and b would both have to equal 0 in order to get f'(x) = 0.
$endgroup$
– Rasheed Amanzai
31 mins ago
add a comment |
$begingroup$
Recently, my Calculus and Vectors (Grade 12) teacher gave our class a thinking question/assignment to work on over the march break, and after working on for some time, I've become stuck on it.
The Question:
Consider f(x), a general quadratic function in standard form, and g(x) its reciprocal. For which values of x are the slopes of their respective tangent lines equal?
Consider two cases: one where it is true for exactly one value of x, and the other where it is true for exactly two values of x. In the latter case, you can assume that the steepness a does not equal 0 of f(x) is equal to both it’s y-intercept and also to the slope of its tangent at x = 1.
Find the required conditions on the parameters a,b,c in terms of a.
My Progress So Far:
So I know that $f(x) = ax^2 +bx + c$ and $g(x) = 1/f(x)$. After I solved for the derivative of each function, I set them both equal to each other and started solving for it. But then I came to a equation of $(ax^2 + bx + c)^2 = -1$ and that doesn't work.
Next I tried something else. Since I know from the 2nd case that $f(x) = c$ and $f(x) = f'(1)$ when a cannot equal 0, I set $c = f'(1)$ and got $c = 2a + b$. But after that, I don't know where to go.
I'm not expecting a full solution, but if anyone could give me a hint for solving this question, I would appreciate it. I know that it says to write everything in terms of a, but I'm not sure how to approach that method.
calculus derivatives quadratics tangent-line slope
edited 50 mins ago
Jon due
948
asked 1 hour ago
Rasheed AmanzaiRasheed Amanzai
213
New contributor
Rasheed Amanzai is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
Recently, my Calculus and Vectors (Grade 12) teacher gave our class a thinking question/assignment to work on over the march break, and after working on for some time, I've become stuck on it.
The Question:
Consider f(x), a general quadratic function in standard form, and g(x) its reciprocal. For which values of x are the slopes of their respective tangent lines equal?
Consider two cases: one where it is true for exactly one value of x, and the other where it is true for exactly two values of x. In the latter case, you can assume that the steepness a does not equal 0 of f(x) is equal to both it’s y-intercept and also to the slope of its tangent at x = 1.
Find the required conditions on the parameters a,b,c in terms of a.
My Progress So Far:
So I know that $f(x) = ax^2 +bx + c$ and $g(x) = 1/f(x)$. After I solved for the derivative of each function, I set them both equal to each other and started solving for it. But then I came to a equation of $(ax^2 + bx + c)^2 = -1$ and that doesn't work.
Next I tried something else. Since I know from the 2nd case that $f(x) = c$ and $f(x) = f'(1)$ when a cannot equal 0, I set $c = f'(1)$ and got $c = 2a + b$. But after that, I don't know where to go.
I'm not expecting a full solution, but if anyone could give me a hint for solving this question, I would appreciate it. I know that it says to write everything in terms of a, but I'm not sure how to approach that method.
calculus derivatives quadratics tangent-line slope
calculus derivatives quadratics tangent-line slope
edited 50 mins ago
Jon due
948
asked 1 hour ago
Rasheed AmanzaiRasheed Amanzai
213
New contributor
Rasheed Amanzai is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 50 mins ago
Jon due
948
asked 1 hour ago
Rasheed AmanzaiRasheed Amanzai
213
New contributor
Rasheed Amanzai is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 50 mins ago
Jon due
948
edited 50 mins ago
Jon due
948
edited 50 mins ago
Jon due
948
948
asked 1 hour ago
Rasheed AmanzaiRasheed Amanzai
213
New contributor
Rasheed Amanzai is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 1 hour ago
Rasheed AmanzaiRasheed Amanzai
213
asked 1 hour ago
Rasheed AmanzaiRasheed Amanzai
213
213
New contributor
Rasheed Amanzai is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Rasheed Amanzai is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Rasheed Amanzai is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$begingroup$
I made a typo in solving and so my prior comment did not make sense. To get your equation $f(x)^2=-1$ you likely made a cancellation on both sides, but what happens if the term you canceled is zero?
$endgroup$
– Michael
1 hour ago
$begingroup$
The term that I cancelled was 2ax+b (which is f'(x)), I don't know if the term cancelled can be zero since I do not know the unknowns. (I apologize if I didn't answer your question).
$endgroup$
– Rasheed Amanzai
1 hour ago
1
$begingroup$
So you got $-f'(x)=f'(x)f(x)^2$ and want to find an $x$ where this is possible. It is not possible if $f'(x) neq 0$ as you already showed and so the only way possible is if $f'(x)=0$. So then for what values of $a,b$ is it possible to have $2ax+b=0$?
$endgroup$
– Michael
55 mins ago
$begingroup$
Now that made a lot more sense to me. So then the values of a and b would both have to equal 0 in order to get f'(x) = 0.
$endgroup$
– Rasheed Amanzai
31 mins ago
add a comment |
$begingroup$
I made a typo in solving and so my prior comment did not make sense. To get your equation $f(x)^2=-1$ you likely made a cancellation on both sides, but what happens if the term you canceled is zero?
$endgroup$
– Michael
1 hour ago
$begingroup$
The term that I cancelled was 2ax+b (which is f'(x)), I don't know if the term cancelled can be zero since I do not know the unknowns. (I apologize if I didn't answer your question).
$endgroup$
– Rasheed Amanzai
1 hour ago
1
$begingroup$
So you got $-f'(x)=f'(x)f(x)^2$ and want to find an $x$ where this is possible. It is not possible if $f'(x) neq 0$ as you already showed and so the only way possible is if $f'(x)=0$. So then for what values of $a,b$ is it possible to have $2ax+b=0$?
$endgroup$
– Michael
55 mins ago
$begingroup$
Now that made a lot more sense to me. So then the values of a and b would both have to equal 0 in order to get f'(x) = 0.
$endgroup$
– Rasheed Amanzai
31 mins ago
$begingroup$
I made a typo in solving and so my prior comment did not make sense. To get your equation $f(x)^2=-1$ you likely made a cancellation on both sides, but what happens if the term you canceled is zero?
$endgroup$
– Michael
1 hour ago
$begingroup$
I made a typo in solving and so my prior comment did not make sense. To get your equation $f(x)^2=-1$ you likely made a cancellation on both sides, but what happens if the term you canceled is zero?
$endgroup$
– Michael
1 hour ago
$begingroup$
The term that I cancelled was 2ax+b (which is f'(x)), I don't know if the term cancelled can be zero since I do not know the unknowns. (I apologize if I didn't answer your question).
$endgroup$
– Rasheed Amanzai
1 hour ago
$begingroup$
The term that I cancelled was 2ax+b (which is f'(x)), I don't know if the term cancelled can be zero since I do not know the unknowns. (I apologize if I didn't answer your question).
$endgroup$
– Rasheed Amanzai
1 hour ago
1
1
$begingroup$
So you got $-f'(x)=f'(x)f(x)^2$ and want to find an $x$ where this is possible. It is not possible if $f'(x) neq 0$ as you already showed and so the only way possible is if $f'(x)=0$. So then for what values of $a,b$ is it possible to have $2ax+b=0$?
$endgroup$
– Michael
55 mins ago
$begingroup$
So you got $-f'(x)=f'(x)f(x)^2$ and want to find an $x$ where this is possible. It is not possible if $f'(x) neq 0$ as you already showed and so the only way possible is if $f'(x)=0$. So then for what values of $a,b$ is it possible to have $2ax+b=0$?
$endgroup$
– Michael
55 mins ago
$begingroup$
Now that made a lot more sense to me. So then the values of a and b would both have to equal 0 in order to get f'(x) = 0.
$endgroup$
– Rasheed Amanzai
31 mins ago
$begingroup$
Now that made a lot more sense to me. So then the values of a and b would both have to equal 0 in order to get f'(x) = 0.
$endgroup$
– Rasheed Amanzai
31 mins ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
If $g(x)=1/f(x)$, then, by the chain rule,
$$
g'(x)=-fracf'(x)f(x)^2
$$
Thus, assuming of course that $f(x)ne0$, we have $g'(x)=f'(x)$ if and only if
$$
-fracf'(x)f(x)^2=f'(x)
$$
which can only happen if $f'(x)=0$. Indeed, if $f'(x)ne0$, the equation becomes $f(x)^2=-1$, which is exactly the condition you find.
You can observe that this is independent of $f$ being a quadratic polynomial.
In your particular case, the condition is $x=-b/(2a)$, provided that $f(-b/(2a))ne0$ (or $g(-b/(2a))$ would be undefined.
answered 1 hour ago
egregegreg
184k1486205
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Rasheed Amanzai is a new contributor. Be nice, and check out our Code of Conduct.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3149858%2fcalculus-derivative-thinking-question%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
If $g(x)=1/f(x)$, then, by the chain rule,
$$
g'(x)=-fracf'(x)f(x)^2
$$
Thus, assuming of course that $f(x)ne0$, we have $g'(x)=f'(x)$ if and only if
$$
-fracf'(x)f(x)^2=f'(x)
$$
which can only happen if $f'(x)=0$. Indeed, if $f'(x)ne0$, the equation becomes $f(x)^2=-1$, which is exactly the condition you find.
You can observe that this is independent of $f$ being a quadratic polynomial.
In your particular case, the condition is $x=-b/(2a)$, provided that $f(-b/(2a))ne0$ (or $g(-b/(2a))$ would be undefined.
answered 1 hour ago
egregegreg
184k1486205
$endgroup$
add a comment |
$begingroup$
If $g(x)=1/f(x)$, then, by the chain rule,
$$
g'(x)=-fracf'(x)f(x)^2
$$
Thus, assuming of course that $f(x)ne0$, we have $g'(x)=f'(x)$ if and only if
$$
-fracf'(x)f(x)^2=f'(x)
$$
which can only happen if $f'(x)=0$. Indeed, if $f'(x)ne0$, the equation becomes $f(x)^2=-1$, which is exactly the condition you find.
You can observe that this is independent of $f$ being a quadratic polynomial.
In your particular case, the condition is $x=-b/(2a)$, provided that $f(-b/(2a))ne0$ (or $g(-b/(2a))$ would be undefined.
answered 1 hour ago
egregegreg
184k1486205
$endgroup$
add a comment |
$begingroup$
If $g(x)=1/f(x)$, then, by the chain rule,
$$
g'(x)=-fracf'(x)f(x)^2
$$
Thus, assuming of course that $f(x)ne0$, we have $g'(x)=f'(x)$ if and only if
$$
-fracf'(x)f(x)^2=f'(x)
$$
which can only happen if $f'(x)=0$. Indeed, if $f'(x)ne0$, the equation becomes $f(x)^2=-1$, which is exactly the condition you find.
You can observe that this is independent of $f$ being a quadratic polynomial.
In your particular case, the condition is $x=-b/(2a)$, provided that $f(-b/(2a))ne0$ (or $g(-b/(2a))$ would be undefined.
answered 1 hour ago
egregegreg
184k1486205
$endgroup$
If $g(x)=1/f(x)$, then, by the chain rule,
$$
g'(x)=-fracf'(x)f(x)^2
$$
Thus, assuming of course that $f(x)ne0$, we have $g'(x)=f'(x)$ if and only if
$$
-fracf'(x)f(x)^2=f'(x)
$$
which can only happen if $f'(x)=0$. Indeed, if $f'(x)ne0$, the equation becomes $f(x)^2=-1$, which is exactly the condition you find.
You can observe that this is independent of $f$ being a quadratic polynomial.
In your particular case, the condition is $x=-b/(2a)$, provided that $f(-b/(2a))ne0$ (or $g(-b/(2a))$ would be undefined.
answered 1 hour ago
egregegreg
184k1486205
answered 1 hour ago
egregegreg
184k1486205
answered 1 hour ago
egregegreg
184k1486205
answered 1 hour ago
egregegreg
184k1486205
184k1486205
add a comment |
add a comment |
Rasheed Amanzai is a new contributor. Be nice, and check out our Code of Conduct.
Rasheed Amanzai is a new contributor. Be nice, and check out our Code of Conduct.
Rasheed Amanzai is a new contributor. Be nice, and check out our Code of Conduct.
Rasheed Amanzai is a new contributor. Be nice, and check out our Code of Conduct.
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3149858%2fcalculus-derivative-thinking-question%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
I made a typo in solving and so my prior comment did not make sense. To get your equation $f(x)^2=-1$ you likely made a cancellation on both sides, but what happens if the term you canceled is zero?
$endgroup$
– Michael
1 hour ago
$begingroup$
The term that I cancelled was 2ax+b (which is f'(x)), I don't know if the term cancelled can be zero since I do not know the unknowns. (I apologize if I didn't answer your question).
$endgroup$
– Rasheed Amanzai
1 hour ago
1
$begingroup$
So you got $-f'(x)=f'(x)f(x)^2$ and want to find an $x$ where this is possible. It is not possible if $f'(x) neq 0$ as you already showed and so the only way possible is if $f'(x)=0$. So then for what values of $a,b$ is it possible to have $2ax+b=0$?
$endgroup$
– Michael
55 mins ago
$begingroup$
Now that made a lot more sense to me. So then the values of a and b would both have to equal 0 in order to get f'(x) = 0.
$endgroup$
– Rasheed Amanzai
31 mins ago