How to Solve this Differential Calculus Problem
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If the equation of the normal line to the curve $y = ax + b/x$ at the point $(2,7)$ is $y+ 2x = 11$, find the value of $a$ and $b$. Given that this normal line meets the curve again at $P$, find the coordinates of $P$.
I found the tangent equation to be $y= 1/2 x + 6$. I inegrated that to get $y = x^2 /4 + 6x + c$. Then I found c by putting the values of the point in as 6.
(I am unsure from here forward) I think $ax + b/x$ would be $ax^2 + yx + b$ , meaning $a$ would be $1/4$ and $b$ would be -7.
I do not know what to do to find the other intersection of the normal, given that I am unsure that my answers for $a$ and $b$ are accurate
calculus integration differential-equations
asked Nov 17 at 21:32
Noel Jonathan Ellis
134
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up vote
1
down vote
favorite
If the equation of the normal line to the curve $y = ax + b/x$ at the point $(2,7)$ is $y+ 2x = 11$, find the value of $a$ and $b$. Given that this normal line meets the curve again at $P$, find the coordinates of $P$.
I found the tangent equation to be $y= 1/2 x + 6$. I inegrated that to get $y = x^2 /4 + 6x + c$. Then I found c by putting the values of the point in as 6.
(I am unsure from here forward) I think $ax + b/x$ would be $ax^2 + yx + b$ , meaning $a$ would be $1/4$ and $b$ would be -7.
I do not know what to do to find the other intersection of the normal, given that I am unsure that my answers for $a$ and $b$ are accurate
calculus integration differential-equations
asked Nov 17 at 21:32
Noel Jonathan Ellis
134
3
Can you show us what you have attempted? Or what is interesting about this question? Usually questions posed with the tone "do this homework problem for me" are not well-received.
– Mason
Nov 17 at 21:39
1
I found the tangent equation to be y= 1/2 x + 6. I inegrated that to get y = x^2 /4 + 6x + c. Then I found c by putting the values of the point in as -6.
– Noel Jonathan Ellis
Nov 17 at 22:00
(I am unsure from here forward) I think ax+b/x would be ax2+yx+b , meaning a would be 1/4 and b would be -6. I do not know what to do to find the other intersection of the normal, given that I am unsure that my answers for a and b are accurate
– Noel Jonathan Ellis
Nov 17 at 22:09
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
If the equation of the normal line to the curve $y = ax + b/x$ at the point $(2,7)$ is $y+ 2x = 11$, find the value of $a$ and $b$. Given that this normal line meets the curve again at $P$, find the coordinates of $P$.
I found the tangent equation to be $y= 1/2 x + 6$. I inegrated that to get $y = x^2 /4 + 6x + c$. Then I found c by putting the values of the point in as 6.
(I am unsure from here forward) I think $ax + b/x$ would be $ax^2 + yx + b$ , meaning $a$ would be $1/4$ and $b$ would be -7.
I do not know what to do to find the other intersection of the normal, given that I am unsure that my answers for $a$ and $b$ are accurate
calculus integration differential-equations
asked Nov 17 at 21:32
Noel Jonathan Ellis
134
If the equation of the normal line to the curve $y = ax + b/x$ at the point $(2,7)$ is $y+ 2x = 11$, find the value of $a$ and $b$. Given that this normal line meets the curve again at $P$, find the coordinates of $P$.
I found the tangent equation to be $y= 1/2 x + 6$. I inegrated that to get $y = x^2 /4 + 6x + c$. Then I found c by putting the values of the point in as 6.
(I am unsure from here forward) I think $ax + b/x$ would be $ax^2 + yx + b$ , meaning $a$ would be $1/4$ and $b$ would be -7.
I do not know what to do to find the other intersection of the normal, given that I am unsure that my answers for $a$ and $b$ are accurate
calculus integration differential-equations
calculus integration differential-equations
asked Nov 17 at 21:32
Noel Jonathan Ellis
134
asked Nov 17 at 21:32
Noel Jonathan Ellis
134
edited Nov 17 at 22:25
asked Nov 17 at 21:32
Noel Jonathan Ellis
134
asked Nov 17 at 21:32
Noel Jonathan Ellis
134
asked Nov 17 at 21:32
Noel Jonathan Ellis
134
134
3
Can you show us what you have attempted? Or what is interesting about this question? Usually questions posed with the tone "do this homework problem for me" are not well-received.
– Mason
Nov 17 at 21:39
1
I found the tangent equation to be y= 1/2 x + 6. I inegrated that to get y = x^2 /4 + 6x + c. Then I found c by putting the values of the point in as -6.
– Noel Jonathan Ellis
Nov 17 at 22:00
(I am unsure from here forward) I think ax+b/x would be ax2+yx+b , meaning a would be 1/4 and b would be -6. I do not know what to do to find the other intersection of the normal, given that I am unsure that my answers for a and b are accurate
– Noel Jonathan Ellis
Nov 17 at 22:09
add a comment |
3
Can you show us what you have attempted? Or what is interesting about this question? Usually questions posed with the tone "do this homework problem for me" are not well-received.
– Mason
Nov 17 at 21:39
1
I found the tangent equation to be y= 1/2 x + 6. I inegrated that to get y = x^2 /4 + 6x + c. Then I found c by putting the values of the point in as -6.
– Noel Jonathan Ellis
Nov 17 at 22:00
(I am unsure from here forward) I think ax+b/x would be ax2+yx+b , meaning a would be 1/4 and b would be -6. I do not know what to do to find the other intersection of the normal, given that I am unsure that my answers for a and b are accurate
– Noel Jonathan Ellis
Nov 17 at 22:09
3
3
Can you show us what you have attempted? Or what is interesting about this question? Usually questions posed with the tone "do this homework problem for me" are not well-received.
– Mason
Nov 17 at 21:39
Can you show us what you have attempted? Or what is interesting about this question? Usually questions posed with the tone "do this homework problem for me" are not well-received.
– Mason
Nov 17 at 21:39
1
1
I found the tangent equation to be y= 1/2 x + 6. I inegrated that to get y = x^2 /4 + 6x + c. Then I found c by putting the values of the point in as -6.
– Noel Jonathan Ellis
Nov 17 at 22:00
I found the tangent equation to be y= 1/2 x + 6. I inegrated that to get y = x^2 /4 + 6x + c. Then I found c by putting the values of the point in as -6.
– Noel Jonathan Ellis
Nov 17 at 22:00
(I am unsure from here forward) I think ax+b/x would be ax2+yx+b , meaning a would be 1/4 and b would be -6. I do not know what to do to find the other intersection of the normal, given that I am unsure that my answers for a and b are accurate
– Noel Jonathan Ellis
Nov 17 at 22:09
(I am unsure from here forward) I think ax+b/x would be ax2+yx+b , meaning a would be 1/4 and b would be -6. I do not know what to do to find the other intersection of the normal, given that I am unsure that my answers for a and b are accurate
– Noel Jonathan Ellis
Nov 17 at 22:09
add a comment |
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
$f(x)=ax+b/x implies f'(x)=a-b/x^2$
And we know that $f(2)=7$ and (by your computation) $f'(2)=1/2$
So then $7= 2a+b/2$ and $1/2=a-b/4$. I think we should be able to solve the system of equations for $a$ and $b$.
We find $(a,b)=(2,6)$ and we can confirm this by checking out the graph.
answered Nov 17 at 22:13
Mason
1,7911426
So I just went the wrong way by integrating the tangent eqaution instead of differentiating the equation of the curve?
– Noel Jonathan Ellis
Nov 17 at 22:24
Yes. Integration isn't the way to go here because while you know your curve is tangent at that point. The tangent line isn't going to communicate any information about the curve in general.
– Mason
Nov 17 at 23:08
I see, thank you
– Noel Jonathan Ellis
Nov 17 at 23:17
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
$f(x)=ax+b/x implies f'(x)=a-b/x^2$
And we know that $f(2)=7$ and (by your computation) $f'(2)=1/2$
So then $7= 2a+b/2$ and $1/2=a-b/4$. I think we should be able to solve the system of equations for $a$ and $b$.
We find $(a,b)=(2,6)$ and we can confirm this by checking out the graph.
answered Nov 17 at 22:13
Mason
1,7911426
So I just went the wrong way by integrating the tangent eqaution instead of differentiating the equation of the curve?
– Noel Jonathan Ellis
Nov 17 at 22:24
Yes. Integration isn't the way to go here because while you know your curve is tangent at that point. The tangent line isn't going to communicate any information about the curve in general.
– Mason
Nov 17 at 23:08
I see, thank you
– Noel Jonathan Ellis
Nov 17 at 23:17
add a comment |
up vote
1
down vote
accepted
$f(x)=ax+b/x implies f'(x)=a-b/x^2$
And we know that $f(2)=7$ and (by your computation) $f'(2)=1/2$
So then $7= 2a+b/2$ and $1/2=a-b/4$. I think we should be able to solve the system of equations for $a$ and $b$.
We find $(a,b)=(2,6)$ and we can confirm this by checking out the graph.
answered Nov 17 at 22:13
Mason
1,7911426
So I just went the wrong way by integrating the tangent eqaution instead of differentiating the equation of the curve?
– Noel Jonathan Ellis
Nov 17 at 22:24
Yes. Integration isn't the way to go here because while you know your curve is tangent at that point. The tangent line isn't going to communicate any information about the curve in general.
– Mason
Nov 17 at 23:08
I see, thank you
– Noel Jonathan Ellis
Nov 17 at 23:17
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
$f(x)=ax+b/x implies f'(x)=a-b/x^2$
And we know that $f(2)=7$ and (by your computation) $f'(2)=1/2$
So then $7= 2a+b/2$ and $1/2=a-b/4$. I think we should be able to solve the system of equations for $a$ and $b$.
We find $(a,b)=(2,6)$ and we can confirm this by checking out the graph.
answered Nov 17 at 22:13
Mason
1,7911426
$f(x)=ax+b/x implies f'(x)=a-b/x^2$
And we know that $f(2)=7$ and (by your computation) $f'(2)=1/2$
So then $7= 2a+b/2$ and $1/2=a-b/4$. I think we should be able to solve the system of equations for $a$ and $b$.
We find $(a,b)=(2,6)$ and we can confirm this by checking out the graph.
answered Nov 17 at 22:13
Mason
1,7911426
edited Nov 17 at 22:19
answered Nov 17 at 22:13
Mason
1,7911426
answered Nov 17 at 22:13
Mason
1,7911426
answered Nov 17 at 22:13
Mason
1,7911426
1,7911426
So I just went the wrong way by integrating the tangent eqaution instead of differentiating the equation of the curve?
– Noel Jonathan Ellis
Nov 17 at 22:24
Yes. Integration isn't the way to go here because while you know your curve is tangent at that point. The tangent line isn't going to communicate any information about the curve in general.
– Mason
Nov 17 at 23:08
I see, thank you
– Noel Jonathan Ellis
Nov 17 at 23:17
add a comment |
So I just went the wrong way by integrating the tangent eqaution instead of differentiating the equation of the curve?
– Noel Jonathan Ellis
Nov 17 at 22:24
Yes. Integration isn't the way to go here because while you know your curve is tangent at that point. The tangent line isn't going to communicate any information about the curve in general.
– Mason
Nov 17 at 23:08
I see, thank you
– Noel Jonathan Ellis
Nov 17 at 23:17
So I just went the wrong way by integrating the tangent eqaution instead of differentiating the equation of the curve?
– Noel Jonathan Ellis
Nov 17 at 22:24
So I just went the wrong way by integrating the tangent eqaution instead of differentiating the equation of the curve?
– Noel Jonathan Ellis
Nov 17 at 22:24
Yes. Integration isn't the way to go here because while you know your curve is tangent at that point. The tangent line isn't going to communicate any information about the curve in general.
– Mason
Nov 17 at 23:08
Yes. Integration isn't the way to go here because while you know your curve is tangent at that point. The tangent line isn't going to communicate any information about the curve in general.
– Mason
Nov 17 at 23:08
I see, thank you
– Noel Jonathan Ellis
Nov 17 at 23:17
I see, thank you
– Noel Jonathan Ellis
Nov 17 at 23:17
add a comment |
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3
Can you show us what you have attempted? Or what is interesting about this question? Usually questions posed with the tone "do this homework problem for me" are not well-received.
– Mason
Nov 17 at 21:39
1
I found the tangent equation to be y= 1/2 x + 6. I inegrated that to get y = x^2 /4 + 6x + c. Then I found c by putting the values of the point in as -6.
– Noel Jonathan Ellis
Nov 17 at 22:00
(I am unsure from here forward) I think ax+b/x would be ax2+yx+b , meaning a would be 1/4 and b would be -6. I do not know what to do to find the other intersection of the normal, given that I am unsure that my answers for a and b are accurate
– Noel Jonathan Ellis
Nov 17 at 22:09