Find the equation of the normal line to the curve $y = sqrt{1+4x}$ at $x=2$.












0












$begingroup$



Find the equation of the normal line to the curve $y = sqrt{1+4x}$ at $x=2$.




So, to begin we get rid of the square root right?



$y=(1+4x)^{-1/2}$



Then, the power rule?



$y’= -1/2(1+4x)$



Then, don’t we multiply by the derivative of the inside of the parenthesis?



$y’ = -1/2 (1+4x) cdot (4)$



$y’ = -2 (1+4x)$



$y’ = -2 - 8x$



Then, we set our derivative to 0 to find the slope.



$0 = -2 - 8x$



$-1/4 = x$ This is our slope!



Then, the normal means it’s the reciprocal of our slope, so it’s actually $-4$.



Then, we plug our $x$ value into our original equation to get our $y$ value.



$y= sqrt{1+4(1/4)}$



$y= sqrt{2}$ (our teacher said we aren’t allowed to have decimals in our answers during this section)



Then we plug everything in.



$sqrt{2} = -4 cdot 2 + b$



$b= sqrt{2} + 8$



Is this correct?










share|cite|improve this question











$endgroup$












  • $begingroup$
    No you cannot get rid of the square root. It is the same thing as raising to the 1/2 power. You raised to the -1/2 power and in addition to that mistake your differientiating is totally off the mark.
    $endgroup$
    – William Elliot
    Dec 3 '18 at 3:04


















0












$begingroup$



Find the equation of the normal line to the curve $y = sqrt{1+4x}$ at $x=2$.




So, to begin we get rid of the square root right?



$y=(1+4x)^{-1/2}$



Then, the power rule?



$y’= -1/2(1+4x)$



Then, don’t we multiply by the derivative of the inside of the parenthesis?



$y’ = -1/2 (1+4x) cdot (4)$



$y’ = -2 (1+4x)$



$y’ = -2 - 8x$



Then, we set our derivative to 0 to find the slope.



$0 = -2 - 8x$



$-1/4 = x$ This is our slope!



Then, the normal means it’s the reciprocal of our slope, so it’s actually $-4$.



Then, we plug our $x$ value into our original equation to get our $y$ value.



$y= sqrt{1+4(1/4)}$



$y= sqrt{2}$ (our teacher said we aren’t allowed to have decimals in our answers during this section)



Then we plug everything in.



$sqrt{2} = -4 cdot 2 + b$



$b= sqrt{2} + 8$



Is this correct?










share|cite|improve this question











$endgroup$












  • $begingroup$
    No you cannot get rid of the square root. It is the same thing as raising to the 1/2 power. You raised to the -1/2 power and in addition to that mistake your differientiating is totally off the mark.
    $endgroup$
    – William Elliot
    Dec 3 '18 at 3:04
















0












0








0





$begingroup$



Find the equation of the normal line to the curve $y = sqrt{1+4x}$ at $x=2$.




So, to begin we get rid of the square root right?



$y=(1+4x)^{-1/2}$



Then, the power rule?



$y’= -1/2(1+4x)$



Then, don’t we multiply by the derivative of the inside of the parenthesis?



$y’ = -1/2 (1+4x) cdot (4)$



$y’ = -2 (1+4x)$



$y’ = -2 - 8x$



Then, we set our derivative to 0 to find the slope.



$0 = -2 - 8x$



$-1/4 = x$ This is our slope!



Then, the normal means it’s the reciprocal of our slope, so it’s actually $-4$.



Then, we plug our $x$ value into our original equation to get our $y$ value.



$y= sqrt{1+4(1/4)}$



$y= sqrt{2}$ (our teacher said we aren’t allowed to have decimals in our answers during this section)



Then we plug everything in.



$sqrt{2} = -4 cdot 2 + b$



$b= sqrt{2} + 8$



Is this correct?










share|cite|improve this question











$endgroup$





Find the equation of the normal line to the curve $y = sqrt{1+4x}$ at $x=2$.




So, to begin we get rid of the square root right?



$y=(1+4x)^{-1/2}$



Then, the power rule?



$y’= -1/2(1+4x)$



Then, don’t we multiply by the derivative of the inside of the parenthesis?



$y’ = -1/2 (1+4x) cdot (4)$



$y’ = -2 (1+4x)$



$y’ = -2 - 8x$



Then, we set our derivative to 0 to find the slope.



$0 = -2 - 8x$



$-1/4 = x$ This is our slope!



Then, the normal means it’s the reciprocal of our slope, so it’s actually $-4$.



Then, we plug our $x$ value into our original equation to get our $y$ value.



$y= sqrt{1+4(1/4)}$



$y= sqrt{2}$ (our teacher said we aren’t allowed to have decimals in our answers during this section)



Then we plug everything in.



$sqrt{2} = -4 cdot 2 + b$



$b= sqrt{2} + 8$



Is this correct?







calculus derivatives tangent-line






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share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 4 '18 at 11:48









N. F. Taussig

44.3k93357




44.3k93357










asked Dec 3 '18 at 2:28









EllaElla

33311




33311












  • $begingroup$
    No you cannot get rid of the square root. It is the same thing as raising to the 1/2 power. You raised to the -1/2 power and in addition to that mistake your differientiating is totally off the mark.
    $endgroup$
    – William Elliot
    Dec 3 '18 at 3:04




















  • $begingroup$
    No you cannot get rid of the square root. It is the same thing as raising to the 1/2 power. You raised to the -1/2 power and in addition to that mistake your differientiating is totally off the mark.
    $endgroup$
    – William Elliot
    Dec 3 '18 at 3:04


















$begingroup$
No you cannot get rid of the square root. It is the same thing as raising to the 1/2 power. You raised to the -1/2 power and in addition to that mistake your differientiating is totally off the mark.
$endgroup$
– William Elliot
Dec 3 '18 at 3:04






$begingroup$
No you cannot get rid of the square root. It is the same thing as raising to the 1/2 power. You raised to the -1/2 power and in addition to that mistake your differientiating is totally off the mark.
$endgroup$
– William Elliot
Dec 3 '18 at 3:04












4 Answers
4






active

oldest

votes


















1












$begingroup$

Find the equation of the normal line using point and slope.



The point on the curve where the normal goes through is $(2,3)$



To get the slope of the normal by taking the negative reciprocal of the slope of the tangent, which you find by differentiating.



$$y=(1+4x)^frac12 implies y' = frac 12 (1+4x)^{-frac 12}(4)$$



at $x=2$ the slope of the tangent will be $frac 23$ so the slope of the normal line will be $-frac 32 $






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Easier to find $frac{dx}{dy}$, doesnt even require any advanced techniques
    $endgroup$
    – Rhys Hughes
    Dec 3 '18 at 3:26



















0












$begingroup$

$x=2to y=3$



$$y=sqrt{1+4x} to x= frac 14y^2-frac 14$$
$$to frac{dx}{dy}=frac y2$$



The gradient of the normal at a point is $-frac{dx}{dy}$ at that point. Then you just need to find the correct constant.






share|cite|improve this answer









$endgroup$





















    0












    $begingroup$

    There seems to be a few gaps in your reasoning. I don't think it's appropriate to address all of them in this reply, but I hope the following solution will help.



    Let's try to work this one out from scratch.



    Suppose I asked you to find out what the equation of a line is, given a point $(x_1,y_1)$ on the line, and the gradient $k$. Would you be able to do it?



    Well this should be straightforward right? You know that the equation of the line should be $y = mx + c$, where m is the gradient, and $c$ is the y intercept. So all you need to do is to find out what $c$ is. Given that you have $(x_1,y_1)$ and the gradient $k$, you have:



    $$
    y_1 = kx_1 + c implies c = y_1 - kx_1
    $$



    Then you'd be able to figure out the equation of the line, given that all the information you need for the line is its gradient and y-intercept, both of which you now know.



    So, going back to your original problem, you can easily find out a point on your normal line by setting
    $$
    begin{align}
    x_1 &= 2 \
    y_1 &= sqrt{1+4x_1} = 3
    end{align}
    $$

    Now, all you need to do is to find out what the gradient of the normal is in order to find out the equation of the normal (since we have shown above that given any point on the line and the gradient, that we'd be able to find the y-intercept)



    To find out what the gradient of the normal is, we will first find out what the gradient of the tangent to the curve $y = sqrt{1+4x}$ is at $x=2$. Using the usual rules of differentiation, we obtain:
    $$
    begin{align}
    frac{d y}{d x} = frac{2}{sqrt{1+4x}}
    end{align}
    $$

    And since we want this quantity at x = 2, we have:
    $$
    begin{align}
    frac{d y}{d x}Big|_{x = 2} &= frac{2}{sqrt{1+4times2}}\
    &= frac{2}{3}
    end{align}
    $$

    Now, using the fact that the slope $m_2$ of a line which is perpendicular to another line with slope $m_1$ is given by:
    $$
    m_2 = -frac{1}{m_1}
    $$

    Thus, we know that the slope $k_1$ of the normal, which is perpendicular (by definition) to the tangent of $y = sqrt{1+4x}$ at $x = 2$ (which has slope $frac{2}{3}$), has to be given by:
    $$
    k_1 = -frac{1}{2/3} = -frac{3}{2}
    $$

    Now that we have a point $(2,sqrt{5})$ on our line, as well as the gradient ($-frac{3}{2}$) of the line, we can find out the y-intercept $c_1$ of the line, which is given by:
    $$
    c_1 = 3 + 2frac{3}{2} = 3 + 3 = 6
    $$

    Thus, the line can be written as:
    $$
    y = -frac{3}{2}x + 6
    $$



    I do think that you should revisit some of the concepts in differentiation to get a clearer (geometrical) understanding of what is happening here. It can help you avoid errors (such as setting the derivative to zero, which is a technique used for finding stationary points, not to calculate gradients).






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      I'm pretty sure you meant $y_1 = sqrt 9 = 3$
      $endgroup$
      – WW1
      Dec 3 '18 at 3:15












    • $begingroup$
      Ah yes, shall edit it accordingly
      $endgroup$
      – Sean Lee
      Dec 3 '18 at 3:27



















    0












    $begingroup$


    The equation of the normal line to $y=f(x)$ at the point $x_0$ is:
    $$y=f(x_0)-frac1{f'(x_0)}(x-x_0)$$




    You want to find the normal line to $y=sqrt{1+4x}$ at the point $x_0=2$. Applying the formula:
    $$f'(x)=(sqrt{1+4x})'=frac{2}{sqrt{1+4x}} Rightarrow f'(2)=frac23;\
    y=f(2)-frac1{f'(2)}(x-2)=3-frac32(x-2)=-frac32x+6.$$




    The equation of the tangent line to $y=f(x)$ at the point $x_0$ is:
    $$y=f(x_0)+f'(x_0)(x-x_0)$$




    If you want to find the tangent line to $y=sqrt{1+4x}$ at the point $x_0=2$. Applying the formula:
    $$f'(x)=(sqrt{1+4x})'=frac{2}{sqrt{1+4x}} Rightarrow f'(2)=frac23;\
    y=f(2)+f'(2)(x-2)=3+frac23(x-2)=frac23x+frac53.$$



    See the graph:



    $hspace{2cm}$![enter image description here






    share|cite|improve this answer









    $endgroup$













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      4 Answers
      4






      active

      oldest

      votes








      4 Answers
      4






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      1












      $begingroup$

      Find the equation of the normal line using point and slope.



      The point on the curve where the normal goes through is $(2,3)$



      To get the slope of the normal by taking the negative reciprocal of the slope of the tangent, which you find by differentiating.



      $$y=(1+4x)^frac12 implies y' = frac 12 (1+4x)^{-frac 12}(4)$$



      at $x=2$ the slope of the tangent will be $frac 23$ so the slope of the normal line will be $-frac 32 $






      share|cite|improve this answer









      $endgroup$













      • $begingroup$
        Easier to find $frac{dx}{dy}$, doesnt even require any advanced techniques
        $endgroup$
        – Rhys Hughes
        Dec 3 '18 at 3:26
















      1












      $begingroup$

      Find the equation of the normal line using point and slope.



      The point on the curve where the normal goes through is $(2,3)$



      To get the slope of the normal by taking the negative reciprocal of the slope of the tangent, which you find by differentiating.



      $$y=(1+4x)^frac12 implies y' = frac 12 (1+4x)^{-frac 12}(4)$$



      at $x=2$ the slope of the tangent will be $frac 23$ so the slope of the normal line will be $-frac 32 $






      share|cite|improve this answer









      $endgroup$













      • $begingroup$
        Easier to find $frac{dx}{dy}$, doesnt even require any advanced techniques
        $endgroup$
        – Rhys Hughes
        Dec 3 '18 at 3:26














      1












      1








      1





      $begingroup$

      Find the equation of the normal line using point and slope.



      The point on the curve where the normal goes through is $(2,3)$



      To get the slope of the normal by taking the negative reciprocal of the slope of the tangent, which you find by differentiating.



      $$y=(1+4x)^frac12 implies y' = frac 12 (1+4x)^{-frac 12}(4)$$



      at $x=2$ the slope of the tangent will be $frac 23$ so the slope of the normal line will be $-frac 32 $






      share|cite|improve this answer









      $endgroup$



      Find the equation of the normal line using point and slope.



      The point on the curve where the normal goes through is $(2,3)$



      To get the slope of the normal by taking the negative reciprocal of the slope of the tangent, which you find by differentiating.



      $$y=(1+4x)^frac12 implies y' = frac 12 (1+4x)^{-frac 12}(4)$$



      at $x=2$ the slope of the tangent will be $frac 23$ so the slope of the normal line will be $-frac 32 $







      share|cite|improve this answer












      share|cite|improve this answer



      share|cite|improve this answer










      answered Dec 3 '18 at 2:56









      WW1WW1

      7,3151712




      7,3151712












      • $begingroup$
        Easier to find $frac{dx}{dy}$, doesnt even require any advanced techniques
        $endgroup$
        – Rhys Hughes
        Dec 3 '18 at 3:26


















      • $begingroup$
        Easier to find $frac{dx}{dy}$, doesnt even require any advanced techniques
        $endgroup$
        – Rhys Hughes
        Dec 3 '18 at 3:26
















      $begingroup$
      Easier to find $frac{dx}{dy}$, doesnt even require any advanced techniques
      $endgroup$
      – Rhys Hughes
      Dec 3 '18 at 3:26




      $begingroup$
      Easier to find $frac{dx}{dy}$, doesnt even require any advanced techniques
      $endgroup$
      – Rhys Hughes
      Dec 3 '18 at 3:26











      0












      $begingroup$

      $x=2to y=3$



      $$y=sqrt{1+4x} to x= frac 14y^2-frac 14$$
      $$to frac{dx}{dy}=frac y2$$



      The gradient of the normal at a point is $-frac{dx}{dy}$ at that point. Then you just need to find the correct constant.






      share|cite|improve this answer









      $endgroup$


















        0












        $begingroup$

        $x=2to y=3$



        $$y=sqrt{1+4x} to x= frac 14y^2-frac 14$$
        $$to frac{dx}{dy}=frac y2$$



        The gradient of the normal at a point is $-frac{dx}{dy}$ at that point. Then you just need to find the correct constant.






        share|cite|improve this answer









        $endgroup$
















          0












          0








          0





          $begingroup$

          $x=2to y=3$



          $$y=sqrt{1+4x} to x= frac 14y^2-frac 14$$
          $$to frac{dx}{dy}=frac y2$$



          The gradient of the normal at a point is $-frac{dx}{dy}$ at that point. Then you just need to find the correct constant.






          share|cite|improve this answer









          $endgroup$



          $x=2to y=3$



          $$y=sqrt{1+4x} to x= frac 14y^2-frac 14$$
          $$to frac{dx}{dy}=frac y2$$



          The gradient of the normal at a point is $-frac{dx}{dy}$ at that point. Then you just need to find the correct constant.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 3 '18 at 3:25









          Rhys HughesRhys Hughes

          6,7901530




          6,7901530























              0












              $begingroup$

              There seems to be a few gaps in your reasoning. I don't think it's appropriate to address all of them in this reply, but I hope the following solution will help.



              Let's try to work this one out from scratch.



              Suppose I asked you to find out what the equation of a line is, given a point $(x_1,y_1)$ on the line, and the gradient $k$. Would you be able to do it?



              Well this should be straightforward right? You know that the equation of the line should be $y = mx + c$, where m is the gradient, and $c$ is the y intercept. So all you need to do is to find out what $c$ is. Given that you have $(x_1,y_1)$ and the gradient $k$, you have:



              $$
              y_1 = kx_1 + c implies c = y_1 - kx_1
              $$



              Then you'd be able to figure out the equation of the line, given that all the information you need for the line is its gradient and y-intercept, both of which you now know.



              So, going back to your original problem, you can easily find out a point on your normal line by setting
              $$
              begin{align}
              x_1 &= 2 \
              y_1 &= sqrt{1+4x_1} = 3
              end{align}
              $$

              Now, all you need to do is to find out what the gradient of the normal is in order to find out the equation of the normal (since we have shown above that given any point on the line and the gradient, that we'd be able to find the y-intercept)



              To find out what the gradient of the normal is, we will first find out what the gradient of the tangent to the curve $y = sqrt{1+4x}$ is at $x=2$. Using the usual rules of differentiation, we obtain:
              $$
              begin{align}
              frac{d y}{d x} = frac{2}{sqrt{1+4x}}
              end{align}
              $$

              And since we want this quantity at x = 2, we have:
              $$
              begin{align}
              frac{d y}{d x}Big|_{x = 2} &= frac{2}{sqrt{1+4times2}}\
              &= frac{2}{3}
              end{align}
              $$

              Now, using the fact that the slope $m_2$ of a line which is perpendicular to another line with slope $m_1$ is given by:
              $$
              m_2 = -frac{1}{m_1}
              $$

              Thus, we know that the slope $k_1$ of the normal, which is perpendicular (by definition) to the tangent of $y = sqrt{1+4x}$ at $x = 2$ (which has slope $frac{2}{3}$), has to be given by:
              $$
              k_1 = -frac{1}{2/3} = -frac{3}{2}
              $$

              Now that we have a point $(2,sqrt{5})$ on our line, as well as the gradient ($-frac{3}{2}$) of the line, we can find out the y-intercept $c_1$ of the line, which is given by:
              $$
              c_1 = 3 + 2frac{3}{2} = 3 + 3 = 6
              $$

              Thus, the line can be written as:
              $$
              y = -frac{3}{2}x + 6
              $$



              I do think that you should revisit some of the concepts in differentiation to get a clearer (geometrical) understanding of what is happening here. It can help you avoid errors (such as setting the derivative to zero, which is a technique used for finding stationary points, not to calculate gradients).






              share|cite|improve this answer











              $endgroup$













              • $begingroup$
                I'm pretty sure you meant $y_1 = sqrt 9 = 3$
                $endgroup$
                – WW1
                Dec 3 '18 at 3:15












              • $begingroup$
                Ah yes, shall edit it accordingly
                $endgroup$
                – Sean Lee
                Dec 3 '18 at 3:27
















              0












              $begingroup$

              There seems to be a few gaps in your reasoning. I don't think it's appropriate to address all of them in this reply, but I hope the following solution will help.



              Let's try to work this one out from scratch.



              Suppose I asked you to find out what the equation of a line is, given a point $(x_1,y_1)$ on the line, and the gradient $k$. Would you be able to do it?



              Well this should be straightforward right? You know that the equation of the line should be $y = mx + c$, where m is the gradient, and $c$ is the y intercept. So all you need to do is to find out what $c$ is. Given that you have $(x_1,y_1)$ and the gradient $k$, you have:



              $$
              y_1 = kx_1 + c implies c = y_1 - kx_1
              $$



              Then you'd be able to figure out the equation of the line, given that all the information you need for the line is its gradient and y-intercept, both of which you now know.



              So, going back to your original problem, you can easily find out a point on your normal line by setting
              $$
              begin{align}
              x_1 &= 2 \
              y_1 &= sqrt{1+4x_1} = 3
              end{align}
              $$

              Now, all you need to do is to find out what the gradient of the normal is in order to find out the equation of the normal (since we have shown above that given any point on the line and the gradient, that we'd be able to find the y-intercept)



              To find out what the gradient of the normal is, we will first find out what the gradient of the tangent to the curve $y = sqrt{1+4x}$ is at $x=2$. Using the usual rules of differentiation, we obtain:
              $$
              begin{align}
              frac{d y}{d x} = frac{2}{sqrt{1+4x}}
              end{align}
              $$

              And since we want this quantity at x = 2, we have:
              $$
              begin{align}
              frac{d y}{d x}Big|_{x = 2} &= frac{2}{sqrt{1+4times2}}\
              &= frac{2}{3}
              end{align}
              $$

              Now, using the fact that the slope $m_2$ of a line which is perpendicular to another line with slope $m_1$ is given by:
              $$
              m_2 = -frac{1}{m_1}
              $$

              Thus, we know that the slope $k_1$ of the normal, which is perpendicular (by definition) to the tangent of $y = sqrt{1+4x}$ at $x = 2$ (which has slope $frac{2}{3}$), has to be given by:
              $$
              k_1 = -frac{1}{2/3} = -frac{3}{2}
              $$

              Now that we have a point $(2,sqrt{5})$ on our line, as well as the gradient ($-frac{3}{2}$) of the line, we can find out the y-intercept $c_1$ of the line, which is given by:
              $$
              c_1 = 3 + 2frac{3}{2} = 3 + 3 = 6
              $$

              Thus, the line can be written as:
              $$
              y = -frac{3}{2}x + 6
              $$



              I do think that you should revisit some of the concepts in differentiation to get a clearer (geometrical) understanding of what is happening here. It can help you avoid errors (such as setting the derivative to zero, which is a technique used for finding stationary points, not to calculate gradients).






              share|cite|improve this answer











              $endgroup$













              • $begingroup$
                I'm pretty sure you meant $y_1 = sqrt 9 = 3$
                $endgroup$
                – WW1
                Dec 3 '18 at 3:15












              • $begingroup$
                Ah yes, shall edit it accordingly
                $endgroup$
                – Sean Lee
                Dec 3 '18 at 3:27














              0












              0








              0





              $begingroup$

              There seems to be a few gaps in your reasoning. I don't think it's appropriate to address all of them in this reply, but I hope the following solution will help.



              Let's try to work this one out from scratch.



              Suppose I asked you to find out what the equation of a line is, given a point $(x_1,y_1)$ on the line, and the gradient $k$. Would you be able to do it?



              Well this should be straightforward right? You know that the equation of the line should be $y = mx + c$, where m is the gradient, and $c$ is the y intercept. So all you need to do is to find out what $c$ is. Given that you have $(x_1,y_1)$ and the gradient $k$, you have:



              $$
              y_1 = kx_1 + c implies c = y_1 - kx_1
              $$



              Then you'd be able to figure out the equation of the line, given that all the information you need for the line is its gradient and y-intercept, both of which you now know.



              So, going back to your original problem, you can easily find out a point on your normal line by setting
              $$
              begin{align}
              x_1 &= 2 \
              y_1 &= sqrt{1+4x_1} = 3
              end{align}
              $$

              Now, all you need to do is to find out what the gradient of the normal is in order to find out the equation of the normal (since we have shown above that given any point on the line and the gradient, that we'd be able to find the y-intercept)



              To find out what the gradient of the normal is, we will first find out what the gradient of the tangent to the curve $y = sqrt{1+4x}$ is at $x=2$. Using the usual rules of differentiation, we obtain:
              $$
              begin{align}
              frac{d y}{d x} = frac{2}{sqrt{1+4x}}
              end{align}
              $$

              And since we want this quantity at x = 2, we have:
              $$
              begin{align}
              frac{d y}{d x}Big|_{x = 2} &= frac{2}{sqrt{1+4times2}}\
              &= frac{2}{3}
              end{align}
              $$

              Now, using the fact that the slope $m_2$ of a line which is perpendicular to another line with slope $m_1$ is given by:
              $$
              m_2 = -frac{1}{m_1}
              $$

              Thus, we know that the slope $k_1$ of the normal, which is perpendicular (by definition) to the tangent of $y = sqrt{1+4x}$ at $x = 2$ (which has slope $frac{2}{3}$), has to be given by:
              $$
              k_1 = -frac{1}{2/3} = -frac{3}{2}
              $$

              Now that we have a point $(2,sqrt{5})$ on our line, as well as the gradient ($-frac{3}{2}$) of the line, we can find out the y-intercept $c_1$ of the line, which is given by:
              $$
              c_1 = 3 + 2frac{3}{2} = 3 + 3 = 6
              $$

              Thus, the line can be written as:
              $$
              y = -frac{3}{2}x + 6
              $$



              I do think that you should revisit some of the concepts in differentiation to get a clearer (geometrical) understanding of what is happening here. It can help you avoid errors (such as setting the derivative to zero, which is a technique used for finding stationary points, not to calculate gradients).






              share|cite|improve this answer











              $endgroup$



              There seems to be a few gaps in your reasoning. I don't think it's appropriate to address all of them in this reply, but I hope the following solution will help.



              Let's try to work this one out from scratch.



              Suppose I asked you to find out what the equation of a line is, given a point $(x_1,y_1)$ on the line, and the gradient $k$. Would you be able to do it?



              Well this should be straightforward right? You know that the equation of the line should be $y = mx + c$, where m is the gradient, and $c$ is the y intercept. So all you need to do is to find out what $c$ is. Given that you have $(x_1,y_1)$ and the gradient $k$, you have:



              $$
              y_1 = kx_1 + c implies c = y_1 - kx_1
              $$



              Then you'd be able to figure out the equation of the line, given that all the information you need for the line is its gradient and y-intercept, both of which you now know.



              So, going back to your original problem, you can easily find out a point on your normal line by setting
              $$
              begin{align}
              x_1 &= 2 \
              y_1 &= sqrt{1+4x_1} = 3
              end{align}
              $$

              Now, all you need to do is to find out what the gradient of the normal is in order to find out the equation of the normal (since we have shown above that given any point on the line and the gradient, that we'd be able to find the y-intercept)



              To find out what the gradient of the normal is, we will first find out what the gradient of the tangent to the curve $y = sqrt{1+4x}$ is at $x=2$. Using the usual rules of differentiation, we obtain:
              $$
              begin{align}
              frac{d y}{d x} = frac{2}{sqrt{1+4x}}
              end{align}
              $$

              And since we want this quantity at x = 2, we have:
              $$
              begin{align}
              frac{d y}{d x}Big|_{x = 2} &= frac{2}{sqrt{1+4times2}}\
              &= frac{2}{3}
              end{align}
              $$

              Now, using the fact that the slope $m_2$ of a line which is perpendicular to another line with slope $m_1$ is given by:
              $$
              m_2 = -frac{1}{m_1}
              $$

              Thus, we know that the slope $k_1$ of the normal, which is perpendicular (by definition) to the tangent of $y = sqrt{1+4x}$ at $x = 2$ (which has slope $frac{2}{3}$), has to be given by:
              $$
              k_1 = -frac{1}{2/3} = -frac{3}{2}
              $$

              Now that we have a point $(2,sqrt{5})$ on our line, as well as the gradient ($-frac{3}{2}$) of the line, we can find out the y-intercept $c_1$ of the line, which is given by:
              $$
              c_1 = 3 + 2frac{3}{2} = 3 + 3 = 6
              $$

              Thus, the line can be written as:
              $$
              y = -frac{3}{2}x + 6
              $$



              I do think that you should revisit some of the concepts in differentiation to get a clearer (geometrical) understanding of what is happening here. It can help you avoid errors (such as setting the derivative to zero, which is a technique used for finding stationary points, not to calculate gradients).







              share|cite|improve this answer














              share|cite|improve this answer



              share|cite|improve this answer








              edited Dec 3 '18 at 3:27

























              answered Dec 3 '18 at 3:09









              Sean LeeSean Lee

              393111




              393111












              • $begingroup$
                I'm pretty sure you meant $y_1 = sqrt 9 = 3$
                $endgroup$
                – WW1
                Dec 3 '18 at 3:15












              • $begingroup$
                Ah yes, shall edit it accordingly
                $endgroup$
                – Sean Lee
                Dec 3 '18 at 3:27


















              • $begingroup$
                I'm pretty sure you meant $y_1 = sqrt 9 = 3$
                $endgroup$
                – WW1
                Dec 3 '18 at 3:15












              • $begingroup$
                Ah yes, shall edit it accordingly
                $endgroup$
                – Sean Lee
                Dec 3 '18 at 3:27
















              $begingroup$
              I'm pretty sure you meant $y_1 = sqrt 9 = 3$
              $endgroup$
              – WW1
              Dec 3 '18 at 3:15






              $begingroup$
              I'm pretty sure you meant $y_1 = sqrt 9 = 3$
              $endgroup$
              – WW1
              Dec 3 '18 at 3:15














              $begingroup$
              Ah yes, shall edit it accordingly
              $endgroup$
              – Sean Lee
              Dec 3 '18 at 3:27




              $begingroup$
              Ah yes, shall edit it accordingly
              $endgroup$
              – Sean Lee
              Dec 3 '18 at 3:27











              0












              $begingroup$


              The equation of the normal line to $y=f(x)$ at the point $x_0$ is:
              $$y=f(x_0)-frac1{f'(x_0)}(x-x_0)$$




              You want to find the normal line to $y=sqrt{1+4x}$ at the point $x_0=2$. Applying the formula:
              $$f'(x)=(sqrt{1+4x})'=frac{2}{sqrt{1+4x}} Rightarrow f'(2)=frac23;\
              y=f(2)-frac1{f'(2)}(x-2)=3-frac32(x-2)=-frac32x+6.$$




              The equation of the tangent line to $y=f(x)$ at the point $x_0$ is:
              $$y=f(x_0)+f'(x_0)(x-x_0)$$




              If you want to find the tangent line to $y=sqrt{1+4x}$ at the point $x_0=2$. Applying the formula:
              $$f'(x)=(sqrt{1+4x})'=frac{2}{sqrt{1+4x}} Rightarrow f'(2)=frac23;\
              y=f(2)+f'(2)(x-2)=3+frac23(x-2)=frac23x+frac53.$$



              See the graph:



              $hspace{2cm}$![enter image description here






              share|cite|improve this answer









              $endgroup$


















                0












                $begingroup$


                The equation of the normal line to $y=f(x)$ at the point $x_0$ is:
                $$y=f(x_0)-frac1{f'(x_0)}(x-x_0)$$




                You want to find the normal line to $y=sqrt{1+4x}$ at the point $x_0=2$. Applying the formula:
                $$f'(x)=(sqrt{1+4x})'=frac{2}{sqrt{1+4x}} Rightarrow f'(2)=frac23;\
                y=f(2)-frac1{f'(2)}(x-2)=3-frac32(x-2)=-frac32x+6.$$




                The equation of the tangent line to $y=f(x)$ at the point $x_0$ is:
                $$y=f(x_0)+f'(x_0)(x-x_0)$$




                If you want to find the tangent line to $y=sqrt{1+4x}$ at the point $x_0=2$. Applying the formula:
                $$f'(x)=(sqrt{1+4x})'=frac{2}{sqrt{1+4x}} Rightarrow f'(2)=frac23;\
                y=f(2)+f'(2)(x-2)=3+frac23(x-2)=frac23x+frac53.$$



                See the graph:



                $hspace{2cm}$![enter image description here






                share|cite|improve this answer









                $endgroup$
















                  0












                  0








                  0





                  $begingroup$


                  The equation of the normal line to $y=f(x)$ at the point $x_0$ is:
                  $$y=f(x_0)-frac1{f'(x_0)}(x-x_0)$$




                  You want to find the normal line to $y=sqrt{1+4x}$ at the point $x_0=2$. Applying the formula:
                  $$f'(x)=(sqrt{1+4x})'=frac{2}{sqrt{1+4x}} Rightarrow f'(2)=frac23;\
                  y=f(2)-frac1{f'(2)}(x-2)=3-frac32(x-2)=-frac32x+6.$$




                  The equation of the tangent line to $y=f(x)$ at the point $x_0$ is:
                  $$y=f(x_0)+f'(x_0)(x-x_0)$$




                  If you want to find the tangent line to $y=sqrt{1+4x}$ at the point $x_0=2$. Applying the formula:
                  $$f'(x)=(sqrt{1+4x})'=frac{2}{sqrt{1+4x}} Rightarrow f'(2)=frac23;\
                  y=f(2)+f'(2)(x-2)=3+frac23(x-2)=frac23x+frac53.$$



                  See the graph:



                  $hspace{2cm}$![enter image description here






                  share|cite|improve this answer









                  $endgroup$




                  The equation of the normal line to $y=f(x)$ at the point $x_0$ is:
                  $$y=f(x_0)-frac1{f'(x_0)}(x-x_0)$$




                  You want to find the normal line to $y=sqrt{1+4x}$ at the point $x_0=2$. Applying the formula:
                  $$f'(x)=(sqrt{1+4x})'=frac{2}{sqrt{1+4x}} Rightarrow f'(2)=frac23;\
                  y=f(2)-frac1{f'(2)}(x-2)=3-frac32(x-2)=-frac32x+6.$$




                  The equation of the tangent line to $y=f(x)$ at the point $x_0$ is:
                  $$y=f(x_0)+f'(x_0)(x-x_0)$$




                  If you want to find the tangent line to $y=sqrt{1+4x}$ at the point $x_0=2$. Applying the formula:
                  $$f'(x)=(sqrt{1+4x})'=frac{2}{sqrt{1+4x}} Rightarrow f'(2)=frac23;\
                  y=f(2)+f'(2)(x-2)=3+frac23(x-2)=frac23x+frac53.$$



                  See the graph:



                  $hspace{2cm}$![enter image description here







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Dec 3 '18 at 4:48









                  farruhotafarruhota

                  20.4k2739




                  20.4k2739






























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