Sunday, February 5, 2017

High School Math Solutions – Derivative Applications Calculator, Normal Lines

Last blog post, we talk about using derivatives to compute the tangent lines of functions at certain points. Another application of derivatives is finding the normal line of a function at a certain point.

A normal line is a line that is perpendicular to a tangent line. The slope of the normal line is the negative reciprocal of the slope of the tangent line.



The steps for finding the equation of a normal line is pretty simple, as long as you’ve mastered finding the equation of a tangent line.

Steps for finding the equation of a normal line:

1.  Find the normal point
  • The normal point is the same as the tangent point
2.  Compute the slope of the function at the x coordinate
  • Compute the derivative of the function at the x coordinate 
  • This is the slope of the tangent line
3.  Compute the slope of the perpendicular line at the x coordinate
  • The slope (m) is the negative reciprocal of the slope (m_1) of the tangent line 
  • m=\frac{-1}{m_1} 
4.  Use the point-slope formula to find the equation of the normal line
  • y-y_1=m(x-x_1)


Let’s see some examples.

First example (click here):

                                          Find the normal line of y=x^2-x-1 at x=2

1. Find the normal point

                                                       y=(2)^2-2-1=1

                                                                  (2,1)

2. Compute the slope of the function at the x coordinate

                                                              y^'=2x-1

                                                          y^'=2(2)-1=3

                                                               m_1=3

3. Compute the slope of the perpendicular line at the x coordinate

                                                             m=\frac{-1}{m_1}
                                                                m=\frac{-1}{3}

4. Use the point-slope formula to find the equation of the normal line

                                                      (2,1)       m=\frac{-1}{3}

                                                       y-y_1=m(x-x_1)

                                                          y-1=\frac{-1}{3}(x-2)

                                                           y=\frac{-1}{3} x+\frac{5}{3}

Next example (click here):

                                           Find the normal line of \f(x)=x^4+2e^x  at (0,2)

1. Find the normal point

      It is already given.

2. Compute the slope of the function at the x coordinate

                                                               \f^' (x)=4x^3+2e^x

                                                         \f^' (0)=4(0)^3+2e^0=2

                                                                        m_1=2

3. Compute the slope of the perpendicular line at the x coordinate

                                                                        m=\frac{-1}{2}

4. Use the point-slope formula to find the equation of the normal line

                                                                 (0,2)      m=\frac{-1}{2}

                                                                   y-2=\frac{-1}{2}(x-0)

                                                                     y=\frac{-1}{2} x+2

Last example (click here):

                                              Find the normal line of \f(x)=\frac{1}{x^2}   at x=-1

1. Find the normal point

                                                            \f(-1)=\frac{1}{(-1)^2 }=1

                                                                          (-1,1)

2. Compute the slope of the function at the x coordinate

                                                                  \f^' (x)=\frac{-2}{x^3}

                                                             \f^' (-1)=\frac{-2}{(-1)^3} =2

                                                                        m_1=2

3. Compute the slope of the perpendicular line at the x coordinate

                                                                        m=\frac{-1}{2}

4. Use point-slope formula to find the equation of the normal line

                                                                (-1,1)     m=\frac{-1}{2}

                                                                y-1=\frac{-1}{2}(x-(-1))

                                                                  y=\frac{-1}{2} x+\frac{1}{2}

Finding the equation of a normal line is very similar to finding the equation of a tangent line. Since the steps are similar, make sure you don’t confuse and mix up the definitions of a tangent line and a normal line. For more practice and help with normal lines, check out Symbolab’s Practice.

Until next time,

Leah

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  58. African InterContinental Group LLC




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  59. African InterContinental Group LLC




    African InterContinental Group,is a Smart Logistics, Freight Forwarding, import and Export management Services.
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  62. African InterContinental Group LLC




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  63. African InterContinental Group LLC




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    ship

    Ocean Freight

    transport

    Air Freight

    In-Land Transportation

    Custom Clearance

    Project Handling

    LCL Consolidation

    Warehouse and Distribution

    Door to Door deliveries





    African InterContinental Group,is a Smart Logistics, Freight Forwarding, import and Export management Services.
    With great expertise in handling shipments to and from ,LAND LOCKED Cameroon,Chad and Central African Republic,Congo,US. African InterContinental Group, serve as
    a one-stop shop for all Logistics and Forwarding needs in the Sub-Region. We are a vibrant and innovative player in the industry,
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  68. Buy top quality handgunss, rifless, shortgunss and other firearmss and have them shipped discreetly to your address.
    We do same day shipment, and tracking information

    is provided as soon as shipment is made.
    All products come with manual and most of them are still in box but not all are brand new.
    These are not stolen and there’s a

    sales document issued for each.
    We also ship to an FFL for those who prefer it that way.
    Please feel free to visit our website.
    Contact us on wickr: thediscreetboss
    Weblink: Buy top quality handgunss, rifless, shortgunss and other firearmss and have them shipped discreetly to your address.
    We do same day shipment, and tracking

    information is provided as soon as shipment is made.
    All products come with manual and most of them are still in box but not all are brand new.
    These are not stolen

    and there’s a sales document issued for each.
    We also ship to an FFL for those who prefer it that way.
    Please feel free to visit our website.
    Contact us on wickr:

    thediscreetboss
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    terminal handling

    containers

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    shipments

    transport

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    AB Universal Logistics is an innovative and integrated service-based company that provides pivotal solutions to the
    logistics and supply chain challenges facing most successful companies around the globe. We provide unbeatable services
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