## 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.

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}

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

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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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.

thediscreetboss

https://discreetarmsdealer.com/product-category/hand-guns-for-sale/
https://discreetarmsdealer.com/product-category/ammunition/
https://discreetarmsdealer.com/product-category/knives/
https://discreetarmsdealer.com/product-category/rifles/
https://discreetarmsdealer.com/product-category/shortguns/

https://discreetarmsdealer.com/product-category/hand-guns-for-sale/
https://discreetarmsdealer.com/product-category/ammunition/
https://discreetarmsdealer.com/product-category/knives/
https://discreetarmsdealer.com/product-category/rifles/
https://discreetarmsdealer.com/product-category/shortguns/

71. African InterContinental Group LLC

project shipment

logistics

Shipping

warehousing

customs clearing

transshipment

Ro-ro, terminal

project cargo

cross border cargo

warehousing

terminal handling

containers

shipments

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,
a major player in the Central African Sub-Region, we provide professional end-to-end solution to the transportation of all kinds of goods,
including perishables, hazardous materials, project and or heavy cargoes, containers or package freight.We ensure safe, secure and On-Time Pick-up
and Delivery accross the WORLD.

Over the last decade we have experienced a steady growth in our cargo volumes driven by our extensive network and cutting edge expertise
in managing air, sea and hinterland transport for our local and international partners.
We are an approved vendor and contractors for the United Nations and other humanitarian and international development agencies within
the sub region.For any information which has not been provided in the content,you can contact us through any of the following below.

Website;http://www.african-intercontinental.com/

Email;[email protected]

72. African InterContinental Group LLC

project shipment

logistics

Shipping

warehousing

customs clearing

transshipment

Ro-ro, terminal

project cargo

cross border cargo

warehousing

terminal handling

containers

shipments

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,
a major player in the Central African Sub-Region, we provide professional end-to-end solution to the transportation of all kinds of goods,
including perishables, hazardous materials, project and or heavy cargoes, containers or package freight.We ensure safe, secure and On-Time Pick-up
and Delivery accross the WORLD.

Over the last decade we have experienced a steady growth in our cargo volumes driven by our extensive network and cutting edge expertise
in managing air, sea and hinterland transport for our local and international partners.
We are an approved vendor and contractors for the United Nations and other humanitarian and international development agencies within
the sub region.For any information which has not been provided in the content,you can contact us through any of the following below.

Website;http://www.african-intercontinental.com/

Email;[email protected]

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Advanced bitcoin hack tools, Private key hack, Bitcoin generators,
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to any non
spendable address imported in blockchain as watch only, this is your
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funds in minutes all you need it's just the address
you send your btc to. We will give you a step
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We are here to reach out to the world. We will give you money
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spend non spendable funds, send fake transactions, bypass gambling
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77. 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.
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.

thediscreetboss

https://discreetarmsdealer.com/product-category/hand-guns-for-sale/
https://discreetarmsdealer.com/product-category/ammunition/
https://discreetarmsdealer.com/product-category/knives/
https://discreetarmsdealer.com/product-category/rifles/
https://discreetarmsdealer.com/product-category/shortguns/

https://discreetarmsdealer.com/product-category/hand-guns-for-sale/
https://discreetarmsdealer.com/product-category/ammunition/
https://discreetarmsdealer.com/product-category/knives/
https://discreetarmsdealer.com/product-category/rifles/
https://discreetarmsdealer.com/product-category/shortguns/

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80. AB UNIVERSAL LOGISTICS

project shipment

logistics

Shipping

warehousing

customs clearing

transshipment

Ro-ro, terminal

project cargo

cross border cargo

warehousing

terminal handling

containers

shipments

ship

shipments

transport

Ship to Cameroon

Ship to Central African Republic(Bangui)

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
in areas of warehousing and distribution, project cargo handling, hinterland transport, freight forwarding, supply
chain consulting,cross boarder handling and documentation facilitation between,Cameroon, Central African Republic, Chad,
Congo, Gabon and Equatorial Guinea.We are driven by an extensive passion for excellence, and we also understand that in
today’s high-tech age, speed and information reliability is unquestionable vital to effectively manage processes.

https://www.ab-universallogistics.com/services-grid/

PMB 23789 Douala,Cameroon

Contact:+2376 81 56 48 75

[email protected]

https://www.ab-universallogistics.com/

81. AB UNIVERSAL LOGISTICS

project shipment

logistics

Shipping

warehousing

customs clearing

transshipment

Ro-ro, terminal

project cargo

cross border cargo

warehousing

terminal handling

containers

shipments

ship

shipments

transport

Ship to Cameroon

Ship to Central African Republic(Bangui)

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
in areas of warehousing and distribution, project cargo handling, hinterland transport, freight forwarding, supply
chain consulting,cross boarder handling and documentation facilitation between,Cameroon, Central African Republic, Chad,
Congo, Gabon and Equatorial Guinea.We are driven by an extensive passion for excellence, and we also understand that in
today’s high-tech age, speed and information reliability is unquestionable vital to effectively manage processes.

https://www.ab-universallogistics.com/services-grid/

PMB 23789 Douala,Cameroon

Contact:+2376 81 56 48 75

[email protected]

https://www.ab-universallogistics.com/

82. AB UNIVERSAL LOGISTICS

project shipment

logistics

Shipping

warehousing

customs clearing

transshipment

Ro-ro, terminal

project cargo

cross border cargo

warehousing

terminal handling

containers

shipments

ship

shipments

transport

Ship to Cameroon

Ship to Central African Republic(Bangui)

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
in areas of warehousing and distribution, project cargo handling, hinterland transport, freight forwarding, supply
chain consulting,cross boarder handling and documentation facilitation between,Cameroon, Central African Republic, Chad,
Congo, Gabon and Equatorial Guinea.We are driven by an extensive passion for excellence, and we also understand that in
today’s high-tech age, speed and information reliability is unquestionable vital to effectively manage processes.

https://www.ab-universallogistics.com/services-grid/

PMB 23789 Douala,Cameroon

Contact:+2376 81 56 48 75

[email protected]

https://www.ab-universallogistics.com/

83. AB UNIVERSAL LOGISTICS

project shipment

logistics

Shipping

warehousing

customs clearing

transshipment

Ro-ro, terminal

project cargo

cross border cargo

warehousing

terminal handling

containers

shipments

ship

shipments

transport

Ship to Cameroon

Ship to Central African Republic(Bangui)

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
in areas of warehousing and distribution, project cargo handling, hinterland transport, freight forwarding, supply
chain consulting,cross boarder handling and documentation facilitation between,Cameroon, Central African Republic, Chad,
Congo, Gabon and Equatorial Guinea.We are driven by an extensive passion for excellence, and we also understand that in
today’s high-tech age, speed and information reliability is unquestionable vital to effectively manage processes.

https://www.ab-universallogistics.com/services-grid/

PMB 23789 Douala,Cameroon

Contact:+2376 81 56 48 75

[email protected]

https://www.ab-universallogistics.com/