Example 1: Tangent Line Approximation

Use linearization to approximate 4 . 14 .

Step 1: Determine the f ( x ) equation that you are trying to approximate.

Since we are trying to approximate the square root of a value, we will use the square root function as our f ( x ) .

f ( x ) = x

Step 2: Determine the x-value that is closest to the actual x-value that you care about, which you will use to create your tangent line.

 

The thought process you will want to use to find the x-value is to ask yourself, “What is an x-value that is very, very, very close to the one I am trying to approximate , and that I could plug into the f ( x ) and solve the math problem in my head?”

 

For this problem you would want an x-value very, very, very close to 4.14 that you could plug into the square root function, the f ( x ) , and do the math in your head.

 

That means you would want to choose x 0 = 4 because 4 is very close to 4.14, and the 4 is a math problem you can do in your head.

x 0 = 4

Step 3: Find the equation of the tangent line to your f ( x )   at the x-value you determined in Step 2 .

 

You will follow the exact same steps that you would to find the equation of any tangent line.

Step 1: Find the tangent point, ( x 1 , y 1 ) .

( x 1 , y 1 ) = ( x 0 , f ( x 0 ) )

Step 2: Find the slope , m , of the tangent line.

m = f ( x 0 )

Step 3: Find the equation of the tangent line.

y = m ( x x 1 ) + y 1

The equation of the tangent line is your linearization equation, your tangent line approximation equation, or your local linear approximation equation, depending on who is asking. ????

Step 1:     f ( x ) = x ,   x 0 = 4

f ( x 0 ) =   f ( 4 ) = 4 = 2

( x 0 , f ( x 0 ) ) = ( 4 , 2 )

( x 1 , y 1 ) =   ( 4 , 2 )

 

Step 2:  f ( x ) = x

f ( x ) = x 1 2

f ( x ) = 1 2 x 1 2

f ( x 0 ) = f ( 4 ) = 1 2 ( 4 ) 1 2 = 1 4

f ( x 0 ) = f ( 4 ) = 1 4

m = 1 4

Step 3: ( x 1 , y 1 ) =   ( 4 , 2 )

m = 1 4

 

y = m ( x x 1 ) + y 1

y = 1 4 ( x 4 ) + 2

Step 4: Use the linearization equation (tangent line equation) that you found in Step 3 to approximate the value you were initially asked to find.

 

What you just created is Step 3 is not the final answer it is now the equation that approximates the actual equation, f ( x ) we care about, the square root function, f ( x ) = x . It is the equation you use to find the final answer.

 

To approximate the value we are trying to find, 4 . 14 , we plug 4.14 in for x in our approximation equation we just created in Step 3.

x 1 4 ( x 4 ) + 2

4 . 14 1 4 ( 4 . 14 4 ) + 2

4 . 14 2 . 035

Final Result:

A linear equation that can provide you an approximation for the square root equation is x 1 4 ( x 4 ) + 2 .

Depending on who is asking you that same equation could be represented as

          Linearization Equation or Local Linear Approximation: L ( x ) = 1 4 ( x 4 ) + 2

          Tangent Line Approximation Equation: f ̂ ( x ) = 1 4 ( x 4 ) + 2

Plugging 4.14 in for x in our approximation equation tells us that 4 . 14 2 . 035 .

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