Use linearization to approximate . |
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Step 1: Determine the 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 . |
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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 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 , and do the math in your head.
That means you would want to choose because 4 is very close to 4.14, and the is a math problem you can do in your head. |
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Step 3: Find the equation of the tangent line to your 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, .
Step 2: Find the slope , m , of the tangent line.
Step 3: Find the equation of the tangent line.
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:
Step 2:
Step 3:
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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, we care about, the square root function, . It is the equation you use to find the final answer.
To approximate the value we are trying to find, , we plug 4.14 in for x in our approximation equation we just created in Step 3. |
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Final Result: A linear equation that can provide you an approximation for the square root equation is . Depending on who is asking you that same equation could be represented as – Linearization Equation or Local Linear Approximation: – Tangent Line Approximation Equation: Plugging 4.14 in for x in our approximation equation tells us that . |