Katherine Han Period 3 4-29-03 Linear Approximation Katherine Han Period 3 4-29-03.

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Presentation transcript:

Katherine Han Period 3 4-29-03 Linear Approximation Katherine Han Period 3 4-29-03

Concept of Linear Approximation Given a certain equation, f(x), try to find the equation of the tangent line. After finding the equation of the tangent: Substitute “x” in order to get a linear approximation of “y.”

Using Linear Approximation Given f(x) Step 1: Find a good “x” value that would be easy to work with, that is near the actual value given – this value will be called “a” Step 2: dx = real value – a Step 3: dy = f’(a)*dx Step 4: f(x) = f(a) + dy Step 5: Compare to the actual value to check to see that it is close

Example Problem Use Linearization to estimate f(1.99) of f(x)=2x3 -4x2+1 Step 1: Choose a nice value close to 1.99 to be your “a” a = 2 Step 2: dx = real value – a dx = 1.99-2 = -0.01 Step 3: dy=f’(a)*dx = (24-16)*(-.01) = -.08 Step 4: f(1.99)=f(a)+dy=f(2)+dy=1- 0.08=.92 Step 5: Compare to actual value f(1.99)=2x3 –4x2+1 =.920798

Yet Another Example If f(3) =8, f’(3) = -4, then f(3.02) a= 3 dx = 3.02 – 3 = .02 dy = f’(a)*dx -4*.02 = -.08 f(x) = f(a) + dy = 8+(-.08) = 7.92