TangentLineApproximations

# TangentLineApproximations - 1 LINEARIZATION AND...

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1 LINEARIZATION AND DIFFERENTIALS LINEARIZATION Consider the function f ( x ) = x 2 + 4. Determine its tangent line at the point (1, 5). f ' (x) = 2 x m = f ' (1) = 2(1) = 2 y - 5 = 2( x - 1) y - 5 = 2 x - 2 y = 2 x + 3 Now, let us look at the graph of the function and its tangent line. Notice that as you get close to the point (1, 5), the function f ' (x) = x 2 + 4 starts to look like y = 2 x + 3. In fact, we can use the tangent line to approximate a value of a function at a point. Thus f ' (x) 2 x + 3 for x near 1. y = x 2 + 4 y = 2 x + 3 DEFINITION: If f is differentiable at x = a, then the approximating function L ( x ) = f (a) + f ' (a)(x - a) is the linearization of f at a. EXAMPLE 1: Find the linearization of f ( x ) = x - 1 at x = 2. SOLUTION: First evaluate the function at x = 2, and then find the first derivative and evaluate it at x = 2. Now substitute the information into the formula for the linearization. For values of

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TangentLineApproximations - 1 LINEARIZATION AND...

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