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Linear Approximation and Differentials Lesson 4.8
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Tangent Line Approximation Consider a tangent to a function at a point x = a Close to the point, the tangent line is an approximation for f(x) a f(a) y=f(x) The equation of the tangent line: y = f(a) + f ‘(a)(x – a)
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Tangent Line Approximation We claim that This is called linearization of the function at the point a. Recall that when we zoom in on an interval of a function far enough, it looks like a line
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New Look at dy = rise of tangent relative to x = dx y = change in y that occurs relative to x = dx x x = dx dy yy x + x
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New Look at We know that then Recall that dy/dx is NOT a quotient it is the notation for the derivative However … sometimes it is useful to use dy and dx as actual quantities
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The Differential of y Consider Then we can say this is called the differential of y the notation is d(f(x)) = f ’(x) * dx it is an approximation of the actual change of y for a small change of x
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Animated Graphical View Note how the "del y" and the dy in the figure get closer and closer
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Try It Out Note the rules for differentials Page 274 Find the differential of 3 – 5x 2 x e -2x
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Differentials for Approximations Consider Use Then with x = 25, dx =.3 obtain approximation
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Propagated Error Consider a rectangular box with a square base Height is 2 times length of sides of base Given that x = 3.5 You are able to measure with 3% accuracy What is the error propagated for the volume? x x 2x
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Propagated Error We know that Then dy = 6x 2 dx = 6 * 3.5 2 * 0.105 = 7.7175 This is the approximate propagated error for the volume
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Propagated Error The propagated error is the dy sometimes called the df The relative error is The percentage of error relative error * 100%
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Assignment Lesson 4.8 Page 276 Exercises 1 – 45 odd
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