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Taylor Series

# Forrest2 45 introduction to taylor series chapter 4

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Unformatted text preview: .01 | = | R2,0 (.01) | f (c) =| (.01 − 0)3 | 6 − cos(c) =| (.01)3 | 6 < 10−6 This shows that the estimate that sin(.01) ∼ .01 is accurate to six decimal places. In fact, = the actual error is approximately −1.666658333 × 10−7 . Finally, we know that for 0 < x < π , the tangent line to the graph of f (x) = sin(x) is above 2 the graph of f (x) since sin(x) is concave downward on this interval. (In fact, the Mean Value Theorem can be used to show that sin(x) ≤ x for every x ≥ 0.) Since the tangent line is the graph of our linear approximation, this means that our estimate is actually too large. DE Math 128 327 (B. Forrest)2 4.5. Introduction to Taylor Series CHAPTER 4. Sequences and Series Taylor’s Theorem can be used to conﬁrm this because sin(.01) − .01 = R1,0 (.01) = − sin(c) (.01)2 < 0 2 since sin(c) > 0 for any c ∈ [0, .01]. 2 We have seen that it is not possible to ﬁnd a nice antiderivative of the function f (x) = e−x . This means that we cannot use the Fundamental Theorem of Calcu...
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