If a population is distributed normally with mean 0 and standard deviation 1, then the probability that a sample will fall within s standard deviations of the mean is S 1 ,-x²/2 dx. Unfortunately, integration techniques you learned in single-variable calculus will not help you take the antiderivative of e-**12. Fortunately, we can approximate this definite integral by replacing f(x) = e-**12 with one of its polynomial approximations. (a) Let f(x) = e-x-12. Find f' (x) and f" (x). Factor e x²/2 out of each answer. (b) Write down the linear and quadratic approximations for f(x) at 0. (c) Use Taylor's formula to find an error bound for your linear approximation that is valid on the interval (-, ). (Hint: f(2) is a product of terms. To find a bound for 2? f(2), start by finding a bound for each factor.)

If a population is distributed normally with mean 0 and standard deviation 1, then the probability that a sample will fall within s standard deviations of the mean is S 1 ,-x²/2 dx. Unfortunately, integration techniques you learned in single-variable calculus will not help you take the antiderivative of e-12. Fortunately, we can approximate this definite integral by replacing f(x) = e-12 with one of its polynomial approximations. (a) Let f(x) = e-x-12. Find f' (x) and f" (x). Factor e x²/2 out of each answer. (b) Write down the linear and quadratic approximations for f(x) at 0. (c) Use Taylor's formula to find an error bound for your linear approximation that is valid on the interval (-, ). (Hint: f(2) is a product of terms. To find a bound for 2? f(2), start by finding a bound for each factor.)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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