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
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Chapter2: Second-order Linear Odes
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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
-x²12
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
f(2), start by finding a bound for each factor.)
2
(d) Use your quadratic approximation in part (b) to find an estimate for
1/2
1
e¯x?12
dx.
Transcribed Image Text: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 -x²12 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 f(2), start by finding a bound for each factor.) 2 (d) Use your quadratic approximation in part (b) to find an estimate for 1/2 1 e¯x?12 dx.
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