3. Consider the usual exponential function f(x) = e. (a) Without using a calculator, use linear approximation to estimate e-0.015 Express your answer as a fraction of two integers. Hint: What is f(x)? What is the "anchor point" = a? What is the point zo at which we are approximating? -0.015 (b) Without using a calculator, determine if your answer in (a) is an overestimate or an underestimate. Circle the best answer choice and justify your answer. Feel free to give a graphical representation. UNDERESTIMATE OVERESTIMATE (c) Use a calculator to find the absolute error Efo(-0.015) that results from using the linear approx- imation to estimate e-0.015. Round your answer to the nearest seventh decimal place. Efo(-0.015) (d) Use a calculator to find the relative percentage error that results from using the linear approxi- mation to estimate e-0.015 Reol-0015)
3. Consider the usual exponential function f(x) = e. (a) Without using a calculator, use linear approximation to estimate e-0.015 Express your answer as a fraction of two integers. Hint: What is f(x)? What is the "anchor point" = a? What is the point zo at which we are approximating? -0.015 (b) Without using a calculator, determine if your answer in (a) is an overestimate or an underestimate. Circle the best answer choice and justify your answer. Feel free to give a graphical representation. UNDERESTIMATE OVERESTIMATE (c) Use a calculator to find the absolute error Efo(-0.015) that results from using the linear approx- imation to estimate e-0.015. Round your answer to the nearest seventh decimal place. Efo(-0.015) (d) Use a calculator to find the relative percentage error that results from using the linear approxi- mation to estimate e-0.015 Reol-0015)
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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Transcribed Image Text:3. Consider the usual exponential function f(x) = e.
(a) Without using a calculator, use linear approximation to estimate e-0.015 Express your answer
as a fraction of two integers. Hint: What is f(x)? What is the "anchor point" = a? What
is the point zo at which we are approximating?
-0.015
(b) Without using a calculator, determine if your answer in (a) is an overestimate or an underestimate.
Circle the best answer choice and justify your answer. Feel free to give a graphical representation.
UNDERESTIMATE
OVERESTIMATE
(c) Use a calculator to find the absolute error Efo(-0.015) that results from using the linear approx-
imation to estimate e-0.015. Round your answer to the nearest seventh decimal place.
Ef.o(-0.015)
(d) Use a calculator to find the relative percentage error that results from using the linear approxi-
mation to estimate e-0.015
Rfo(-0.015)=
Transcribed Image Text:Upper Bound for the Error Using Linear Approximation. Suppose that f is a function
such that f" is bounded on some interval centered at x = a; that is, there exist positive real
numbers K and u such that f"(x)| ≤K for all a in I = (a-u, a+u). Then for every ro € I,
we have that Ef,a (20) = |Lf₁a(xo) - f(xo)| ≤ K | 20-a².
(e) Use the upper bound for the error in linear approximation to find an interval of the form T =
(-6, 6) contained in I = (-1,1) guaranteeing that Efo(ro) <1 whenever ro € T.
(f) Use the upper bound for the error in linear approximation to find an interval T = (-6, 6) contained
in I = (-1, 1) guaranteeing that Efo(ro) < 106 whenever To T.
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