b. Write the third-order Taylor Series for f(x) = 2 sin(x) + x3 about xo terms up to and including the third derivative of f (x). Include your simplified third-order Taylor series approximation: f(x)Taylor =? · Hint 1: f(x)raylor = f (xo) + f'(xo)(x – xo) + f(n) (xo) O showing the f() (xo) (x – xo) 3+ + z(°x – x). f"(xo) 2! 3! (x - xo)" ... п! Hint 2: 3!=6 Hint 3: f'(x) = 2 cos(x) + 3x² f"(x) = -2sin(x) + 6x f(®)(x) = -2 cos(x) + 6 c. Use the result above to evaluate f(2)Taylor (show only four decimals) d. Calculate the true error for the Taylor Series approximation to the value at f (2) Hint 1: Using a calculator set to Radian, the true value of f (2) can be calculated as: 9.8186 Hint 2: The true error is defined as |f(x) – f(x)raylor|
b. Write the third-order Taylor Series for f(x) = 2 sin(x) + x3 about xo terms up to and including the third derivative of f (x). Include your simplified third-order Taylor series approximation: f(x)Taylor =? · Hint 1: f(x)raylor = f (xo) + f'(xo)(x – xo) + f(n) (xo) O showing the f() (xo) (x – xo) 3+ + z(°x – x). f"(xo) 2! 3! (x - xo)" ... п! Hint 2: 3!=6 Hint 3: f'(x) = 2 cos(x) + 3x² f"(x) = -2sin(x) + 6x f(®)(x) = -2 cos(x) + 6 c. Use the result above to evaluate f(2)Taylor (show only four decimals) d. Calculate the true error for the Taylor Series approximation to the value at f (2) Hint 1: Using a calculator set to Radian, the true value of f (2) can be calculated as: 9.8186 Hint 2: The true error is defined as |f(x) – f(x)raylor|
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:b. Write the third-order Taylor Series for f(x) = 2 sin(x) + x³ about x, = 0 showing the
terms up to and including the third derivative of f (x). Include your simplified third-order
Taylor series approximation: f(x)Taylor =? ·
Hint 1: f(x)raylor = f (xo) + f'(xo(x – xo) +
f(m) (xo) (x – xo)"“
%3D
f"(xo)
(x – xo)² +
f(3) (xo)
(x – xo) 3 +
2!
3!
+
п!
Hint 2: 3!=6
Hint 3: f'(x) = 2 cos(x) + 3x² f"(x) = –2sin(x) + 6x f()(x) = -2 cos(x) + 6
c. Use the result above to evaluate f(2)Taylor (show only four decimals)
d. Calculate the true error for the Taylor Series approximation to the value at f(2)
Hint 1: Using a calculator set to Radian, the true value of f (2) can be calculated as:
9.8186
Hint 2: The true error is defined as |f(x) – f(x)Taylor|
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