Question : Determine the first four non-zero terms of the Taylor series at x = 0 for f(x) = sinx^3. : 2163558
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
26) Find an infinite series that converges to the value of 
dx. Is (1/3) - (1/6) + (1/18) - (1/72) ... correct?
Enter "yes" or "no".
27) Find an infinite series that converges to the value of 
dx. Is 1 - (2/3) + (1/4) - (1/15) ... correct?
Enter "yes" or "no".
28) Find the Taylor series at x = 0 of f(x) = e3x by computing four derivatives and using the definition of the Taylor series. Is 1 - 3x + (32/2!)x2 - (33/3!)x3 + (34/4!)x4 - ... the correct answer?
Enter "yes" or "no".
29) Determine the first four non-zero terms of the Taylor series at x = 0 for f(x) = sinx3.
Is f(x) = x3 + (x9/3!) + (x15/5!) + (x21/7!) the correct answer?
Enter "yes" or "no".
30) Determine the first three non-zero terms of the Taylor series at x = 0 for f(x) = xcosx - sinx.
Is ((3! - 2!)x3/2! 3!) + ((5! - 4!)x5/4! 5!) + ((7! - 6!)x7/6! 7!) the correct answer?
Enter "yes" or "no".
31) Determine the first four non-zero terms of the Taylor series at x = 0 for f(x) = xe(1/2)x.
Is x - (x2/2) + (x3/22⋅2!) - (x4/23⋅3!) the correct answer?
Enter "yes" or "no".
32) Find the first four non-zero terms of the Taylor series at x = 0 for f(x) = 1 + xex.
Is 1 + x + x2 + (x3/2!) the correct answer?
Enter "yes" or "no".
33) Find the Taylor series at x = 0 of f(x) = x2e2x. Use enough terms to calculate 0.25e to two decimal places of accuracy.
Enter just a real number to 2 decimal places.
34) Find the first four non-zero terms of the Taylor series at x = 0 of f(x) = cos3x + sin2x.
Is 1 + 2x - (9x2/2!) - (8x3/3!) + ... the correct answer?
Enter "yes" or "no".
35) Find the first four non-zero terms of the Taylor series at x = 0 of f(x) = sin2x.
Is f(x) = 2x - (8x3/3!) + (32x5/5!) - (128x7/7!) the correct answer?
Enter "yes" or "no".
36) Find the first four non-zero terms of the Taylor series at x = 0 of f(x) = √(4 - x) .
Is f(x) = 2 - (x/4) - (x2/32⋅2!) - (3x3/256⋅3!) correct?
Enter "yes" or "no".
37) Find the first four non-zero terms of the Taylor series at x = 0 of f(x) = ex .
Is f(x) = 1 - x + (x2/2!) - (x3/3!) correct?
Enter "yes" or "no".
38) Find the first four non-zero terms of the Taylor series at x = 0 of f(x) = ln(x + 1) .
Is f(x) = x + (x2/2!) + (2x3/3!) + (6x4/4!) correct?
Enter "yes" or "no".
39) Find the first four non-zero terms of the Taylor series at x = 0 of f(x) = e-2x.
Is f(x) = 1 + 2x + (4x2/2!) + (8x3/3!) correct?
Enter "yes" or "no".
40) Find the first three non-zero terms of the Taylor series at x = 0 of f(x) = exsinx.
Is f(x) = x + x2 + (x3/3) correct?
Enter "yes" or "no".
41) The Taylor series at x = 0 for f(x) = ln((1 + x/1 - x)) is 2x + (2/3)x3 + (2/5)x5 + (2/7)x7 + ..., |x| < 1. Find f(6)(0).
Enter just an integer.
42) The Taylor series at x = 0 for f(x) = tanx is x + (1/3)x3 + (2/15)x5 + (17/315)x7 + ... . Find f(5)(0).
Enter just an integer.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
43) Find the Taylor series at x = 0 of the function f(x) = (1/1 - 3x) by computing three or four derivatives and using the definition of the Taylor series.
A) 1 + 3x + 9x2 + 27x3 + ...
B) 1 + (3x)2 + (3x)4 + (3x)6 + ...
C) 1 + (3x/1)x + (9x/2)x2 + ...
D) 1 - 3x + 9x2 - 27x3 + ...
E) none of these
44) Find the Taylor series at x = 0 of the function f(x) = xln(1 + 2x) by computing three or four derivatives and using the definition of the Taylor series.
A) 2x2 + (4x3/2!)
B) 2x2 - (4x3/2)
C) 2x - (4x2/2!) + (8x3/3!)
D) 2x2 - (4x3/3)
E) none of these
45) Find the Taylor series expansion for f(x) = (x/1 - x) and use it to determine which of the following is false?
A) - (3/5) = ((3/2)) + ((3/2))2 + ((3/2))3 +...
B) (x/1 - x) = x + x2 + x3 + x4 +...
C) (1/2) = (1/3) + ((1/3))2 + ((1/3))3 + ...
D) - (1/3) = (- (1/2)) + (- (1/2))2 + (- (1/2))3 + ...
E) All the statements are true.
46) The Bessel function f(x) of order zero has the Taylor series at x = 0 given by f(x) = 1 - (x2/4) + (x4/64) - (x6/2304) + ... . What is f(4)(0) ?
A) (1/4)
B) (3/8)
C) (1/64)
D) none of these
47) The Taylor Series for (1/(1 - x)2) at x = 0 is given by f(x) = 1 + 2x + 3x2 + 4x3 + ... . Find f(3)(0).
A) 3
B) (2/3)
C) 4
D) none of these
48) Find the Taylor Series at x = 0 for f(x) = (sinx/x).
A) x - (x2/3!) + (x4/5!) - (x6/7!) + ...
B) x2 - (x3/3!) + (x5/5!) - (x7/7!) + ...
C) 1 - (x2/3!) + (x4/5!) - (x6/7!) + ...
D) none of these
49) Find the Taylor Series at x = 0 for f(x) = ex + 1.
A) (1/e) + (x/e) + (x2/e2!) + (x3/e3!) + ...
B) e - ex + (ex2/2!) - (ex3/3!) + ...
C) e + ex + (ex2/2!) + (ex3/3!) + ...
D) none of these
50) The Bessel function of order 1 has the Taylor Series at x = 0 given by f(x) = (x/2) - (x3/16) + (x5/384) - ..., find f(5)(0).
A) (1/384)
B) (5/16)
C) (5/384)
D) none of these
