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Quiz 8: Taylor polynomials

Last unanswered question  Question  Next unanswered question
 

Question 1

 
 
Find the Taylor polynomial of degree 3 about x = 0 for the function f(x) = e-x.
a) 1 2 1 3 1 - x+ 2x - 6x   b) 1 + x+ 1x2 - 1x3 2 6
c) 1 - x+ x2 - x3   d) 1 + x+ x2 + x3
e) 1 - x+ 1x2 - 1x3 2 3

 

Your answer is correct.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.
 

Question 2

 
 
Find the Taylor polynomial of degree 3 about x = 0 for the function f(x) = ln(1 + x).
a) 1 - x+ x2 - x3   b) x - x2 + x3
c) 1 1 1 - x+ -x2 - -x3 2 6   d) 1 2 1 3 x - 2x + 3x
e) x - 1-x2 + 1--x3 24 120

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
Not correct. Choice (e) is false.
 

Question 3

 
 
Find the Taylor polynomial of degree 2 about x = 0 for 2 f (x) = e-x .
a) 1 + x+ x22   b) 1 - x2
c) 1 + x2   d) 2 1 + x+ x
e) 1 - 2x - x2

 

Not correct. Choice (a) is false.
Your answer is correct.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.
 

Question 4

 
 
Find the Taylor polynomial of degree 3 about x = 0 for f(x) = sin2x.
a) 2x - 8x3   b) x - x3 6
c) 2x + 4x3- 3   d) 2x - 4x3- 3
e) x3 x + 6

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
Not correct. Choice (e) is false.
 

Question 5

 
 
Find the Taylor polynomial of degree 3 about x = 0 for f(x) = xcosx.
a) x - x2 + x3 3   b) 2 x3 x - x - 3
c) x3 x - 2   d) x - x3
e) x - x3 6

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.
 

Question 6

 
 
Find the Taylor polynomial of degree 4 about x = 1 for f(x) = 1 x .
a) 2 3 4 1 - x+ x - x +x
b) 1 + x+ x2 + x3 +x4
c) 1 - (x - 1)+ (x - 1)2 - (x - 1)3 + (x- 1)4
d) 1 + (x - 1)+ (x - 1)2 + (x - 1)3 + (x- 1)4

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
Not correct. Choice (d) is false.
 

Question 7

 
 
Find the Taylor polynomial of degree 3 about x = π∕4 for f(x) = tan x .
a) 2 3 0.01+ (x- π∕4)+ 0.02(x - π∕4) + 0.33(x- π∕4)
b) 2 8 3 1 + 2(x - π∕4)+ 2(x - π∕4) + 3(x- π∕4)
c) 1 + 2(x - π∕4)+ 2(x - π∕4)2 + 2(x- π∕4)3
d) 1 + 2(x - π∕4)+ (x- π∕4)2 + 1(x- π∕4)3 6

 

Not correct. Choice (a) is false.
Your answer is correct.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
 

Question 8

 
 
If f(x) = 1+ 3x - 5x2 + 4x3 + 2x4 , which of the following are true statements?
a) The Taylor polynomial of degree 6 about x = 3 for f(x) is equal to f(x) .
b) The Taylor polynomial of degree 6 about x = 3 for f(x) does not exist.
c) The remainder term R5(x) for f(x) is equal to zero.
d) The remainder term R3(x) for f(x) is equal to zero.
e) The remainder term R3(2) for f(x) is equal to 32.

 

There is at least one mistake.
For example, choice (a) should be true.
There is at least one mistake.
For example, choice (b) should be false.
There is at least one mistake.
For example, choice (c) should be true.
There is at least one mistake.
For example, choice (d) should be false.
There is at least one mistake.
For example, choice (e) should be true.
Your answers are correct
  1. True.
  2. False.
  3. True.
  4. False.
  5. True.
 

Question 9

 
 
Find the difference between the Taylor polynomial of degree 4 about the point 0 for sinx evaluated at x = 1, and sin1.
a) cos1 5!
b) cosc --5! for some c between 0 and 1.
c) --cosc 5! for some c between 0 and 1.
d) cos1 ---- xn+1 5! for some x between 0 and 1.
e) --cos1 n+1 5! x for some x between 0 and 1.

 

Not correct. Choice (a) is false.
Your answer is correct.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.
 

Question 10

 
 
Find the coefficient of xn in the Taylor polynomial of degree n (n ≥ 2) for f(x) = √1-+-x- about x = 0.
a)
(- 1)n3.5.7....(2n-- 1) 2nn!
b)
(- 1)n3.5.7....(2n-- 1) 2n
c)
(- 1)n+13.5.7....(2n-- 3) 2n
d)
(- 1)n+13.5.7....(2n-- 3) 2nn!

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.