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Quiz 9: Taylor series and Eulers formula

Last unanswered question  Question  Next unanswered question
 

Question 1

 
 
Find the coefficient of x2 in the Taylor series about x = 0 for f(x ) = e-x2 .
a) 1/4   b) -1
c) 1/2   d) -2
e) 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 2

 
 
Find the coefficient of x3 in the Taylor series about x = 0 for f(x) = sin2x.
a) -2/3   b) -4/3
c) 4/3   d) -8/3
e) 2/3

 

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 3

 
 
Which one of the following is the Taylor series for sinx?
a) ∑∞ (--1)nx2n-1 n=1 (2n - 1)!   b) ∞∑ (--1)n+1x2n+1- n=0 (2n+ 1)!
c) ∞∑ (--1)nx2n+1- n=1 (2n+ 1)!   d) ∞∑ (--1)nx2n+1- n=0 (2n+ 1)!

 

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

Question 4

 
 
Which of the following are true statements?
a) Any function f(x) is equal to its Taylor series for all real x.
b) There exist functions f(x) which are equal to their Taylor series for all real x.
c) There exist functions f(x) which are equal to their Taylor series for some, but not all, real numbers x.
d) A function f(x) can never equal its Taylor series. The Taylor series is only ever an approximation to the function.

 

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

Question 5

 
 
∑∞ (- x)n n=0 is the Taylor series for which function?
a) sinx   b) cosx
c) --1-- 1 - x   d) --1-- 1 + x
e) ex

 

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 6

 
 
Find the Taylor series about x = 0 for ---1--- f(x) = (1- x)2 .
a) ∞ ∑ n (n + 1)x n=0   b) ∑∞ (- 1)n(n+ 1)xn n=0
c) ∑∞ nxn n=0   d) ∑∞ 2n x n=0
e) ∑∞ - x2n n=0

 

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 7

 
 
The simplest possible cartesian form of 8eiπ is
a) 8   b) 8π
c) 8 - i   d) 8i
e) -8

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Your answer is correct.
Note iπ e = cosπ+ isin π = - 1+ 0i so 8eiπ = -8.
 

Question 8

 
 
If z = -1 + i then z expressed in exponential notation equals
a) √2 cis π 4   b) √ - i3π∕4 2e
c) -1- -iπ∕4 √2-e   d) exp π- 4
e) None of the above

 

Not correct. Choice (a) is false.
Your answer is correct.
√ - 2ei3π∕4 = √ - 2(cos3π -4- + isin3π 4--)
= √ - 2(- 1 √-- 2 + i 1 √-- 2)
= -1 + i.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.
 

Question 9

 
 
Find uv if u = 2exp(- π-i) 2 and v = 1 exp(5π-i) 2 3 .
a) (5π2) exp -6-   b) (7-π-) exp 6 i
c) ( ) exp - 5πi 6   d) (13π--) exp 6 i

 

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

Question 10

 
 
Which of the following are equal to (1 + i)10?
a) 0   b) (5πi) 32exp 2
c) 32i   d) √ - ( 5πi) 2exp 2
e) ( ) exp 5πi 2

 

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