Find the Maclaurin Series for questions 1 and 2.

1. f(x) = ℮^5x
A)     Σ (5^n / n!) (x^n)
B)     Σ (5 / n!) (x^n)
C)     Σ (5x / n!) (x^n)
D)     Σ (5^x / n!) (x^n)

2. f(x) = x℮^x
A)     Σ x^(n^2) / n!
B)     Σ x^2n / n!
C)     Σ x^(n+1) / n!
D)     Σ x^n / n

Find the Taylor Series for questions 3 and 4.

3. f(x) = 1 + x + x^2     a = 2
A) 7 + 7x + 7x^2
B) 7 + 7(x-2) + (x-2)^2
C) 7 + 5(x-2) +(x-2)^2
D) 7 + 5(x-2) + 3(x-2)^2

4. f(x) = cos(x)    a = π
A) Σ ((-1)^(n + 1) (x - π)^n) / n!
B) Σ ((-1)^n (x - π)^n) / n!
C) Σ ((-1)^ (n + 1) (x - π)^2n) / (2n)!
D) Σ ((-1)^n (x - π)^2n) / (2n)!

5. Find the Maclaurin series for f(x) = 1/x using the Maclaurin series for f(x) = ln(x).
A) Σ (-1)^n (x - 1)^n
B) Σ (-1)^n (x - 1)^(n + 1)
C) Σ (-1)^(n + 1) (x - 1)^n
D) Σ (-1)^n x^n

6. What is the third-degree Maclaurin series polynomial for f(x) = sin(x)?
A)     x + (x^2/2!) - (x^3/3!)
B)     x – (x^2/2!) + (x^4/4!)
C)     x + (x^3/3!) - (x^5/5!)
D)     x - (x^3/3!) + (x^5/5!)

7. What is the third-degree Taylor series polynomial for f(x) = cos(x) at a = 8?
A)     1 – ((x – 8)^3 / 3!) + ((x – 8)^4 / 4!) 
B)     1 – ((x – 8)^2 / 2!) + ((x – 8)^4 / 4!)  
C)     1 + ((x – 8)^2 / 2!) - ((x – 8)^4 / 4!)  
D)     1 – ((x – 8)^3 / 3!) + ((x – 8)^5 / 5!)  

8. What is the function f(x) for the Maclaurin series Σ ((-1)^n x^(2n + 1)) / (2n)! ?
A) xcos (x)
B) sin (x)
C) cos (x^2)
D) cos (x)

9. If f(x) = Σ bn (x - 5)^n for all x, what is b8 (n = 8)?
A) f^5 (5) / 8!
B) f^8 (5) / 8
C) f^5 (8) / 5!
D) f^8 (5) / 8!

10. Use the Maclaurin series to calculate е^-0.2 correct to five decimal places.
A) 0.98124
B) 0.88723
C) 0.81873
D) 0.81855