Review for Test 4

Test on April 17, 2000

Instructions.  Show your work and/or explain your answers
  1. Find T3( x) for f(x) = Öx centered at p = 4.
  2. Find the quadratic approximation centered at 0 of the solution to
    3. y' =  x
    y    		 ,    y(0) = 2
  3. Find the general form of Tn(x) centered at p = 2 when f(x) = ln(x) .
  4. Find the general form of Tn(x) centered at p = p when f(x) = sin(x) .
  5. Use Taylor's theorem to find the maximum error in approximating f(x) = x3 by L0(x) = x over the interval [ -0.1,0.1] . Sketch the graphs of f(x) and L0(x) over [ -0.1,0.1] . What error estimate do you obtain from the graphs?
  6. Use Taylor's theorem to find the maximum error in approximating f(x) = x4 by T4(x) centered at p = 1 over the interval      [-2,2] .
  7. What is the Maclaurin series expansion of
    8. 2x
    x    2-1
  8. Use the geometric series and the fact that
    9. d
    dx    		 ln( 1-x2) =   2x
    x    2-1
    to find the Maclaurin series expansion of ln(1-x2) . Where does it converge?
  9. Find the Maclaurin series expansion of cosh( x) using the fact that
    10. cosh( x) =  ex+e-x
    2
  10. Use the fact that i2 = -1, i4 = 1, i6 = -1, i8 = 1, ... to show that
    11. cos( ix) = cosh( x) 
  11. Compute the product
    12. ( 1+x) ( x2+x3+x4+x5+¼) 
    Bonus: How many ways are there of choosing 5 letters from a group of 2 letters if the first letter can occur at most once and the second letter must appear at least twice?
  12. Find the open interval of convergence of the series
    13. ¥
    å
    n = 0    		 n
    2n    		 ( x-1) n
  13. Find the open interval of convergence of the series
    14. ¥
    å
    n = 0    		 ( 2n) !
    ( n!) 2    		 xn