Calculus I
Test #1
September 26, 2001
Name____________________ 
R.  Hammack
Score _______


(1)
  (a)   sin 30π = 0
  
(b)    cos [Graphics:Images/1F01sol_gr_1.gif]= [Graphics:Images/1F01sol_gr_2.gif]
  (c)   Find all values of  θ  for which  cos θ = [Graphics:Images/1F01sol_gr_3.gif]
  
  
  [Graphics:Images/1F01sol_gr_4.gif] + 2nπ    and    -[Graphics:Images/1F01sol_gr_5.gif] + 2nπ   where n is an integer.
  
  
  (2) In this problem   0 < x < [Graphics:Images/1F01sol_gr_6.gif] and   sin x = [Graphics:Images/1F01sol_gr_7.gif]
  
  (a)    cos x  = [Graphics:Images/1F01sol_gr_8.gif] (use the identity [Graphics:Images/1F01sol_gr_9.gif]x = 1 to solve for cos x.)

(b)    tan x = [Graphics:Images/1F01sol_gr_10.gif]= [Graphics:Images/1F01sol_gr_11.gif] (here we used the fact   tan x = [Graphics:Images/1F01sol_gr_12.gif])
  
(3) In this problem  f(x) = [Graphics:Images/1F01sol_gr_13.gif]  and   g(x) = x - 4.
  
  (a)    ([Graphics:Images/1F01sol_gr_14.gif])(x)  = [Graphics:Images/1F01sol_gr_15.gif]= [Graphics:Images/1F01sol_gr_16.gif]
  
  
  (b)   (f º g)(x) = [Graphics:Images/1F01sol_gr_17.gif]= [Graphics:Images/1F01sol_gr_18.gif]= [Graphics:Images/1F01sol_gr_19.gif]
  
  (c)   (g º g)(x) = g(x) - 4 = x - 4 - 4 = x - 8

(4) This problem concerns the function  f(x) = [Graphics:Images/1F01sol_gr_20.gif]

(a)    f(x + 2) =  [Graphics:Images/1F01sol_gr_21.gif]

(b)    Find the domain of f.  Be sure to show your work.

We require that[Graphics:Images/1F01sol_gr_22.gif] [Graphics:Images/1F01sol_gr_23.gif] ≥ 0 and x is not 0.
i.e. we require  [Graphics:Images/1F01sol_gr_24.gif]≥ 0 and x ≠ 0.



Thus the domain is [-2, 0) [2, ∞)


(c)   Write f as a composition of two functions.
Let h(x) = [Graphics:Images/1F01sol_gr_25.gif]
and g(x) = [Graphics:Images/1F01sol_gr_26.gif]
Then f(x) = h(g(x))

(d)     [Graphics:Images/1F01sol_gr_27.gif]f(x) =  [Graphics:Images/1F01sol_gr_28.gif][Graphics:Images/1F01sol_gr_29.gif]= [Graphics:Images/1F01sol_gr_30.gif]= [Graphics:Images/1F01sol_gr_31.gif]= [Graphics:Images/1F01sol_gr_32.gif]

(e)    [Graphics:Images/1F01sol_gr_33.gif]f(x) =  [Graphics:Images/1F01sol_gr_34.gif][Graphics:Images/1F01sol_gr_35.gif]=  [Graphics:Images/1F01sol_gr_36.gif]= [Graphics:Images/1F01sol_gr_37.gif]= [Graphics:Images/1F01sol_gr_38.gif]
     = [Graphics:Images/1F01sol_gr_39.gif]= 0

  (f)    [Graphics:Images/1F01sol_gr_40.gif] f(x) =   [Graphics:Images/1F01sol_gr_41.gif] [Graphics:Images/1F01sol_gr_42.gif] = ∞  (numerator gets close to -4, denominator is close to 0, negative)


(5) Calculate the limits.
  
(a)    [Graphics:Images/1F01sol_gr_43.gif](3[Graphics:Images/1F01sol_gr_44.gif]- 4x + π) = 3[Graphics:Images/1F01sol_gr_45.gif]- 4(3) + π = 15 + π

(b)    [Graphics:Images/1F01sol_gr_46.gif][Graphics:Images/1F01sol_gr_47.gif] =  [Graphics:Images/1F01sol_gr_48.gif]= [Graphics:Images/1F01sol_gr_49.gif]= 0


(c)     [Graphics:Images/1F01sol_gr_50.gif][Graphics:Images/1F01sol_gr_51.gif] =  [Graphics:Images/1F01sol_gr_52.gif][Graphics:Images/1F01sol_gr_53.gif]=   [Graphics:Images/1F01sol_gr_54.gif]

(d)   [Graphics:Images/1F01sol_gr_55.gif][Graphics:Images/1F01sol_gr_56.gif] = ∞  (note, top approaches 18 while bottom is positive, approaching 0)

(e)   [Graphics:Images/1F01sol_gr_57.gif][Graphics:Images/1F01sol_gr_58.gif] =  [Graphics:Images/1F01sol_gr_59.gif][Graphics:Images/1F01sol_gr_60.gif]=  [Graphics:Images/1F01sol_gr_61.gif][Graphics:Images/1F01sol_gr_62.gif]= [Graphics:Images/1F01sol_gr_63.gif][Graphics:Images/1F01sol_gr_64.gif]=  [Graphics:Images/1F01sol_gr_65.gif][Graphics:Images/1F01sol_gr_66.gif]= [Graphics:Images/1F01sol_gr_67.gif] = 1

(f)   [Graphics:Images/1F01sol_gr_68.gif][Graphics:Images/1F01sol_gr_69.gif] =  [Graphics:Images/1F01sol_gr_70.gif][Graphics:Images/1F01sol_gr_71.gif][Graphics:Images/1F01sol_gr_72.gif]= [Graphics:Images/1F01sol_gr_73.gif][Graphics:Images/1F01sol_gr_74.gif]=  [Graphics:Images/1F01sol_gr_75.gif][Graphics:Images/1F01sol_gr_76.gif]= [Graphics:Images/1F01sol_gr_77.gif]= [Graphics:Images/1F01sol_gr_78.gif]