Calculus I
Quiz # 8
November 19, 2003
Name____________________
Score ______
R.  Hammack 

(1)  Simplify the following expressions as much as possible.

(a)   sin^(-1)(3^(1/2)/2) = π/3

(b)   tan ( sin^(-1)(x) ) =OPP/ADJ = x/(1 - x^2)^(1/2)

 

(c)    sin^(-1) ( sin(π) ) = sin^(-1) ( 0 ) = 0

 [Graphics:HTMLFiles/quiz8sol_5.gif]

Note:  sin ( sin^(-1)(x) ) = x  for all x in the domain of sin^(-1), but the equation sin^(-1) ( sin(x) ) = x  is not true for all values of x. This is an example where it is not true.

(d)   ln(e/25) + 2ln(5e) = ln(e/25) + ln((5e)^2) = ln(e/25) + ln(25e^2) = ln(e/2525e^2) = ln(e^3) = 3

(e) d/dx[ln(2)] =0
  


  
(2)
(a)   d/dx[ln(x^2 + 1)^(1/2)] =d/dx[ ( ln(x^2 + 1)   )^(1/2)] = 1/2 ( ln(x^2 + 1)   )^(-1/2) (2x)/(x^2 + 1) = x/(ln(x^2 + 1)^(1/2) (x^2 + 1))

(b)  d/dx[ x e^(-x) ] =(1) e^(-x) + x e^(-x)(-1) = e^(-x) - x e^(-x) = e^(-x) (1 - x )



(3)
(a)  If f(x) = tan^(-1)(x),  find f ' (6).

f ' (x) = 1/(1 + x^2), so  f ' (6) = 1/(1 + 6^2) = 1/37


(b) d/dx[sec^(-1)(x^7)] =1/(| x^7 | ((x^7)^2 - 1)^(1/2)) 7x^6 = (7x^6)/(| x^7 | (x^14 - 1)^(1/2)) = (7x^6)/(x^6 | x | (x^14 - 1)^(1/2)) = 7/(| x | (x^14 - 1)^(1/2))