Calculus I
Test #2
October  22, 2001
Name____________________
R.  Hammack
Score ______


(1)   Suppose   f(x) = 1/(2x).     Use the limit definition of the derivative to find  f '(x).
  
  


(2) Find the derivatives of the following functions. You may use any applicable rule.
  
(a)    f(x) = 4x^10+ 3x^3+   π^2


  
(b)   f(x) = (x + sin x)/(cos x)



(c)  y = x tan(x)



(d) d/dx[ 25 + cos( x ^4) ] =


(e) d/dx[ (x^2 + 4)^(1/2) ] =


(3) Suppose f(x) equals the number of dollars it costs to erect an x-foot-high transmitting tower.
(a) What are the units of f '(x)?



(b) Suppose that f '(100) = 105.    Explain, in ordinary English, what this means.



(4) This problem concerns the function f that is graphed below
[Graphics:HTMLFiles/2F01sol_38.gif]


(a)    Sketch the graph of f '(x). (Use the same coordinate axis)

(b)    Suppose g(x) = sin(f(x)). Find g '(4).



(c)   Suppose h(x) = 4 + x^2+ x^2f(x).   Find h'(2).



(5) Sketch the graph of a function  f  whose derivative has the following properties:
f(0) = 2,     f '(0) = 0,    f '(3) = 0,   and  f '(x) ≤ 0 for all values of x.

  



(6) Consider the function  f(x) = x + x^(1/2)

(a) Find the slope of the tangent line to the graph of f at the point where x = 4.


(b) Find the equation of the tangent line to the graph of f at the point where x = 4.




(7) Find all values of x for which the slope of the tangent to the graph of   y = sin x   at the point   x   is  1/2 .

 



(8)  Find the slope of the tangent to the graph of   tan(xy) = y^2 at the point (π/4,  1).

(9) Suppose a 10-foot-long ladder is sliding down a wall in such a way that the base of the ladder moves away from the wall at a constant rate of 2 feet per second.  How fast is the top of the ladder moving down the wall when it is 6 feet above the floor?