Calculus I
Test #3
November 19, 2001
R.  Hammack
Score _______

(1) Consider the function    f(x) = [Graphics:Images/3F01sol_gr_1.gif].

(a)  Find the inverse of f.
y = [Graphics:Images/3F01sol_gr_2.gif]
  y(2x - 4) = 1
  2xy -4y = 1
  2xy = 1 + 4y
  y = [Graphics:Images/3F01sol_gr_3.gif]= [Graphics:Images/3F01sol_gr_4.gif]+ 2
  Thus  [Graphics:Images/3F01sol_gr_5.gif][Graphics:Images/3F01sol_gr_6.gif]+ 2

(b)  Find the range of [Graphics:Images/3F01sol_gr_7.gif].
Range of  [Graphics:Images/3F01sol_gr_8.gif]  =  (Domain of f) = all real numbers except 2
(By looking at the formula for f.)

(c)  Find the domain of  [Graphics:Images/3F01sol_gr_9.gif].
(domain of  [Graphics:Images/3F01sol_gr_10.gif]) = all real numbers except  0
(By looking at the formula for [Graphics:Images/3F01sol_gr_11.gif].)

(d)  Find the range of f.
Range of f )= (Domain of [Graphics:Images/3F01sol_gr_12.gif])  =  all real numbers except 0
(By part c above.)

(2)   Simplify the following expressions. Your answers should contain neither an e nor an ln.

(a)  [Graphics:Images/3F01sol_gr_13.gif]

(b)   ln( ln(e) ) [Graphics:Images/3F01sol_gr_14.gif]

(c)   [Graphics:Images/3F01sol_gr_15.gif] [Graphics:Images/3F01sol_gr_16.gif]

d)  [Graphics:Images/3F01sol_gr_17.gif]  [Graphics:Images/3F01sol_gr_18.gif]

(a) [Graphics:Images/3F01sol_gr_19.gif][Graphics:Images/3F01sol_gr_20.gif]/4

(b) [Graphics:Images/3F01sol_gr_21.gif][Graphics:Images/3F01sol_gr_22.gif][Graphics:Images/3F01sol_gr_23.gif]
(4)  Differentiate the following functions.

(a)   [Graphics:Images/3F01sol_gr_24.gif][Graphics:Images/3F01sol_gr_25.gif]

(b)  [Graphics:Images/3F01sol_gr_26.gif][Graphics:Images/3F01sol_gr_27.gif]

(c)  [Graphics:Images/3F01sol_gr_28.gif]

(5) This problem involves the function [Graphics:Images/3F01sol_gr_29.gif].

(a)   Find all critical points of f.
[Graphics:Images/3F01sol_gr_30.gif][Graphics:Images/3F01sol_gr_31.gif] [Graphics:Images/3F01sol_gr_32.gif]
The derivative is defined for all values of x but equals zero when x is 1 or  -1.
  Therefore, the critical points of f are 1 and -1.
(b) Find the interval(s) on which f increases and on which it decreases.
          -1           1
  + + + | - - - - - | + + + + f  '(x)
  f increases on (-∞, -1] and [1,∞)
  f decreases on [-1, 1]

(c) Identify the locations of any extrema of f.  Classify them as relative maxima or minima.
Using the first derivative test,
f has a relative maximum at x= -1.
f has a relative minimum at x = 1.

(6) The graph of the derivative f ' of a function f is sketched.Supply the following information about the function f.


(a)    List the critical points of f.
These occur where f ' (x) is zero or undefind. Thus the critical points are -3, 1, 4

(b)   State the interval(s) on which f is decreasing.
This happens where f ' is negative, namely an (-∞,-3], [-3,1] and [4,∞)

(c)   State the intervals(s) on which f is concave up.
This happens where f ' is increasing, namely on  (-∞,-3], [-1,2]

(d)   At which value of x does f have a relative minimum?
Where f ' switches from neagtive to positive, namely at x=1.

(e)  Using the same coordinate axis, sketch a possible graph of f.
The possible sketch of f is drawn in heavy line.

(7) The graph of the second derivative g'' of a function g is sketched. Suppose you also know that the first derivative g '  has x-intercepts at -3, 0, and 4.  Supply the following information about the function g.


(a) At what values of x does g have a relative maximum?
It must happen at a critical point of g, namely at one of the x values -3, 0, and 4.
Of these points, the second derivative is negative at -3 and 4.
Therefore, by the second derivative test, g has a relative maximum at x=-3 and x=4.

(b) State the intervals on which g is increasing.
This happens where the first derivative g ' (x) is positive. We know the x-intercepts of g' are -3, 0, and 4, and we can tell where it's increasing and decreasing by the sign of its derivative g ''(x), which is sketched. Based on this information, a very rough sketch of g ' is drawn above in bold. You can see that the intervals on which g increases are (-∞,-3] , [0,4].