November 19, 2001
(1) Consider the function f(x)
(a) Find the inverse of f.
y(2x - 4) = 1
2xy -4y = 1
2xy = 1 + 4y
y = =
(b) Find the range of .
Range of = (Domain
of f) = all real numbers except 2
(By looking at the formula for f.)
(c) Find the domain of .
(domain of )
= all real numbers except 0
(By looking at the formula for .)
(d) Find the range of f.
(Range of f )= (Domain of ) = all
real numbers except 0
(By part c above.)
(2) Simplify the following expressions. Your answers should
contain neither an e nor an ln.
(b) ln( ln(e) )
(4) Differentiate the following functions.
(5) This problem involves the function .
(a) Find all critical points of f.
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.
+ + + | - - - - - | + + + + 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
(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].