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Calculus I                                            Test #3                             May 4, 2004

Name__________________           R.  Hammack                   Score ______
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(1)  Sketch the graph of the function .
The function y = is drawn dashed.
The function y = is the previous graph shifted one unit right. It's drawn dotted.
The function y = is the previous graph shifted 2 unit down. It's drawn solid.

(2) Find the inverse of the function .

(3) Does the function    have an inverse?  Explain.

Notice that   can have either positive or  negative values, depending on x.
Examples:

Thus the function g increases and decreases. It fails the Horizontal Line Test.
There is no inverse.

(4) Solve the equation   .  (It's OK to have logarithms  in your final answer.)

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

(a)

(b)

(c)

(d)

(6)  Find the following derivatives.

(a)

(b)

(c)

(d)

(7) The graph of the derivative of a function f(x) is drawn. Answer the following questions about the function f(x).

(a) Find the interval(s) on which f(x) increases.
f(x)  INCREASES on and [2,3], because that's where its derivative is positive.

(b)  Find the critical points of .
-3, 0, 2, 3

(c)  Find the locations of the relative extrema of , and identify them as relative maxima or minima.
Relative Max at 0 and 3 (Derivative switches from + to -)
Relative Min at 2 (Derivative switches from - to +)

(d)  Find the intervals on which the graph of f(x) is concave up/down.
CONCAVE UP on [-3 ,-1] & [1, 2.5] (That's where f  ' increases, so f " is positive.)
CONCAVE DOWN on & [-1, 1] & (That's where f  ' decreases, so f " is negative.)

(a) Find the critical points of .

The critical points (which make the derivative 0) are 0 and 1

(b)  Find the intervals of increase/decrease of f(x).
0     1
----+---+-----
- - - - -  | + + f '(x)

f increases on

f decreases on

(c)  Find the locations of the relative extrema of , and identify them as relative maxima or minima.
Relative minimum at x = 1,
No relative maximum.

(d)  Find the intervals on which the graph is concave up/down.

0     2/3
----+---+-----
+ +| - - | + + f ''(x)

f concave up on

f concave down on