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Calculus I Test
#3 November
24, 2003

Name____________________ R. Hammack Score
______

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(1) Sketch the graph of the
function .

The graph of y = ln(x)
is drawn dashed.

The graph of y = -ln(x)
is drawn dotted. (Previous graph reflected across the x-axis.)

The graph of y = 2-ln(x)
is drawn solid. (Previous graph moved up 2 units.)

(2) Find the inverse of the function
.

(interchange x and y)

Thus, the inverse is

[Just as a quick check, note .]

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

Notice that is
negative for all values of x.

This means that the graph of g(x)
decreases, and never increases.

Consequently the function g(x)
passes the horizontal line test, and is therefore invertible.

(4) The graph of a one-to-one function
is given. Using the same coordinate axis, sketch the graph of

Reflecting across the line y = x,
gives the following graph of the inverse, drawn in bold.

(5) The function is
invertible. Find the number a
for which .

We seek an a for which .

Take f of both sides to get .

Now this becomes

(6) Solve the equation . Hint:
Notice that this has the form of a quadratic.

This equation is true if one of the two factors is 0,

that is if
or if .

Solving
gives ; ;
;
x
= 0

Solving
gives ; ;
;
x
= -ln(2)

The solutions are x = 0 and x = -ln(2).

(7)

(8)
-ln(e) = -1

(9)

(10)

(11) Use logarithmic differentiation
to find the derivative of the function .

(12)

(13)

(14)

(15)

(16)

The problems on this page are about the function .

(17) Find the intervals of increase/decrease.

Looking at this, you can see that f
'(x) is positive when x
is negative and negative when x is
positive.

Thus, f increases on
and decreases on

(18) Find the critical points
of .

x = 0 is the only critical point.

(19) Find the locations of
the relative extrema of ,
and identify them as relative maxima or minima.

Since f ' changes from positive
to negative at 0, f has a
relative maximum at 0.

This is the only extremum: no relative minimum.

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

-1 1

-----*------*-------

+ + | - - - | + + + f ' '(x)

f is
concave up on
and

f is
concave down on