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Calculus I Test
#3 December
1, 2004

Name____________________ R. Hammack Score
______

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(1) Find the inverse of the function

(2) Is the function invertible
or not? Explain.

Since is
positive for all x, the
function g(x)
increases and never decreases.

Hence it passes the Horizontal Line Test, so it's invertible.

(3) Find the equation of the tangent
line to the graph of
at the point where .

Slope at x is y
' =.
Thus the slope of the tangent is

Point of tangency is

Point-Slope formula:

ANSWER:

(4) Solve the equation .

(5) Simplify each expression as much
as possible.

(a)

(b)

(c)
0

(d)

(e)

(6) The graph of the
derivative of a function
f is given.

In each case, indicate whether the ? should be replaced with the symbol ,
, or
=.

(a) f(1) ? f(3) ANSWER:
> , because f decreases between
1 and 3 (its derivative is negative there).

(b) f
'(1) ? f
'(3) ANSWER: =, by reading straight from the graph.

(c) f "(1) ? f
"(3) ANSWER: <, by looking at slope on the graph of
f '

(7) Find the derivatives.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h) ,
using logarithmic differentiation, as below.

(8) Consider the function
.

(a) List all critical points of f.

From this you can read off the critical points as 0
and -3

(b) Find the intervals on which f
increases/decreases.

-3 0

---|------|-----

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

f increases between 0 and infinity.

f decreases between negative infinity
and 0

(c) Find
the intervals on which f is
concave up/down.

-3 -1

---|------|-----

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

f is concave down on [-3,-1]

Elsewhere, f is concave up

(d) Locate and
identify all extrema of f .

By first derivative test (see part b above) there is a relative minimum at x
= 0.

There is no relative maximum.

(e) List the
locations (x-values) of all inflection
points of f.

By part c above, the locations are -3 and -1.