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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.

No. It's not one-to-one:

,
and ,

so g(1) = g(-1).

It fails the H.L.T.

(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 .

SOLUTIONS: 2, 3

(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.