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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:

(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.