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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 f(x) = 2 - ln(x).


[Graphics:HTMLFiles/T3F03Dsol_2.gif]


(2) Find the inverse of the function f(x) = 2 - ln(x).

 

 



(3) Does the function  g(x) = -2x^5 - 4x^3 - 7x + 20  have an inverse? Explain.

 



(4) The graph of a one-to-one function h(x) is given.  Using the same coordinate axis, sketch the graph of h^(-1)(x).
[Graphics:HTMLFiles/T3F03Dsol_16.gif]



(5) The function f(x) = x/(x + 1)is invertible.  Find the number a for which f^(-1)(a) = 3.

 



(6) Solve the equation   e^(-2x) - 3e^(-x) = -2.  Hint: Notice that this has the form of a quadratic.

 

 



(7)  FormBox[RowBox[{RowBox[{16, ^, RowBox[{(, RowBox[{-, 1.75}], )}]}], =}], TraditionalForm]
(8) ln(1/e) =

(9)  cos^(-1)(1/2) =

(10) sin(tan^(-1)(x)) =

(11) Use logarithmic differentiation to find the derivative of the function y = sin(x)^x.

 

 

 



(12)   d/dx[ x^3e^x ] =

(13)   d/dx[ π^sin(x) ] =

(14)   d/dx[   (e^(5x) + 3x)^(1/2)    + ln(x) + 4  ] =


(15)   d/dx[  ln ( sin^(-1)(x) ) ] =

(16)    d/dx[tan^(-1)(1/x)] =

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The problems on this page are about the function f(x) = e^(-x^2/2).

(17) Find the intervals of increase/decrease.

 


(18)  Find the critical points of f(x).

 



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

 



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