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Calculus I                                                                 Final Exam                                   December 10, 2002

Name____________________                             R.  Hammack                                        Score ______
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(1) Evaluate the following limits.

(a)    Underscript[lim , x3] (x^2 - 2x )/(x + 1) =

(b)    Underscript[lim , x4] (x^2 - 16)/(x - 4) =

(c)   Underscript[lim , x4] (4 - x)/(2 - x^(1/2)) =

(d)    Underscript[lim , xπ^+] cot(x) =

(e)    Underscript[lim , x∞] (6x^5 + x)/(3x^5 - 8) =

(f)    Underscript[lim , x2] sin(2x - 4)/(x^2 - 4) =


(2) Use the limit definition of the derivative to find the derivative of the function f(x) = (x + 1)^(1/2).






(3) Find the following derivatives.

(a)    d/dx[  3x^8 + 2x + 1 ] =

(b)    d/dx[  sin(π) + ln(x) ] =

(c)    d/dx[  x^(1/2) + 1/x ] =

(d)    d/dx[   (2x - 1)/(x + 3) ] =

(e)    d/dx[   (x sin(x))/(x + 1) ] =

(f)   d/dx[   x e^x] =

(g)    d/dx[ cos(5x)^(1/2)   ] =

(h)    d/dx[   x^3sec(1/x) ] =

(i)    d/dx[   ln(1 - x e^(-x)) ] =

(j)    d/dx[ x^x  ] =




(k)    d/dx[ e^( sin^(-1)(x))   ] =


(4) Simplify the following expressions as much as possible.

(a)    (-8)^(-2/3) =

(b)     e^( 3ln(2)) =

(c)     FormBox[RowBox[{RowBox[{log_10, (, 0.001, )}], =}], TraditionalForm]

(d)     ln(e^(1/2)) =  

(e)    ln(e)^(1/2) =

(f)     ln ( sin(π/2) ) =  

(g)     cos^(-1)(1/2) =

(h)    tan ( cos^(-1)(x)) =

(i)    cos(5π) =

(j)    sec(-π/4) =


(5)   Sketch the graph of y = 1 - e^x


[Graphics:HTMLFiles/ExF02_68.gif]


(6)   Solve the equation   ln(1/x) + ln(2x^3) = ln(3)   for x.

 


  

(7) The questions on this page concern the one-to-one function f(x) = x/(2 - x)

(a)    Underscript[lim , x2^+] f(x) =

(b)    Underscript[lim , x∞] f(x) =

(c)   List the vertical asymptotes of f  (if any).

(d)   List  the horizontal asymptotes of f  (if any).

(e)   Find the inverse of f.

 




(f) State the domain of f.  All values of x except 2.

(g)   State the domain of f^(-1).

(h)  State the range of  f. 

(i)   State the range of f^(-1).

(j) Find the equation of the tangent line to the graph of y = f(x) at the point where x = 1.

 



(8) This problem concerns the function f(x) = ln(x^2 + 1).

(a) f ' (x) =

(b) f '' (x) =

(c)  Find the interval(s) on which  f  is increasing and on which it is decreasing.



(d)  Find the interval(s) on which  f  is concave up and on which it is concave down.

 

 



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


(f)  List all the critical numbers of f.

(g)  List the x-coordinates of the relative extrema of  f  (if any) and say whether there is a relative minima or a relative maxima.



(9) Given the equation   y + sin(y) = x,  find  dy/dx.

 

 

 



(10) A rectangular box with with two square sides and an open top is to have a volume of 36 cubic feet. Find the dimensions of the container with minimum surface area.
[Graphics:HTMLFiles/ExF02_118.gif]

 

 



(11) A spherical balloon is deflated in such a way that its radius decreases at a constant rate of 15 cm/min. At what rate is air escaping when the radius is 2 cm?

Hint: The volume of a sphere of radius r is V = 4/3π r^3.